Lie supergroups vs. super Harish-Chandra pairs: a new equivalence
aa r X i v : . [ m a t h . R A ] O c t LIE SUPERGROUPS vs.SUPER HARISH-CHANDRA PAIRS:A NEW EQUIVALENCE
Fabio GAVARINI
Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”via della ricerca scientifica 1 — I-00133 Roma, Italy e-mail: [email protected]
Abstract
It is known that there exists a natural functor Φ from Lie supergroups to super Harish-Chandrapairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair,yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by otherauthors, to different levels of generality, but always elaborating on Koszul’s original idea.In this paper, I provide two new backwards equivalences, i.e. two different functors Ψ ˝ and Ψ e thatconstruct a Lie supergroup (thought of as a special group-valued functor) out of a given super Harish-Chandra pair, so that any Lie supergroup is recovered from its naturally associated super Harish-Chandrapair; more precisely, both Ψ ˝ and Ψ e are quasi-inverse to the functor Φ . Contents
MSC : Primary 14M30, 14A22 / 58A50, 58C50; Secondary 17B20 / 81Q60
Keywords:
Lie supergroups; Lie superalgebras; super Harish-Chandra pairs.Partially supported by a MIUR grant PRIN 2012, n. 2012KNL88Y 002, and by the MIUR
Excellence DepartmentProject awarded to the Department of Mathematics, University of Rome “Tor Vergata”, CUP E83C18000100006. From super Harish-Chandra pairs to Lie supergroups 27 G ˝ P as a Lie supergroup . . . . . . . . . . . . . . . . . . . . . 315.3 Supergroup functors out of super Harish-Chandra pairs: second recipe . . . . . . . . 365.4 The supergroup functor G e P as a Lie supergroup . . . . . . . . . . . . . . . . . . . . . 39 ( sHCp ) – ( Lsgrp ) . 42 ˝ Ψ – id ( sHCp ) . . . . . . . . . . 426.2 The functor Ψ as a quasi-inverse to Φ : proof of Ψ ˝ Φ – id ( Lsgrp ) . . . . . . . . . . 43 G to G . . . . . . . . . . . . . . . . . . . . . . . 45 The study of supergroups is a chapter of supergeometry, i.e. geometry in a Z –graded sense. Inparticular, the relevant structure sheaves of (commutative) algebras sitting on top of the topologicalspaces one works with are replaced with sheaves of (commutative) superalgebras .When dealing with differential supergeometry, our “superspaces” are supermanifolds, that arereal smooth, real analytic or complex holomorphic (depending on the context): any such super-manifold can be considered as a classical (i.e. non-super) manifold – of the appropriate type —endowed with a suitable sheaf of commutative superalgebras. The supergroups in this context arethen Lie supergroups, of smooth, analytic or holomorphic type according to the chosen setup.For every Lie supergroup G there exists a special pair of objects, say ` G , g ˘ , that is naturallyassociated with it: G is the classical Lie group underlying G — roughly given by “killing the oddpart” of the structure sheaf on G — while g “ Lie p G q is the tangent Lie superalgebra of G , andthese two objects are “compatible” in a natural sense. More in general, any similar pair ` K ` , k ˘ made by a Lie group K and a Lie superalgebra k obeying the same compatibility constraints iscalled “super Harish-Chandra pair” (a terminology first found in [8]), or just ”sHCp” for short: infact, this notion is tailored in such a way that mapping G ÞÑ ` G , g ˘ yields a functor, call it Φ ,from the category of Lie supergroups — either smooth, analytic or holomorphic — to the categoryof super Harish-Chandra pairs — of smooth, analytic or holomorphic type respectively.The key question is: can one recover a Lie supergroup out of its associated sHCp ? Moreprecisely: is there any functor Ψ from sHCp’s to Lie supergroups which be a quasi-inverse for Φ ,so that the two categories be equivalent? And how much explicit such a functor (if any) is ?A first answer to this question was given by Kostant and by Koszul in the real smooth case(see [17] and [18]), with equivalent methods, providing an explicit quasi-inverse for Φ . Lateron, Vishnyakova (see [24]) fixed the complex holomorphic case, and her proof works for the realanalytic case as well. More recently, this result was increasingly extended to the setup of algebraicsupergeometry , i.e. for algebraic supergroups (and corresponding sHCp’s), over fields and then overrings (see [7], [21], [22]). It is worth remarking, though, that all these subsequent results were, inthe end, further improvements of the original idea by Koszul (while Kostant’s method was a slightvariation of that): indeed, Koszul defines a Lie supergroup out of a sHCp ` K ` , k ˘ as a super-ringed space, just defining the “proper” sheaf of (commutative) superalgebras onto K ` by meansof k ; the authors of the successive, above mentioned papers just re-worked this same recipe.2n this paper I present a new method to solve that problem, i.e. I provide a different, moreconcrete functor Ψ from sHCp’s to Lie supergroups that is quasi-inverse to Φ . The starting ideais to follow a different approach to supergeometry, `a la Grothendieck: namely instead of thinkingof supermanifolds as being super-ringed manifolds (i.e. classical manifolds endowed with a sheafof commutative superalgebras), one studies (or directly defines) them through their “functor ofpoints”. Thus, if M is a supermanifold, then for each commutative superalgebra A one has themanifold M p A q of A –points of M ; in fact, in order to recover the full supermanifold M one canrestrict this functor to a smaller category, namely that of Weil superalgebras — roughly, those whichare direct sum of a copy of our ground field plus a finite-dimensional nilpotent ideal. Conversely,functors from Weil superalgebras to manifolds enjoying some additional properties do correspondto Lie supergroups (i.e., they are the functor of points of some Lie supergroup): so one can directlycall “Lie supergroup” any such special functor. This functorial point of view allows to unify severaldifferent approaches to supergeometry (see [2]) and also to treat infinite-dimensional supermanifolds(see [1]). For a broader discussion of the interplay between different approaches to supergeometrywe refer to classical sources as [3], [8], [19], [25] or more recent ones like [2], [4], [6], [23].Now, if we want a functor Ψ from sHCp’s to Lie supergroups, we need a Lie supergroup G P foreach sHCp P ; using the functorial point of view, in order to have G P as a functor we need a Liegroup G P p A q for each Weil superalgebra A , and their definition must be natural in A : moreover,one still has to show that the resulting functor have those additional properties that make it into aLie supergroup. Finally, all this should aim to find a Ψ that is quasi-inverse to Φ — and this fixesultimate bounds to the construction we aim to.Bearing all this in mind, the construction that I present goes as follows. Given a super Harish-Chandra pair P “ p G ` , g q , for each Weil superalgebra, say A , I define a group G P p A q abstractly,by generators and relations: this definition is uniform and natural with respect to A , hence it yieldsa functor from Weil algebras to (abstract) groups, call it G P . As key step in the work, one provesthat G P admits a “global splitting”, i.e. it is the direct product of G ` times a totally odd affinesuperspace (isomorphic to g , the odd part of g ): as both these are supermanifolds, it turns outthat G P itself is a supermanifold as well, and in fact it is a Lie supergroup because (as a functor) itis group-valued too. One more step proves that the construction of G P is natural in P , so it yieldsa functor Ψ from sHCp’s to Lie supergroups: this is our candidate to be a quasi-inverse to Φ .It is immediate to check that Φ ˝ Ψ is isomorphic to the identity functor onto sHCp’s; on theother hand, proving that Ψ ˝ Φ is isomorphic to the identity functor onto Lie supergroups is muchmore demanding. In fact, to get the latter we need to know that every Lie supergroup G has a“global splitting” on its own: this implies that G and Ψ ` Φ p G q ˘ share the same structure, in thatboth are the direct product of G and g “ ` Lie p G q ˘ . Now, the fact that a “Global SplittingTheorem” for Lie supergroups does hold true is (more or less) known among specialists; however,we need it stated in a genuine geometrical form, while it is usually given in sheaf-theoretic terms, soin the end we work it out explicitly. In fact, we find two different formulations of such a result: thisis why, building upon them, we can provide two versions, Ψ ˝ and Ψ e , of a functor Ψ as required.Two last words are still in order: (a) The recipe given here — for Ψ ˝ — was originally presented in [15] to solve the sameproblem in the context of algebraic supergeometry; adapting this idea to the differential setup (i.e.to Lie supergroups and their sHCp’s), however, is definitely not straightforward. The second recipeinstead — introducing the functor Ψ e — is entirely original; with some work, it can be adapted tothe setup of algebraic supergroups and sHCp’s too. (b) We deal here with Lie supergroups (and sHCp’s) of finite dimension; nevertheless, ourconstruction of the functor Ψ is perfectly fit for the infinite dimensional case too — still followingthe functorial approach, as in [1]. This requires extra technicalities which go beyond our scopes,so we do not fulfill that task; however, the core strategy to follow is already displayed hereafter.3inally, the paper is organized as follows. In section 2, I fix language and notations. Section3 introduces the notion of super Harish-Chandra pair and the natural functor Φ from Lie super-groups to super Harish-Chandra pairs. Section 4 presents structure results for Lie supergroups, inparticular about “global splittings”: more or less, these results are known (or should be known),but I could not find them in literature (in the form I need them), so I wrote them down myself.The core of the paper is in sections 5 and 6 . In section 5, I introduce two definitions of functorΨ , namely Ψ ˝ and Ψ e , and I prove key structure results for the Lie supergroups G P : “ Ψ p P q — with Ψ P Ψ ˝ , Ψ e ( ; in fact, in both cases the very definition of G P and the results about itsstructure are “prescribed” by the structure results of section 4 for Lie supergroups in general. Insection 6 then I prove, using the structure results of sections 4 and 5 (mainly the “Global SplittingTheorems”), that both versions of functor Ψ are indeed quasi-inverse to Φ , as expected.Finally, section 7 treats special cases and applications. Acknowledgements
The author thanks Alexander Alldridge, Claudio Carmeli and Rita Fioresi for many valuable conversations.
In all this work, K will denote the field R or C of real or complex numbers, respectively, followingthe context. All modules (i.e., vector spaces), algebras etc. will be considered over K . We call K –supermodule , or K –super vector space , any K –module V endowed with a Z –grading V “ V ‘ V , where Z “ t , u is the group with twoelements. Then V and its elements are called even , while V and its elements odd . By | x | or p p x qpP Z q we denote the parity of any homogeneous element, defined by the condition x P V | x | .We call K –superalgebra any associative, unital K –algebra A which is Z –graded (as a K –algebra):so A has a Z –splitting A “ A ‘ A , and A a A b Ď A a ` b ; any such A is said to be commutative — in “super sense” — iff x y “ p´ q | x | | y | y x for all homogeneous x , y P A and z “ z P A . All K –superalgebras form a category, whose morphisms are those of unital K –algebraspreserving the Z –grading; inside it, commutative K –superalgebras form a subcategory, that wedenote by ( salg ) , or also by ( salg ) K . Moreover, we shall denote by ( alg ) — or ( alg ) K , sometimes —the category of (associative, unital) commutative K –algebras, and by (mod) K that of K –modules.For A P ( salg ) , n P N , we call A r n s the A –submodule of A spanned by all products ϑ ¨ ¨ ¨ ϑ n with ϑ i P A for all i . We need also the following constructions: if J A : “ p A q is the ideal of A generated by A , then J A “ A r s ‘ A , and A : “ A L J A is a commutative superalgebra which is totally even , i.e. A P ( alg ) ; also, there is an obvious isomorphism A : “ A L p A q – A L A r s .Finally, the constructions of A , of A p n q and of A all are functorial in A . We now introduce a special class of commutative superalgebras, the
Weil superalgebras , or “super Weil algebras” , to be used later on (cf. [2] and references therein).4 efinition 2.1.3.
We call
Weil superalgebra any finite-dimensional commutative K –superalgebra A such that A “ K ‘ ˝ A where K is even and ˝ A “ ˝ A ‘ ˝ A is a Z –graded nilpotent ideal (the nilradical of A ). By construction, every Weil superalgebra A is automatically endowed with thecanonical (super)algebra morphisms p A : A ÝÑ K and u A : K ÝÑ A associated with the directsum splitting A “ K ‘ ˝ A ; in particular p A ˝ u A “ id K , so that p A is surjective and u A is injective.Weil superalgebras over K form a full subcategory of ( salg ) K , denoted ( Wsalg ) K or ( Wsalg ) . ♦ A special class of Weil superalgebras is given by
Grassmann algebras : namely, for any n P N by Grassmann algebra in n variables over K we mean the polynomial K –algebra Λ K p ξ , . . . , ξ n q : “ K “ ξ , . . . , ξ n ‰ in n mutually anticommuting indeterminates ξ , . . . , ξ n . Giving degree to every ξ i all these Grassmann algebras turn into commutative K –superalgebras; we denote by ( Grass ) K , orjust ( Grass ) , the full subcategory of (
Wsalg ) K whose objects are isomorphic to some Λ n ( n P N ). The infinitesimal counterpart of Lie supergroups is given by the notion of Lie superalgebras.We shall see their link later on, while now we fix that notion, as well as its functorial formulation.
Definition 2.2.1.
Let g “ g ‘ g be a K –supermodule. We say that g is a Lie superalgebra ifwe have a (Lie super)bracket r ¨ , ¨ s : g ˆ g ÝÑ g , p x, y q ÞÑ r x, y s , which is K –bilinear, Z –gradedand satisfies the following properties (for all x, y, z P g Y g ): (a) r x, y s ` p´ q | x | | y | r y, x s “ (anti-symmetry) ; (b) p´ q | x | | z | r x, r y, z ss ` p´ q | y | | x | r y, r z, x ss ` p´ q | z | | y | r z, r x, y ss “ (Jacobi identity). (c) In this situation, we shall write Y x y : “ ´ r Y, Y s ` P g ˘ for all Y P g .All Lie K –superalgebras form a category, denoted ( sLie ) K or just ( sLie ) , whose morphisms arethe K –linear, graded maps that preserve the bracket. ♦ Note that if g is a Lie K –superalgebra, then its even part g is automatically a Lie K –algebra. Example 2.2.2.
Let V “ V ‘ V be a K –supermodule, and consider End p V q , the endomorphismsof V as an ordinary K –module. This is in turn a K –supermodule, End p V q “ End p V q ‘ End p V q ,where End p V q are the morphisms which preserve the parity, and End p V q those which reverseit. If V has finite dimension and we choose a basis for V of homogeneous elements, writingfirst the even ones, then End p V q is the set of all diagonal block matrices, while End p V q isthe set of all anti-diagonal block matrices. Thus End p V q is a Lie K –superalgebra with bracket r A, B s : “ AB ´ p´ q | A || B | BA for homogeneous A, B P End p V q ; and then Y x y “ Y for odd Y .The standard example is V : “ K p ‘ K q , with V : “ K p and V : “ K q . In this case we alsowrite End ` K p | q ˘ : “ End p V q or gl p | q : “ End p V q . (cid:7) Let (
Wsalg ) K be the category of Weil K –superalgebras (see § Lie ) K the category of Lie K –algebras and by mod K the category of K –modules. Every Lie K –superalgebra g P ( sLie ) K yields a functor L g : ( Wsalg ) K ÝÝÝÑ ( Lie ) K , A ÞÑ L g p A q : “ ` A b g ˘ “ p A b g q ‘ p A b g q Indeed, A b g is a Lie superalgebra (in a suitable, more general sense, on the Weil K –superalgebra A ) on its own, with Lie bracket “ a b X , a b X ‰ : “ p´ q | X | | a | a a b “ X, X ‰ , given by the5o-called “sign rules”; now L g p A q is the even part of the Lie superalgebra A b g , hence it is a Liealgebra on its own (see [6] for details). In particular, all this applies to g : “ End p V q .More in general, the following holds. Every K –supermodule m “ m ‘ m defines a functor L m : ( Wsalg ) K ÝÝÝÑ (mod) K , A ÞÑ L m p A q : “ ` A b K m ˘ “ p A b m q À p A b m q and then m is a Lie K -superalgebra ðñ L m takes values in ( Lie ) K .In fact, this holds true also if we replace ( Wsalg ) K with its (smaller) subcategory ( Grass ) K .This “functorial presentation” of Lie superalgebras can be adapted to representations too. In-deed, let V be a g –module, with representation map φ : g ÝÑ End p V q — a Lie superalgebramorphism. Scalar extension induces a morphism id A b φ : A b g ÝÑ A b End p V q for each A P ( Wsalg ) K , whose restriction to the even part gives a morphism ` A b g ˘ ÝÑ ` A b End p V q ˘ ,that is a morphism L g p A q ÝÑ L End p V q p A q in ( Lie ) K . The whole construction is natural in A ,hence it induces a natural transformation of functors L g ÝÑ L End p V q .The same considerations apply as well if ( Wsalg ) K is replaced with ( Grass ) K .In the sequel, we shall call quasi-representable any functor L : ( Wsalg ) K ÝÑ ( Lie ) K for whichthere exists g P ( sLie ) K such that L – L g ; the same applies with ( Grass ) K instead of ( Wsalg ) K . In this subsection we introduce the “supermanifolds” — real smooth, real analytic or complexholomorphic ones — as well as the corresponding group objects. This material is more or lessstandard; we follow two equivalent approaches: we refer to [2] — possibly with different terminologyand notation — where all needed details can be found, as well as the original references.
Definition 2.3.1. A superspace is a pair S “ ` | S | , O S ˘ of a topological space | S | and a sheaf ofcommutative superalgebras O S on it such that the stalk of O S at each point x P | S | , denoted by O S,x , is a local superalgebra. If S and T are two superspaces, a morphism φ : S ÝÝÑ T betweenthem is a pair ` | φ | , φ ˚ ˘ where | φ | : | S | ÝÝÑ | T | is a continuous map of topological spaces and φ ˚ : O T ÝÝÑ | φ | ˚ ` O S ˘ is a morphism of sheaves on | T | such that φ ˚ x p m | φ |p x q q Ď m x , where m | φ |p x q and m x denote the maximal ideals in the stalks O T, | φ |p x q and O S,x respectively. ♦ Examples 2.3.2.
Fix any p, q P N ` . (a) The real smooth local model — The superspace R p | q is the topological space R p endowedwith the following sheaf of commutative superalgebras: O R p | q p U q : “ C R p p U q b R Λ R p ξ , . . . , ξ q q forany open set U Ď R p , where C R p is the sheaf of smooth functions on R p . (b) The real analytic local model — The superspace R p | qω is the topological space R p endowedwith the following sheaf of commutative superalgebras: O R p | qω p U q : “ C ω R p p U q b R Λ R p ξ , . . . , ξ q q forany open set U Ď R p , where C ω R p is the sheaf of analytic functions on R p (c) The holomorphic local model — The superspace C p | q is the topological space C p endowedwith the following sheaf of commutative superalgebras: O C p | q p U q : “ C C p p U q b C Λ C p ϑ , . . . , ϑ q q forany open set U Ď C p , where C C p is the sheaf of holomorphic functions on C p .Patching together these local models one makes up “supermanifolds”, defined as follows: Definition 2.3.3. (a) A (real) smooth supermanifold of (super)dimension p | q is a superspace M “ ` | M | , O M ˘ such that | M | is Hausdorff and second-countable and M is locally isomorphic to R p | q , i. e. for each x P | M | there is an open set V x Ď | M | with x P V x and U Ď R p such that O M ˇˇ V x – O R p | q ˇˇ U .6 b) A (real) analytic supermanifold of (super)dimension p | q is defined like in (a) but with R p | qω replacing R p | q everywhere: in particular, it is locally isomorphic to R p | qω . (c) A (complex) holomorphic supermanifold of (super)dimension p | q is defined like in (a) butwith C p | q replacing R p | q everywhere: in particular, it is locally isomorphic to C p | q . (d) A morphism of smooth, analytic or holomorphic supermanifolds is, by definition, a mor-phism of the underlying superspaces.In either case above, the sheaf O M is called the structure sheaf of M : to simplify notation, weshall write O p M q instead of O M ` | M | ˘ to denote the superalgebra of global sections of O M .We denote the category of (real) smooth, (real) analytic, or (complex) holomorphic superman-ifolds by ( ssmfd ) , ( asmfd ) , or ( hsmfd ) , respectively. ♦ In most cases later on the distinction between the smooth, the analytic or the holomorphic case isimmaterial: therefore, in order to minimize repetitions, I shall often refer only to “supermanifolds”.
Let M be a smooth supermanifold and U an open subset in | M | . Let I M p U q be the ideal of O M p U q generated by the odd part of the latter:then O M L I M is a presheaf, whose sheafification Č O M L I M is a sheaf of purely even superalgebrasover | M | , locally isomorphic to C ` R p ˘ . Then M : “ ´ | M | , Č O M L I M ¯ is a classical smoothmanifold, called the reduced smooth manifold associated with M ; the standard, built-in projection s ÞÑ ˜ s : “ s ` I M p U q (for all s P O M p U q ) at the sheaf level corresponds to a natural embedding M ÝÑ M , so that M can be thought of as an embedded sub(super)manifold of M itself.A similar construction applies when the supermanifold M is analytic, resp. holomorphic, yieldingthe notion of “reduced analytic manifold” , resp. “reduced holomorphic manifold” , M of M .A key feature of this construction is that it is natural, i.e. it provides a functor from the categoryof supermanifolds (of either type: smooth, etc.) to the category of manifolds (of the correspondingtype), defined on objects by M ÞÑ M and on morphisms in a natural way — cf. [2] for details.Finally, by the very definition of supermanifolds (smooth or analytic or holomorphic), one seesat once that also “classical” manifolds (of either type) can be seen as “supermanifolds”, simplyobserving that their structure sheaf is one of superalgebras that are actually totally even , i.e. withtrivial odd part. Conversely, any supermanifold enjoying the latter property is actually a “classical”manifold, nothing more. In other words, classical manifolds identify with those supermanifolds M that actually coincide with their reduced (sub)manifolds M .We finish this subsection introducing the notion of “Lie supergroup”: Definition 2.3.5. A group object in the category ( ssmfd ) , or ( asmfd ) , or ( hsmfd ) , is called (real smooth) Lie supergroup , resp. (real) analytic Lie supergroup , resp. (complex) holomorphicLie supergroup . These objects, together with the obvious morphisms, form a subcategory amongsupermanifolds, denoted ( Lsgrp ) R , resp. ( Lsgrp ) ω R , resp. ( Lsgrp ) ω C . ♦ In this subsection we introduce the language of “functor(s) of points” for supermanifolds andLie supergroups, whose basic idea goes back to Weil’s and Grothendieck’s approach to algebraicgeometry. It will be just a quick reminder, for more details the interested reader can refer to [2].We begin with some notation. For any two categories A and B , with r A , B s we denote the cat-egory of all functors between A and B , the morphisms in r A , B s being the natural transformations.As usual A op will denote the opposite category to A , so that “ A op , B ‰ is nothing but the categoryof contravariant functors from A to B . 7 .4.1. The functor of points of a supermanifold. Our first kind of “functor of points” isthe following. Given a (real) smooth supermanifold M P ( ssmfd ) , its associated functor of points F M : ( ssmfd ) op ÝÝÑ ( set ) is defined on objects by F M p S q : “ Hom p S, M q and on morphismsby F M p φ q : F M p S q ÝÝÝÑ F M p T q , f ÞÑ ` F M p φ q ˘ p f q : “ f ˝ φ , for all S, T P ( ssmfd ) and φ P Hom p S, T q . The elements in F M p S q are called the S –points of M . A similar definition holdsfor analytic, resp. holomorphic, supermanifolds, with ( asmfd ) , resp. ( hsmfd ) , replacing ( ssmfd )wherever it occurs in the previous definition.Now consider two supermanifolds M and N : in order to write precise formulas we assume themto be both smooth, but the discussion hereafter makes sense the same if M and N are both analyticor both holomorphic. By definition of “functor of points” Yoneda’s lemma yields a bijection Hom ( ssmfd ) p M, N q ÐÝÝÝÑ
Hom r ( ssmfd ) op , ( set ) s ` F M , F N ˘ between morphisms M ÝÝÑ N and natural transformations F M ÝÝÑ F N (cf. [20], ch. 3, or[9], ch. 6). Thus we have an immersion Y : ( ssmfd ) ÝÝÝÑ “ ( ssmfd ) op , ( set ) ‰ of ( ssmfd ) into “ ( ssmfd ) op , ( set ) ‰ that is full and faithful. The objects in “ ( ssmfd ) op , ( set ) ‰ that lie in the imageof this immersion — i.e., that arise as functors of points of some supermanifold — are exactlythose which are representable; indeed, not all objects in “ ( ssmfd ) op , ( set ) ‰ are representable, butimportant representability criteria exist — e.g., see [2], Theorem 2.13.Finally, in this functorial approach the Lie supergroups are characterized as follows: any su-permanifold M is actually a Lie supergroup if and only if its functor of points F M is actuallygroup-valued, i.e. its target category is ( group ) rather than ( set ) . A –points. We introduce now the notion of A –points of asupermanifold, following the presentation in [2], § M one is forced to endow each set of A –points of M with an extra structure, namely that of an “ A –manifold ”, that we shall see later.In the constructions we mainly refer to smooth supermanifolds but, with minimal changes, theyadapt to analytic and holomorphic supermanifolds too (cf. [2]). We begin with the key definitions: Definition 2.4.3.
Let M be a supermanifold (either smooth, etc.). (a) For every point x P | M | and every Weil superalgebra A P ( Wsalg ) we define the set of A –points near x as given by M A,x : “ Hom ( salg ) ` O M,x , A ˘ and the set of (all) A –points as givenby M A : “ Ů x P| M | M A,x . If x A P M A,x we call ˜ x A : “ p A ˝ x A the base point of x A , where p A : A ÝÑ K is the built-in projection of A onto K ; in fact, ˜ x A canonically identifies with x . (b) We denote by W M : ( Wsalg ) ÝÑ ( set ) the functor defined by A ÞÑ W M p A q : “ M A on objects and on morphisms by ρ ÞÑ W M p ρ q : “ ρ p M q for every A, B P ( Wsalg ) and every ρ P Hom ( salg ) p A, B q with ρ p M q : M A ÝÑ M B , x A ÞÑ ρ ˝ x A . ♦ The above mentioned constructionof W M for a supermanifold M is clearly natural in M : in other words, it gives rise to a functor B : ( ssmfd ) ÝÑ r ( Wsalg ) , ( set ) s given on objects by M ÞÑ B p M q : “ W M . As it is explained in[2], § ssmfd ) , as it does not identify the latter with a full subcategory of r ( Wsalg ) , ( set ) s . Oneinstead has to understand what is exactly the (non-full) subcategory of r ( Wsalg ) , ( set ) s which isthe image of B , and then consequently adapt B itself to a “nicer” functor. I shall now shortlysketch the construction that is needed to solve this problem, referring to [2], §
4, for further details.8ne starts by introducing the notion of A –manifold. Roughly speaking, given a finite dimen-sional commutative algebra A P ( alg ) K , an A –manifold is a (smooth, etc.) manifold M endowedwith an L –atlas, i.e. an atlas of local charts each of which is diffeomorphic (or bianalytic, or biholo-morphic) with some open subset of a given finite dimensional A –module L in such a way that thedifferential of every change of charts is an A –module isomorphism (between local copies of L ).Given two A –manifolds M and N , possibly modelled on different A –modules, an A –smooth (oranalytic, or holomorphic) morphism φ : M ÝÑ N is any morphism from M to N in the standardsense (smooth, etc.) such that its differential at each point is A –linear. The resulting categorythen is denoted ( A – smfd ), resp. ( A – amfd ), resp. ( A – hmfd ), possibly with a subscript K .For the second step, we gather together all possible A –manifolds for all finite dimensional A P ( alg ) K . Now consider A , A P ( alg ) K , a morphism ρ : A ÝÑ A , an A –manifold M and an A –manifold M . By scalar restriction (through ρ ), M is also an A –manifold, thus we define a morphism from M to M as being a morphism of A –manifolds — in particular, its differential is A –linear (through ρ ). With this notion of “morphism”, the various A –manifolds (for all A ’s)altogether form a new category, denoted by ( A – smfd ) , resp. ( A – amfd ) , resp. ( A – hmfd ) , inthe smooth, resp. analytic, resp. holomorphic case. We can now introduce our next definition: Definition 2.4.5.
We denote ““ ( Wsalg ) , ( A – smfd ) ‰‰ the subcategory of “ ( Wsalg ) , ( A – smfd ) ‰ whose objects are those in “ ( Wsalg ) , ( A – smfd ) ‰ and whose morphisms are all natural transfor-mations φ : G ÝÝÑ H such that for every A P ( Wsalg ) the induced φ A : G p A q ÝÝÑ H p A q is A –smooth. Similarly we define the categories ““ ( Wsalg ) , ( A – amfd ) ‰‰ and ““ ( Wsalg ) , ( A – hmfd ) ‰‰ respectively in the analytic and in the holomorphic case. ♦ The motivation for introducing the notion of A –manifold lies in the following three results: (1) if M is any supermanifold, for each A P ( Wsalg ) the set W M p A q can be naturally endowedwith a canonical structure of A –manifold; (2) if M is a supermanifold and ρ : A ÝÝÝÝÑ B a morphism in ( Wsalg ) , then the map W M p A q ρ p M q ÝÝÑ W M p B q ` x A ÞÑ ρ ˝ x A ˘ is a morphism in ( A – smfd ), resp. ( A – amfd ), resp. ( A – hmfd ). (3) if φ : M ÝÝÑ N is a morphism of supermanifolds, then for all A P ( Wsalg ) the map φ A : W M p A q ÝÝÑ W N p A q ` x A ÞÑ x A ˝ φ ˚ ˘ is a morphism in ““ ( Wsalg ) , ( A – smfd ) ‰‰ , resp. in ““ ( Wsalg ) , ( A – amfd ) ‰‰ , resp. in ““ ( Wsalg ) , ( A – hmfd ) ‰‰ .Thanks to the above, we can correctly introduce Weil-Berezin functors and Shvarts embedding: Definition 2.4.6. (a)
For every smooth supermanifold M P ( ssmfd ) , we call Weil-Berezin (local) “functor of A –points” of M the functor W M : ( Wsalg ) ÝÝÑ ( A – smfd ) defined as in Definition 2.4.3 (b) — which makes sense thanks to the previous remarks. The same terminology applies, mutatismutandis , in the case of any analytic or holomorphic supermanifold. (b) We call
Shvarts embedding the functor S : ( ssmfd ) ÝÑ ““ ( Wsalg ) , ( A – smfd ) ‰‰ , in the smoothcase, defined on objects by M ÞÑ W M ; and similarly in the analytic and the holomorphic case. ♦ The key point here is that the Shvarts embedding is a full and faithful functor , so that for anytwo supermanifolds, say smooth, M and N one has Hom ( ssmfd ) p M, N q –
Hom rr ( Wsalg ) , ( A – smfd ) ss ` S p M q , S p N q ˘ “ Hom rr ( Wsalg ) , ( A – smfd ) ss ` W M , W N ˘ hence in particular M – N if and only if S p M q – S p N q , that is W M – W N .Therefore one can correctly study supermanifolds via their Weil-Berezin functors. However, todo that one still has to be able to characterize those objects in ““ ( Wsalg ) , ( A – smfd ) ‰‰ — in the9mooth case, and similarly in the other cases — that actually are (isomorphic to) the Weil-Berezinfunctors of some supermanifolds; in other words, one needs a characterization of the image ofShvarts embedding, which is actually not all of its target category, but a proper subcategory of it.This is the “representability problem”, which we do not really care so much for the present work.What is still relevant to us, is that the Shvarts embedding S preserves products, hence alsogroup objects. This means, in the end, that the following holds true (cf. [2], § Proposition 2.4.7.
A supermanifold M is a Lie supergroup if and only if S p M q : “ W M takesvalues in the subcategory (among A –manifolds) of group objects, that we call “Lie A –groups” . Aswe saw in § totally even , i.e. with trivial oddpart; conversely, any supermanifold with this property is actually “classical”. In other words, theseare all those supermanifolds M which coincide with their (classical) reduced subsupermanifold M .From the functorial point of view, it is clear that a supermanifold M is classical if and only if theassociated Weil-Berezin functor S p M q : “ W M P ““ ( Wsalg ) , ( set ) ‰‰ coincides with its restriction tothe subcategory ( alg ) Ş ( Wsalg ) ; in short, these are those M such that W M p A q “ W M p A q forall A P ( Wsalg ) K . If instead one deals with a (general) supermanifold M , then the restriction to( alg ) Ş ( Wsalg ) of its Weil-Berezin functor coincides with the Weil-Berezin functor (for classicalmanifolds) of its associated reduced submanifold M : in a nutshell, S p M q ˇˇ ( alg ) Ş ( Wsalg ) “ S p M q . Λ –points. As it is explained in [2], § Wsalg ) K with itsfull subcategory ( Grass ) K . Thus for each smooth supermanifold M P ( ssmfd ) R the restriction to( Grass ) R of its Weil-Berezin functor yields a new functor W p Λ q M : ( Grass ) R ÝÑ ( A – smfd ) , whichwe call “(Weil-Berezin) functors of Λ –points of M ” . As M ranges in ( ssmfd ) R all these W p Λ q M ’s givea new functor S p Λ q : ( ssmfd ) R ÝÝÝÑ ““ ( Grass ) R , ( A – smfd ) ‰‰ whose main feature is that it isagain a full and faithful embedding, which we call again “Shvarts embedding”.Similarly one does with analytic or holomorphic supermanifolds.The outcome is that to study supermanifolds it is enough to consider their functors of Λ–points;in particular, two supermanifolds (of either type) are isomorphic if and only if their correspondingfunctors of points are. Moreover, the new Shvarts embedding S p Λ q again preserves products, so ittakes group objects to group objects: therefore, the characterization of Lie supergroups stated inProposition 2.4.7 still makes sense reading “ S p Λ q p M q : “ W p Λ q M ” instead of “ S p M q : “ W M ”.A direct consequence of all this is the following. In the rest of the paper, we shall work with Liesupergroups considered as special functors, i.e. we study M using its Weil-Berezin functor of points W M ; or even, conversely, we consider special functors W : ( Wsalg ) R ÝÑ ( A – smfd ) — in thesmooth case, say — and then prove that there exists some (smooth) Lie supergroup M such that W M “ W . Now, due to the above discussion it makes sense to (try to) follow the same strategyusing ( Grass ) instead of (
Wsalg ) and the functor of Λ–points W p Λ q M instead of W . The good newsare, indeed, that this is actually feasible, in that all our discussion in the sequel will perfectly makessense and will be equally correct in both approaches. Thus one can choose to work in the largerframework of Weil superalgebras (as we do) or in the simpler setup of Grassmann algebras, and inboth cases our procedure and results will apply and hold true exactly the same.10 From Lie supergroups to super Harish-Chandra pairs
In this section we present the notion of super Harish-Chandra pair, showing how it naturallyarises from that of Lie supergroup. Indeed, here “naturally” means that one has a functorialconstruction that, starting from any Lie supergroup, leads to a special “pair”, whose properties arethen singled out to set down the very definition of “super Harish-Chandra pairs”.
We present now the notion of super Harish-Chandra pair , introduced by Koszul in [18] — butthis terminology is first found in [8]; in the next subsection we shall also see its (natural) motivation.
Definition 3.1.1.
We call (smooth, analytic or holomorphic) super Harish-Chandra pair — or just “sHCp” , in short — over K any pair p G ` , g q such that (a) G ` is a (smooth, analytic or holomorphic) Lie group over K , and g P ( sLie ) K ; (b) Lie p G ` q “ g ; (c) there is a (smooth, analytic or holomorphic) G ` –action on g by Lie K –superalgebraautomorphisms, hereafter denoted by Ad : G ` ÝÝÝÑ
Aut p g q , such that its restriction to g isthe adjoint action of G ` on Lie p G ` q “ g and the differential of this action is the restriction to Lie p G ` q ˆ g “ g ˆ g of the adjoint action of g on itself.If ` G , g ˘ and ` G , g ˘ are two super Harish-Chandra pairs over K , a morphism amongthem is any pair p φ ` , ϕ q : ` G , g ˘ ÝÑ ` G , g ˘ where φ ` : G ÝÑ G is a morphism ofLie groups (in the smooth, analytic or holomorphic sense), ϕ : g ÝÑ g is a morphism of Liesuperalgebras, and the two are compatible with the additional structure, that is to say (d) ϕ ˇˇ g “ dφ ` , Ad ` φ ` p g q ˘ ˝ ϕ “ ϕ ˝ Ad p g q @ g P G ` .All super Harish-Chandra pairs over K along with their morphisms form a category, denoted( sHCp ) K . When we have to specify its type, we write ( sHCp ) R if this type is real smooth, ( sHCp ) ω R if it is real analytic, and ( sHCp ) ω C if it is complex holomorphic. ♦ In this subsection we show how one can naturally associate a significant super Harish-Chandrapair with any Lie supergroup; indeed, this is the reason why the very notion of super Harish-Chandra pair was introduced. What follows is well-known, for details and proofs we refer to [6].
Let G be a Lie supergroup (of eithertype: smooth, etc.). As it is a supermanifold, from § G of G . Taking the functorial point of view, we know that S G “ S G ˇˇ ( alg ) Ş ( Wsalg ) (cf. § S G takes values in the subcategory of Lie A –groups, (by Proposition 2.4.7); butthen the latter is true for S G as well, hence — by Proposition 2.4.7 again — we argue that G itself is indeed a Lie group (either smooth, etc., like G is).Moreover, when φ : G ÝÑ G is a morphism of Lie supergroups, the morphism of manifolds φ : G ÝÑ G — induced by the functoriality of the construction G ÞÑ G — is in addition aLie group morphism. Therefore, we conclude that G ÞÑ G and φ ÞÑ φ define a functor from Liesupergroups (of either type) to Lie groups (of the same type).11 .2.2. The tangent Lie superalgebra of a Lie supergroup. We now quickly recall how toassociate with a Lie supergroup its “tangent Lie superalgebra”, referring to [6] for further details.For any A P ( Wsalg ) K , we let A r ε s : “ A r x s L` x ˘ be the so-called superalgebra of dual numbers over A , in which ε : “ x mod ` x ˘ is taken to be even . Then A r ε s “ A ‘ Aε , and there are twonatural morphisms i A : A ÝÑ A r ε s , a i A ÞÑ a , and p A : A r ε s ÝÑ A , ` a ` a ε ˘ p A ÞÑ a , such that p A ˝ i A “ id A . Note also that it follows by construction that A r ε s P ( Wsalg ) K again. Definition 3.2.3.
Given a functor G : ( Wsalg ) K ÝÑ ( group ) , let G p p A q : G p A p ε qq ÝÑ G p A q bethe morphism associated with the morphism p A : A r ε s ÝÝÑ A in ( Wsalg ) K . Then there exists aunique functor Lie p G q : ( Wsalg ) K ÝÑ ( set ) given on objects by Lie p G qp A q : “ Ker ` G p p A q ˘ . ♦ The key fact is that when the functor G as above is a (smooth, analytic or holomorphic) Liesupergroup, then Lie p G q is Lie algebra valued, i.e. it is a functor Lie p G q : ( Wsalg ) K ÝÑ ( Lie ) K .This requires a non-trivial proof (like in the classical case), for which we refer to [6], Ch. 11 (withthe few adaptations needed for the present setup), and only quickly sketch here the main steps.The Lie structure on any object Lie p G qp A q is introduced as follows. First, define the adjointaction of G on Lie p G q as given, for every A P ( Wsalg ) K , by Ad : G p A q ÝÑ GL ` Lie p G qp A q ˘ , Ad p g qp x q : “ G p i A qp g q ¨ x ¨ ` G p i A qp g q ˘ ´ for all g P G p A q , x P Lie p G qp A q . Second, define the adjoint morphism ad as ad : “ Lie p Ad q : Lie p G q ÝÑ Lie p GL p Lie p G qqq : “ End p Lie p G qq and finally define r x, y s : “ ad p x qp y q for all x, y P Lie p G qp A q . Then we have the following: Proposition 3.2.4.
Given a (smooth, analytic or holomorphic) Lie supergroup G , let g : “ T e p G q be the tangent K –supermodule to G at the unit point e P G .(a) Lie p G q with the bracket r ¨ , ¨ s above is Lie algebra valued, i.e. Lie p G q : ( Wsalg ) K ÝÑ ( Lie ) K ;(b) Lie p G q is quasi-representable (see § Lie p G q “ L g , where g is endowed with acanonical structure of Lie K –superalgebra, and it is also representable, namely represented by g ˚ ;(c) for every A P ( Wsalg ) K one has Lie p G qp A q “ Lie ` G p A q ˘ , the latter being the tangent Liealgebra to the Lie group G p A q . Note that the previous proposition also collects all that was shortly explained in § N.B.: due to the previous proposition, in the sequel we shall freely identify the functor
Lie p G q “ L g with the tangent superspace g — now thought of as a Lie superalgebra — calling this commonobject “the tangent Lie superalgebra of (or “to”) the Lie supergroup G ”.There exist several other realizations of the tangent Lie superalgebra to G , and canonicalidentifications among all of them. We will only occasionally need some of them, so we do not gointo further details, but refer instead to the literature, in particular [7] (especially § G ÞÑ Lie p G q for Lie supergroups is actually natural, in that anymorphism φ : G ÝÝÑ G of Lie supergroups induces a morphism Lie p φ q : Lie ` G ˘ ÝÝÑ
Lie p G q of Lie superalgebras. Eventually, all this together provides functors Lie : (
Lsgrp ) R ÝÝÝÑ ( sLie ) R , Lie : (
Lsgrp ) ω R ÝÝÝÑ ( sLie ) R and Lie : (
Lsgrp ) ω C ÝÝÝÑ ( sLie ) C ; see [6] and [2] for details. Gathering together the previousresults we get the core of the present section. Namely, if G is any Lie supergroup then ` G , Lie p G q ˘ is a super Harish-Chandra pair, and this construction is functorial, as the following claims:12 heorem 3.2.6. (see for instance [7]) There exist functors
Φ : (
Lsgrp ) R ÝÝÑ ( sHCp ) R , Φ : (
Lsgrp ) ω R ÝÝÑ ( sHCp ) ω R , Φ : (
Lsgrp ) ω C ÝÝÑ ( sHCp ) ω C that are given on objects by G ÞÑ ` G , Lie p G q ˘ and on morphisms by φ ÞÑ ` φ , Lie p φ q ˘ . In this section we present some results on the possibility to split Lie supergroups in two specialways: “Boseck’s splittings” and “global splittings”. These are essentially well-known, but usuallystated in different ways; so we provide independent proofs for them, so to fill any possible gap in lit-erature and to have a self-contained presentation (for all cases: smooth, analytic and holomorphic).
This subsection is devoted to a first kind of splitting that we refer to as “Boseck’s splitting”, asit was first mentioned in Boseck’s work [5]. The starting point is the following easy result:
Lemma 4.1.1.
Let p : A ÝÝÑ A and u : A ÝÝÑ A be morphisms in ( Wsalg ) K such that p ˝ u “ id A (hence p A is surjective and u A injective), and let G : ( Wsalg ) K ÝÑ ( group ) be anyfunctor. Then G p A q canonically splits into a semi-direct product G p A q “ Im ` G p u q ˘ ˙ Ker ` G p p q ˘ – G ` A ˘ ˙ Ker ` G p p q ˘ Proof.
From p ˝ u “ id A we get G p p q ˝ G p u q “ G p p ˝ u q “ G ` id A ˘ “ id G p A q ; thus G p u q , resp. G p p q , is a section of G p p q , resp. a retraction of G p u q , in the category of groups: in particular, G p u q is injective and G p p q surjective. The claim then follows by standard group-theoretic arguments.When the functor G is in fact a Lie supergroup , we have the following, interesting outcome:
Proposition 4.1.2. (cf. [5], §
2, Proposition 7)
Let G be any (smooth, analytic or holomorphic) Lie supergroup over K . Then for every Weilsuperalgebra A P ( Wsalg ) K there exists a canonical splitting of Lie groups G p A q – G p K q ˙ N G p A q (4.1) where G p K q “ G p K q is nothing but the classical, ordinary Lie group underlying G (i.e., the Liegroup of K –points of G ) and N G p A q : “ Ker ` G ` p A ˘˘ with p A : A ÝÝÑ K as in Definition 2.1.3.Proof. By assumption, for the given A P ( Wsalg ) K we have morphisms p A : A ÝÝÑ K and u A : K ÝÝÑ A in ( Wsalg ) K such that p A ˝ u A “ id K (cf. Definition 2.1.3). Then Lemma 4.1.1yields a group-theoretic splitting G p A q “ Im ` G ` u A ˘˘ ˙ Ker ` G ` p A ˘˘ . As G takes values into thecategory of Lie groups, this is actually a splitting of Lie groups (actually, even one of A –Lie groupsindeed). Moreover, we clearly have Im ` G ` u A ˘˘ – G p K q “ G p K q , whence (4.1) is proved.This result also applies to totally even supergroups (i.e., classical Lie groups) as follows:13 roposition 4.1.3. Let G ` be any (smooth, analytic or holomorphic) Lie group over K . Thenfor every A ` P ( Wsalg ) K Ş ( alg ) K there exists a canonical splitting of Lie groups G ` ` A ` ˘ – G ` p K q ˙ N G ` ` A ` ˘ (4.2) where G ` p K q is the ordinary Lie group underlying G (i.e., the Lie group of K –points of G ` ) and N G ` ` A ` ˘ : “ Ker ` G ` p A ` ˘˘ with p A ` : A ` ÝÝÑ K as in Definition 2.1.3.Proof. The same arguments as for the proof of Proposition 4.1.2 apply again.
Remark 4.1.4.
To the best of the author’s knowledge, the (canonical) splitting (4.1) of G p A q wasfirst mentioned by Boseck (dealing with Lie supergroups defined over ( Grass ) K , but the idea is thesame): cf. [5], §
2, Proposition 7; thus we shall refer to (4.1) or (4.2) as to “Boseck’s splitting(s)”.The same result was considered by other authors too, e.g. Molotkov: see (7.4.1) in § The notion of “Boseck’s splitting” for Liesupergroups has a natural counterpart for Lie superalgebras, when thought of as functors.Indeed, consider a Lie K –superalgebra g “ g ‘ g and the functor L g : ( salg ) K ÝÑ ( Lie ) K given by L g p A q : “ ` A b g ˘ “ p A b g q ‘ p A b g q , for all A P ( salg ) K , as in § A P ( Wsalg ) K has built-in morphisms p A : A ÝÝÑ K and u A : K ÝÝÑ A such that p A ˝ u A “ id K .Then applying L g we get L g ` p A ˘ ˝ L g ` u A ˘ “ L g ` p A ˝ u A ˘ “ L g ` id K ˘ “ id L g p K q , a relation regardingmorphisms of Lie algebras. By standard arguments this yields a Lie algebra splitting L g p A q “ Im ` L g ` u A ˘˘ i Ker ` L g ` p A ˘˘ (4.3)where the symbol “ i ” denotes the (internal) semi-direct sum of Im ` L g ` u A ˘˘ — a Lie subalgebrainside L g p A q — with Ker ` L g ` p A ˘˘ — a Lie ideal in L g p A q . Now, on the one hand definitions give Im ` L g ` u A ˘˘ – L g p K q : “ ` K b K g ˘ “ g ; on the other hand, to simplify a bit the notation wewrite n g p A q : “ Ker ` L g ` p A ˘˘ . Then (4.3) reads also L g p A q “ g i n g p A q @ A P ( Wsalg ) K (4.4)In the following, we shall refer to (4.4) as to “Boseck’s splitting for L g ” — or simply “for g ” itself.It is still worth remarking that one has a non-trivial Boseck’s splitting also when g “ g , i.e. g is a classical (=totally even) Lie algebra. Indeed, if g ` is just a Lie K –algebra then (4.4) reads L g ` ` A ` ˘ “ g ` i n g ` ` A ` ˘ @ A ` P ( Wsalg ) K Ş ( alg ) K (4.5)and will again be called “Boseck’s splitting for L g ` ” — or “for g ` ”.We will now give an explicit description of n g p A q . By definition, n g p A q : “ Ker ` L g ` p A ˘˘ where p A : A “ K ‘ ˝ A ÝÝ ։ K is the canonical projection of A “ K ‘ ˝ A onto its left-hand side summand.Now, A “ A ‘ A with A “ K ‘ ˝ A and A “ ˝ A , hence L g p A q : “ ` A b K g ˘ “ ` A b K g ˘ ‘ ` A b K g ˘ “ g ‘ ` ˝ A b K g ˘ ‘ ` A b K g ˘ from which it clearly follows that n g p A q : “ Ker ` L g ` p A ˘˘ “ ` ˝ A b K g ˘ ‘ ` ˝ A b K g ˘ @ A P ( Wsalg ) K (4.6)In the case when g “ g is just a “classical” Lie algebra, say g ` , this reads slightly simpler, namely n g ` ` A ` ˘ “ ˝ A ` b K g ` @ A ` P ( Wsalg ) K Ş ( alg ) K (4.7)This entails the following: 14 roposition 4.1.6. (a) Let g be a Lie K –superalgebra and A P ( Wsalg ) K . Then the Lie algebra n g p A q is nilpotent.(b) Let g ` be a Lie K –algebra and let A ` P ( Wsalg ) K Ş ( alg ) K . Then n g ` ` A ` ˘ is nilpotent.Proof. Claim (a) follows at once from (4.6) and the fact that ˝ A is nilpotent, and likewise for (b) . Letagain G be a Lie supergroup over K , and g : “ Lie p G q be its tangent Lie superalgebra. For any A P ( Wsalg ) K , the Lie group G p A q and the Lie algebra g p A q : “ L g p A q — also equal to Lie p G qp A q “ Lie ` G p A q ˘ , cf. Proposition 3.2.4 (c) — are linked by the exponential map exp : g p A q ÝÝÑ G p A q which is a local isomorphism (either in the smooth, analytic or holomorphic sense, as usual).Similarly for the “even counterparts” we have also the local isomorphism exp : g ÝÝÑ G p K q with exp “ exp ˇˇ g if we think at g as embedded into g p A q : “ L g p A q “ g i n g p A q — cf. (4.4).Now, since the Lie algebra n g p A q is nilpotent — cf. Proposition 4.1.6 — its image exp ` n g p A q ˘ for the exponential map is a (closed, connected) nilpotent Lie subgroup of G p A q . Furthermore, letus use notation g ` p A ˘ : “ L g ` p A ˘ and g ` u A ˘ : “ L g ` u A ˘ , and consider the diagram g p A q exp / / g p p A q (cid:15) (cid:15) G p A q G p p A q (cid:15) (cid:15) g exp / / g p u A q K K G p K q G p u A q S S This diagram is commutative, hence in particular G ` p A ˘ ˝ exp “ exp ˝ g ` p A ˘ , which in turnimplies at once G ` p A ˘´ exp ` n g p A q ˘¯ “ exp ´ g ` p A ˘` n g p A q ˘¯ “ exp ` g (˘ “ G p K q ( because n g p A q : “ Ker ` g ` p A ˘˘ ; so in the end exp ` n g p A q ˘ Ď Ker ` G ` p A ˘˘ “ : N G p A q — cf. Proposition 4.1.The fact that exp : g p A q ÝÝÑ G p A q is a local isomorphism, together with Boseck’s splittings— namely, g p A q “ g i n g p A q and G p A q “ G p K q ˙ N G p A q — and dim ` g ˘ “ dim ` G p K q ˘ ,jointly imply dim ` n g p A q ˘ “ dim ` N G p A q ˘ . On the other hand, as n g p A q is nilpotent, its exponentialmap — i.e., just the restriction to n g p A q of exp : g p A q ÝÑ G p A q — is actually a global isomorphism of K –manifolds from n g p A q to exp ` n g p A q ˘ . It then follows that dim ` exp ` n g p A q ˘˘ “ dim ` N G p A q ˘ .Let now N G p A q ˝ be the connected component of N G p A q , so dim ` N G p A q ˝ ˘ “ dim ` N G p A q ˘ ;our previous analysis yields exp ` n g p A q ˘ Ď N G p A q ˝ , the former being a closed Lie subgroup of thelatter. But dim ` exp ` n g p A q ˘˘ “ dim ` N G p A q ˝ ˘ too, so eventually we get exp ` n g p A q ˘ “ N G p A q ˝ .We shall now analyze exp ` n g p A q ˘ “ N G p A q ˝ , eventually proving that it coincides with N G p A q . N G p A q . Let G be a Lie supergroup, as before. As we saw in § A P ( Wsalg ) K the group G p A q is defined as G p A q : “ G A “ Ů g P| G | G A,g where | G | is theunderlying topological space of G and G A,g : “ Hom ( salg ) K ` O G,g , A ˘ , with O G,g being the stalk(a local superalgebra) of the structure sheaf of G at the point g P | G | . We adopt the canonicalidentification | G | “ G p K q via g ÞÑ ev g with ev g : O G,g
ÝÝÑ K given by f ÞÑ ev g p f q : “ f p g q .For every g A P G A,g we have r g A : “ p A ˝ g A (cf. Definition 2.4.3) which coincides with ev g ;moreover, the very definition gives also r g A : “ p A ˝ g A “ G ` p A ˘ p g A q , in short r g A “ G ` p A ˘ p g A q .Finally, due to the splitting A “ K ‘ ˝ A , for every g A P G p A q , say g A P G A,g , there exists also aunique map p g A : O G,g
ÝÝÑ ˝ A such that g A “ r g A ` p g A .15ow assume g A P N G p A q : “ Ker ` G ` p A ˘˘ . Then G ` p A ˘ p g A q “ GA P G p A q ; therefore — bythe previous analysis — we have r g A “ g A “ ` p g A — which can be read as the sum,in the natural sense, of maps from O G, to A . We can re-write our g A as g A “ ` p g A “ exp ` X g A ˘ with X g A : “ log ` g A ˘ “ `8 ř n “ p´ q n ` p g n A n (4.8)where exp ` X g A ˘ : “ `8 ř n “ X ng A M n ! and all powers in these formulas are given by X ng A p f q : “ ` X g A p f q ˘ n , p g n A p f q : “ `p g A p f q ˘ n , etc. All this makes sense because Im `p g A ˘ P ˝ A , by construction; thus p g A isnilpotent, hence X g A is given by a finite sum and it is nilpotent, so exp ` X g A ˘ is a finite sum too.By formal properties of exponential and logarithm, since g A : O G, ÝÑ A is a (superalgebra)morphism it follows from (4.8) that X g A : O G, ÝÑ A is in turn a (superalgebra) derivation; thus— cf. § Proposition 3.2.4 (c) — we have X g A P Lie ` G p A q ˘ “ ` Lie p G q ˘ p A q “ L g p A q “ : g p A q . Finally,by construction we have also Im ` X g A ˘ P ˝ A . Along with Boseck’s splitting g p A q “ g i n g p A q —see (4.4) for g p A q : “ L g p A q — and with n g p A q “ ` ˝ A b K g ˘ ‘ ` ˝ A b K g ˘ — as in (4.6) — allthis together eventually leads to X g A P n g p A q . Tiding everything up, we come now to the end: Proposition 4.1.9.
For any Lie supergroup G and A P ( Wsalg ) K we have N G p A q “ exp ` n g p A q ˘ .In particular, N G p A q is connected nilpotent, and (globally) isomorphic, as a manifold, to n g p A q .Proof. Our analysis above shows that each g A P N G p A q can be realized as g A “ exp ` X g A ˘ with X g A P n g p A q ; hence N G p A q Ď exp ` n g p A q ˘ ; conversely, § ` n g p A q ˘ “ N G p A q ˝ Ď N G p A q . Thus N G p A q “ exp ` n g p A q ˘ as claimed. The last part of the claim then is clear. This subsection is devoted to finding two remarkable splittings for the groups G p A q of A –pointsof any Lie supergroup G ; these are natural in A , hence overall they give noteworthy splittings for G as a functor, known as “global splittings” of G . Such a result is often stated in a form which isnot as “geometric” as we wished (typically, as a splitting of the structure sheaf — cf. for instance:[3], Ch. 2 §
2; [23], § §
2) so we provide an independent proof, with a geometrical statement.The inspiring idea is that we look for a splitting of the form “ G p A q “ G p A q ˆ G p A q ” whichhas to be, somehow, a group-theoretic counterpart of the splitting g p A q “ ` A b K g ˘ ‘ ` A b K g ˘ .Indeed, we will achieve such a goal, in two versions, relying upon Boseck’s splitting of § As above let G be a (smooth, analytic or holomorphic) Lie supergroup over K , whose tangent Lie superalgebra is g : “ Lie p G q , and let A P ( Wsalg ) K be any Weil superalgebra. The powers ˝ A d of the nilradical ˝ A of A form a descending sequence such that ˝ A N “ N " n p d q g p A q : “ ´ ˝ A d b K g ¯ “ ´` ˝ A d ˘ b K g ¯ ‘ ´` ˝ A d ˘ b K g ¯ @ d P N ` this in turn yields a decreasing filtration of Lie subalgebras of n g p A q , with n p N q g p A q “ N " η Y : “ η b Y P n g p A q with η P A , Y P g . By definition η “ p η Y q as a formal series we actuallyhave exp p η Y q “ ` η Y . Similarly, for every c X “ c b X P n g p A q with c P A , X P g if c “ p c X q reads exp p c X q “ ` c X .16or later use, we fix a K –basis B of g of the form B : “ B Ů B with B “ X j ( j P J , resp. B “ Y i ( i P I , being a K –basis of g , resp. of g . Moreover, we fix any total order ĺ on both I and J , so that both B and B are totally ordered and, declaring elements from B to be less thanthose of B — in a nutshell, setting B ĺ B — overall the whole basis B is totally ordered too.Now consider the K –algebra K xx Z , Z yy of formal power series in the non-commutative vari-ables Z and Z . The well-known Campbell-Baker-Hausdorff formula (see, e.g., [16]) in K xx Z , Z yy is exp p Z q ¨ exp p Z q “ exp p Z ˚ Z q with Z ˚ Z : “ log ` exp p Z q ¨ exp p Z q ˘ P K xx Z , Z yy . Moreprecisely, the formal series expansion of Z ˚ Z can be re-arranged in the shape of a formal series Z ˚ Z “ ř `8 n “ L n p Z , Z q where each L n p Z , Z q is a homogeneous Lie monomial of degree n inthe free Lie K –algebra @ Z , Z D K Lie generated by Z and Z . In particular, if one replaces Z and Z with elements z and z sitting in some nilpotent Lie algebra, then all but finitely many of the L n p z , z q ’s do vanish, hence z ˚ z can be written as a finite sum.Our next goal is another description of N G p A q “ exp ` n g p A q ˘ . We need an auxiliary result: Lemma 4.2.2.
Let S , . . . , S ℓ P n g p A q with S i P n p d i q g p A q for some d i P N ` ( i “ , . . . , ℓ ). Thenthere exist T , . . . , T k P n g p A q such that T j P n pB j q g p A q with B j ě d a j ` d b j for some a j , b j P t , . . . , ℓ u (for all j “ , . . . , k ), and exp ` S ` ¨ ¨ ¨ ` S ℓ ˘ “ exp p S q ¨ ¨ ¨ exp p S ℓ q exp p T q ¨ ¨ ¨ exp p T k q Proof.
Writing all exponentials as formal series (actually finite sum , because of the nilpotency ofall elements in n g p A q , cf. § ℓ viaa straightforward application of Baker-Campbell-Hausdorff formula.We can now provide our new description of the subgroup N G p A q “ exp ` n g p A q ˘ : Proposition 4.2.3.
The subgroup N G p A q “ exp ` n g p A q ˘ of G p A q is generated by the set Γ B : “ ! exp ` t j X j ˘ , exp ` η i Y i ˘ ˇˇˇ t j P ˝ A , η i P ˝ A “ A , @ j P J, i P I ) where X j ( j P J Ů Y i ( i P I “ B Ů B “ B is the K –basis of g chosen in § Let n P N G p A q “ exp ` n g p A q ˘ , say n “ exp p Z q with Z P n g p A q ; clearly we can assume Z “ K –basis B of g our Z expands into Z “ ř j P J t j X j ` ř i P I η i Y i for some t j P ˝ A and η i P ˝ A , by the very definition of n g p A q . By Lemma 4.2.2, this implies thatexp p Z q “ exp ´ř j P J t j X j ` ř i P I η i Y i ¯ “ ÝÑ ś j P J exp ` t j X j ˘ ÝÑ ś i P I exp ` η i Y i ˘ ¨ exp ´ Z p q ¯ ¨ ¨ ¨ exp ´ Z p q k ¯ for some Z p q , . . . , Z p q k P n g p A q , where the symbols ÝÑ ś j P J and ÝÑ ś i P I denotes ordered products.Even more, the Lemma ensures that the new terms Z p q h ’s actually “lie deeper”, in the decreasingfiltration of n g p A q given by the n p d q g p A q ’s, than the initial Z we started with, and this allows us toget to the end iterating this argument, in finitely many steps.Indeed, formally speaking we define d p T q P N ` by the condition T P n p d p T qq g p A q z n p d p T q` q g p A q .Then in our construction Lemma 4.2.2 ensures that for the newly occurring elements Z p q , . . . , Z p q k we have d ` Z p q h ˘ ą d p Z q for all h “ , . . . , k . Now we can repeat our argument, with the Z p q h ’splaying the role of Z , and find similar results, i.e. a new expression for exp p Z q of the formexp p Z q “ ÝÑ ś j P J exp ` t j X j ˘ ÝÑ ś i P I exp ` η i Y i ˘ ¨ k ÝÑ ś s “ ˆ ÝÑ ś j P J exp ` t s,j X j ˘ ÝÑ ś i P I exp ` η s,i Y i ˘˙ ¨ k ÝÑ ś r “ exp ` Z p q r ˘ Z p q , . . . , Z p q k P n g p A q such that d p Z p q r q ą min d ` Z p q h ˘( h “ ,...,k for all r “ , . . . , k .Iterating this process we find at each step new factors belonging to Γ B and possibly new factors ofthe form exp ` Z p c q q ˘ , q “ , . . . , k c , such that the sequence n c : “ min d ` Z p c q q ˘( q “ ,...,k c is strictlyincreasing. Then, since we have n p N q g p A q “ t u for N " Z p c q q ” n “ exp p Z q is eventually written as a product ofelements in Γ B ; thus the latter is indeed a generating set for N G p A q “ exp ` n g p A q ˘ , as claimed. G p A q . Before going on, let us consider elements in G p A q of theform exp ` t X ˘ or exp ` η Y ˘ — with t P A such that t “ t P ˝ A indeed), X P g , η P ˝ A , Y P g — like those (see above) that generate exp ` n g p A q ˘ “ N G p A q . Since both t and η have square zero, the formal power series expansion of both exp ` t X ˘ and exp ` η Y ˘ is actuallytruncated at first order, i.e. it reads exp ` t X ˘ “ ` ` t X ˘ and exp ` η Y ˘ “ ` ` η Y ˘ respectively.More in general, we consider elements of the form exp p X q , exp p Y q P exp ` n g p A q ˘ “ N G p A q with X P ˝ A b g and Y P ˝ A b g “ A b g . As both ˝ A and A are nilpotent, the formal powerseries expansion of both exp p X q and exp p Y q can be seen again as a (finite) polynomial.In the next Lemma we collect some identities in G p A q involving “special exponentials” as thosementioned above. This will be crucial all over the sequel. Lemma 4.2.5.
Let A P ( Wsalg ) K , η, η , η P A , η i P A (for all i P I ), Y, Y P g , X P g and g P G p A q . Then inside G p A q we have (notation of Definition 2.2.1)(a) ` ` η η r Y, Y s ˘ “ exp ` η η r Y, Y s ˘ P G p A q (b) p ` η Y q g “ g ` ` η Ad ` g ´ ˘ p Y q ˘ , exp ` ř i P I η i Y i ˘ g “ g exp ` ř i P I η i Ad ` g ´ ˘ p Y i q ˘ (c) ` ` η Y ˘ ` ` η Y ˘ “ ` ` η η r Y , Y s ˘ ` ` η Y ˘ ` ` η Y ˘ (d) ` ` η Y ˘ ` ` η Y ˘ “ ` ` η p Y ` Y q ˘ “ ` ` η Y ˘ ` ` η Y ˘ (e) ` ` η Y ˘ ` ` η Y ˘ “ ` ` η η Y x y ˘ ` ` p η ` η q Y ˘ (f ) p ` η Y q p ` η η X q “ p ` η η X q ` ` η η η r Y, X s ˘ p ` η Y q ““ p ` η η X q p ` η Y q ` ` η η η r Y, X s ˘ (g) Let p h, k q : “ h k h ´ k ´ be the commutator of elements h and k in a group. Then `` ` η Y ˘ , ` ` η Y ˘˘ “ ` ` η η r Y, Y s ˘ , `` ` η Y ˘ , ` ` η Y ˘˘ “ ` ` η p Y ` Y q ˘`` ` η Y ˘ , ` ` η Y ˘˘ “ ` ` η η Y x y ˘ “ ` ` η η Y x y ˘ “ ` ` η η r Y, Y s ˘ (N.B.: taking the rightmost term in the last identity, the latter is a special case of the first).(h) For any n P N ` , there exist unique T p n q , T p n q P @ Z , Z D K Lie , independent of A , such that:— T p n q is a K –linear combination of Lie monomials of even degree greater than n ,— T p n q is a K –linear combination of Lie monomials of odd degree greater than n ,— setting d : “ dim p g q , for any Y , Y P A b K g we have exp ` Y ˘ exp ` Y ˘ “ exp ´ P p d q ` Y , Y ˘¯ exp ´ Y ` Y ` P p d q ` Y , Y ˘¯ with P p d q : “ T p q ˚ T p q ˚ ¨ ¨ ¨ ˚ T p d ´ q and P p d q : “ T p d ´ q ` ¨ ¨ ¨ ` T p q ` T p q .Proof. Writing all exponentials as formal power series (actually finite sum , as noticed above),claims (a) through (g) follow at once from definitions, via straightforward applications of theBaker-Campbell-Hausdorff formula. Claim (a) is even simpler, since ` ` η η r Y, Y s ˘ is just theformal power series expansion of exp ` η η r Y, Y s ˘ , and the latter belong to exp ` ˝ A g ˘ Ď G p A q .18laim (h) requires some more work. An equivalent formulation of it is that the identity Y ˚ Y “ P p d q ` Y , Y ˘ ˚ ´ Y ` Y ` P p d q ` Y , Y ˘¯ (4.9)holds true for some uniquely determined Lie polynomials P p d q : “ T p q ˚ T p q ˚ ¨ ¨ ¨ ˚ T p d ´ q and P p d q : “ T p d ´ q ` ¨ ¨ ¨ ` T p q ` T p q with the T p i q { ’s having the properties mentioned above.We start working with the product “ ˚ ” in @ Z , Z D K Lie . As a matter of terminology, we call order of any non-zero Lie polynomial P in two variables the least degree of a homogeneous monomialoccurring with non-zero coefficient in the standard K –linear expansion of P (accordingly, the orderof the zero polynomial will be ´8 ).First of all, we need three technical results. For formal symbols F, G there exist unique
R , S P @ F, G D K Lie such that F ˚ G “ R ˚ ` F ` G ˘ (4.10) F ` G “ F ˚ S ˚ G (4.11)and R , S are Lie polynomials (in F and G ) of order greater than 1. In fact, R is the uniquesolution of the equation exp p F q exp p G q “ exp p R q exp p F ` G q , given by R “ log ´ exp p F q exp p G q exp p F ` G q ´ ¯ “ F ˚ G ˚ p´ F ´ G q while S is the unique solution of the equation exp p F ` G q “ exp p F q exp p S q exp p G q , given by S “ log ´ exp p F q ´ exp p F ` G q exp p G q ´ ¯ “ p´ F q ˚ p F ` G q ˚ p´ G q Then the explicit expression of the product “ ˚ ” implies that both R and S have order greaterthan 1, as claimed; moreover, both are independent of A and g whatsoever. Finally, for any Liepolynomial T in two variables there exist unique T and T such that T “ T ` T (4.12)where T , resp. T , is a K –linear combination of Lie monomials of even , resp. odd , degree.With repeated applications of (4.10), (4.11) and (4.12) we find Z ˚ Z “ R ˚ ` Z ` Z ˘ p T p q : “ R q “ T p q ˚ ` Z ` Z ˘ (4.12) “ ´ T p q ` T p q ¯ ˚ ` Z ` Z ˘ (4.11) “ (4.11) “ T p q ˚ S ˚ T p q ˚ ` Z ` Z ˘ (4.10) “ T p q ˚ S ˚ R ˚ ´ T p q ` Z ` Z ¯ p T p q : “ S ˚ R q “ p T p q : “ S ˚ R q “ T p q ˚ T p q ˚ ´ T p q ` Z ` Z ¯ (4.12) “ T p q ˚ ´ T p q ` T p q ¯ ˚ ´ T p q ` Z ` Z ¯ (4.11) “ (4.11) “ T p q ˚ T p q ˚ S ˚ T p q ˚ ´ T p q ` Z ` Z ¯ (4.10) “ (4.10) “ T p q ˚ T p q ˚ S ˚ R ˚ ´ T p q ` T p q ` Z ` Z ¯ p T p q : “ S ˚ R q “ p T p q : “ S ˚ R q “ T p q ˚ T p q ˚ T p q ˚ ´ T p q ` T p q ` Z ` Z ¯ “ ¨ ¨ ¨¨ ¨ ¨ “ T p q ˚ T p q ˚ ¨ ¨ ¨ ˚ T p n ´ q ˚ T p n q ˚ ´ T p n ´ q ` ¨ ¨ ¨ ` T p q ` T p q ` Z ` Z ¯ so that in the end we get Z ˚ Z “ T p q ˚ T p q ˚ ¨ ¨ ¨ ˚ T p n ´ q ˚ T p n q ˚ ´ T p n ´ q ` ¨ ¨ ¨ ` T p q ` T p q ` Z ` Z ¯ n P N , where each Lie polynomial T s — by its very construction — has ordergreater than s . Finally, we can re-write this last formula as Z ˚ Z “ P p n q ` Z , Z ˘ ˚ T p n q ˚ ´ Z ` Z ` P p n q ` Z , Z ˘¯ (4.13)for all n P N , with P p n q : “ T p q ˚ T p q ˚ ¨ ¨ ¨ ˚ T p n ´ q and P p n q : “ T p n ´ q ` ¨ ¨ ¨ ` T p q ` T p q .Now observe that every Lie monomial of degree m ą d : “ dim p g q vanishes when computedon A b K g ; therefore T p m ´ q vanishes as well. It then follows that for n “ d replacing Y for Z and Y for Z in (4.13) we eventually get (4.9), q.e.d.We still need to introduce some auxiliary objects associated with G : Definition 4.2.6.
Let G be a Lie supergroup, as above. For any A P ( Wsalg ) K , we define: (a) G ´ p A q : “ ! ś ns “ ` ` η s Y s ˘ ˇˇˇ n P N , p η s , Y s q P A ˆ g @ s P t , . . . , n u ) ` Ď G p A q ˘ (b) exp ` A b K g ˘ : “ exp ` Y ˘ ˇˇ Y P A b K g ( ´ Ď G ´ p A q ¯ (c) N r s G p A q : “ exp ´ A r s b K “ g , g ‰¯ ` Ď N G ` A ˘ “ N G p A q Ş G p A q ˘ (d) for any fixed K –basis Y i ( i P I of g (for some index set I ) and any fixed total order in I , G ă ´ p A q : “ " Ñ ś i P I ` ` η i Y i ˘ ˇˇˇˇ η i P A @ i P I * ` Ď G ´ p A q ˘ where Ñ ś i P I denotes an ordered product — with respect to the fixed total order in I . ♦ Remark 4.2.7.
By definition, exp ` A b K g ˘ contains the set of generators of G ´ p A q ; therefore,the former generates a subgroup A exp ` A b K g ˘E of G p A q that contains G ´ p A q . On the otherhand, for any ř ns “ η s Y s P A b K g , the formal series expansion of exp ` ř ns “ η s Y s ˘ yieldsexp ´ ř ns “ η s Y s ¯ “ ś σ P S n ` ` η σ p q Y σ p q L n ! ˘ ¨ ` ` η σ p q Y σ p q L n ! ˘ ¨ ¨ ¨ ` ` η σ p n q Y σ p n q L n ! ˘ that implies A exp ` A b K g ˘E Ď G ´ p A q . The outcome is that G ´ p A q “ A exp ` A b K g ˘E .From now on, we fix a K –basis Y i ( i P I of g (for some index set I ) and we fix in I a totalorder, as in Definition 4.2.6 (d) . Our first result provides new, interesting factorizations for G p A q : Proposition 4.2.8.
Let G be a Lie supergroup as above, let Y i ( i P I be a totally ordered K –basisof g (for some total order in the set I ) and let A P ( Wsalg ) K be any Weil superalgebra. Then:(a) G ´ p A q coincides with the subgroup @ G ă ´ p A q D of G p A q generated by G ă ´ p A q and with thesubgroup @ exp ` A b K g ˘D generated by exp ` A b K g ˘ ;(b) there exist set-theoretic factorizations (with respect to the group product “ ¨ ”) G ´ p A q “ N r s G p A q ¨ G ă ´ p A q , G ´ p A q “ G ă ´ p A q ¨ N r s G p A q (4.14) N G p A q “ N G ` A ˘ ¨ G ă ´ p A q , N G p A q “ G ă ´ p A q ¨ N G ` A ˘ (4.15) G p A q “ G p A q ¨ G ă ´ p A q , G p A q “ G ă ´ p A q ¨ G p A q (4.16)20 c) there exist set-theoretic factorizations (with respect to the group product “ ¨ ”) G ´ p A q “ N r s G p A q ¨ exp ` A b K g ˘ , G ´ p A q “ exp ` A b K g ˘ ¨ N r s G p A q (4.17) N G p A q “ N G ` A ˘ ¨ exp ` A b K g ˘ , N G p A q “ exp ` A b K g ˘ ¨ N G ` A ˘ (4.18) G p A q “ G p A q ¨ exp ` A b K g ˘ , G p A q “ exp ` A b K g ˘ ¨ G p A q (4.19) Proof. (a)
In Remark 4.2.7 above we proved that @ exp ` A b K g ˘D “ @ G ă ´ p A q D , and we are leftto prove that @ G ă ´ p A q D “ G ´ p A q . It is clear by definition that G ´ p A q is the subgroup in G p A q generated by p ` η Y q ˇˇ η P A , Y P g ( : thus it is enough to prove that each of its generators p ` η Y q actually belongs to @ G ă ´ p A q D .Now, given Y P g let Y “ ř i P I c i Y i (with c i P K ) be its K –linear expansion with respectto the K –basis Y i ( i P I of g . Then repeated applications of the identity in Lemma 4.2.5 (d) yield p ` η Y q “ ` ` η ř i P I c i Y i ˘ “ ` ` ř i P I p c i η q Y i ˘ “ ÝÑ ź i P I ` ` p c i η q Y i ˘ P G ă ´ p A q , q.e.d. (b) We begin with the proof of (4.14): by left-right symmetry, it is enough to prove the left-hand side, that is G ´ p A q “ N r s G p A q ¨ G ă ´ p A q , so we focus on that. By claim (a) , any element of G ´ p A q can be written as a (unordered) product of the form ś Nk “ ` ` η k Y i k ˘ with η k P A and i k P I for all k . Our goal is to prove the following Claim : Any (unordered) product of the form ś Nk “ ` ` η k Y i k ˘ can be “re-ordered”, namely itcan be re-written as an element of N r s G p A q ¨ G ă ´ p A q . To prove this
Claim , let a be the (two-sided) ideal of A generated by the η k ’s, and denote by a n its n –th power ( n P N ); as the η k ’s are N elements and are odd , we have a n “ t u for n ą N .Let us denote by ĺ our fixed total order in I . Given the product ś Nk “ ` ` η k Y i k ˘ , we defineits inversion number as being the number of occurrences of two consecutive indices k s and k s ` forwhich i k s ł i k s ` : then the product itself is ordered if and only if its inversion number is zero.Now assume the product g : “ ś Nk “ ` ` η k Y i k ˘ is unordered: then there exists at least aninversion, say i k s ł i k s ` , i.e. either i k s ą i k s ` or i k s “ i k s ` . Once again, using some of therelations considered in Lemma 4.2.5 we can re-write the product of these two “unordered factors” ` ` η k s Y i ks ˘ ` ` η k s ` Y i ks ` ˘ in either form (depending on whether i k s ą i k s ` or i k s “ i k s ` ) ` ` η k s Y i ks ˘ ` ` η k s ` Y i ks ` ˘ “ ` ` η k s ` η k s “ Y i ks ` , Y i ks ‰˘ ` ` η k s ` Y i ks ` ˘ ` ` η k s Y i ks ˘` ` η k s Y i ks ˘ ` ` η k s ` Y i ks ˘ “ ´ ` η k s ` η k s Y x y i ks ¯ ` ` p η k s ` η k s ` q Y i ks ˘ according to whether i k s ą i k s ` or i k s “ i k s ` respectively. In either case, re-writing in this waythe product of the k s –th and the k s ` –th factor in the original product g : “ ś Nk “ ` ` η k Y i k ˘ ,we end up with another product expression were we did eliminate one inversion, but we “payedthe price” of inserting a new factor . However, in both cases the newly added factor is of the form ` ` a X ˘ for some X P r g , g s and a P a , so that ` ` a X ˘ P N r s G p A q .By repeated use of relations of the form ` ` η Y ˘ ¨ g “ g ¨ ` ` η Ad ` g ´ ˘ p Y q ˘ we canshift the newly added factor ` ` a X ˘ to the leftmost position in g — now re-written oncemore in yet a different product form — at the cost of inserting several new factors of the form ` ` b t Z t ˘ for some Z t P g and b t P a . Moreover, by repeated use of relations of the form ` ` η Y ˘ ¨ ` ` η Y ˘ “ ` ` η ` Y ` Y ˘˘ we can re-write each of these new factors as a productof factors of the form ` ` η h Y i h ˘ where η h P A is a multiple of some b t , so that η h P a too.Eventually, we find a new factorization of the original element g : “ ś Nk “ ` ` η k Y i k ˘ in thenew form g : “ g ¨ ś N h “ ` ` η h Y i h ˘ , where now g P N r s G p A q and the factors ` ` η h Y i h ˘ satisfythe following conditions: 21 (a) each factor ` ` η h Y i h ˘ is either one of the old factors ` ` η k Y i k ˘ or a truly new factor; — (b) in every (truly) new factor ` ` η h Y i h ˘ one has η h P a ; — (c) the number of inversions among factors ` ` η h Y i h ˘ “ ` ` η k Y i k ˘ of the old type isone less than before.Iterating this procedure, after finitely many steps we obtain a new factorization of the initialelement g : “ ś Nk “ ` ` η k Y i k ˘ of the form g “ g ¨ ś N h “ ` ` η h Y i h ˘ where g P N r s G p A q and thefactors ` ` η h Y i h ˘ enjoy properties (a) and (b) above plus the “optimal version” of (c) , namely — (c+) the number of inversions among factors of the old type is zero.Now we apply the same “reordering operation” to the product ś N h “ ` ` η h Y i h ˘ . By construc-tion, an inversion now can occur only among two factors of new type or among an old and a newfactor; but then the two coefficients η h involved by this inversion belong to a and at least one ofthem is in a . It follows that when one performs the “reordering operation” onto the pair of factorsinvolved in the inversion the new factor which pops up necessarily involves a coefficient in a . Asthis applies for any possible inversion, in the end we shall find a new factorization of g of the form g “ g ¨ p g ¨ ś p Nt “ ` ` p η t Y p i t ˘ in which p g P G ` p A q and the factors ` ` p η t Y p i t ˘ are either old factors ` ` η k Y i k ˘ , with noinversions among them , or new factors such that p η t P a .In order to conclude, we can iterate at will this procedure: then — as a n “ t u for n ą N —after finitely many steps we shall no longer find any new factor coming in; thus, we eventually find g “ r g ¨ ś r Nℓ “ ` ` r η ℓ Y r i ℓ ˘ “ r g ¨ Ñ ś r Nℓ “ ` ` r η ℓ Y r i ℓ ˘ in which r g P N r s G p A q and ś r Nℓ “ ` ` r η ℓ Y r i ℓ ˘ “ Ñ ś r Nℓ “ ` ` r η ℓ Y r i ℓ ˘ P G ă ´ p A q is an ordered product ,as required: this means exactly g P N r s G p A q ¨ G ă ´ p A q , thus the Claim is proved.Our
Claim above ensures that G ´ p A q Ď N r s G p A q ¨ G ă ´ p A q . Now we have to prove the converseinclusion. To this end, recall that N r s G p A q is generated by elements of the form p ` c X q with c P A r s and X P r g , g s — still with notation of type ` ` c X ˘ : “ exp p c X q , as before —therefore c “ ř s α s α s and X “ ř r “ Y r , Y r ‰ for some α s , α s P A and some Y r , Y r P g . Now,inside G ` A ˘ we always have relations of the form ` ` a Z ˘ ¨ ` ` a Z ˘ “ ` ` ` a ` a ˘ Z ˘ “ ` ` a Z ˘ ¨ ` ` a Z ˘ for all Z P g and a , a P A such that a “ “ a . Applying this repeatedly to ` ` c X ˘ “ ` ` ` ř s α s α s ˘ X ˘ yields ` ` c X ˘ “ ` ` ` ř s α s α s ˘ X ˘ “ ś s ` ` α s α s X ˘ (4.20)where the factors in the final product can be taken in any order , as they mutually commute.Now recall instead that X “ ř r “ Y r , Y r ‰ , hence each factor ` ` α s α s X ˘ in (4.20) reads ` ` α s α s X ˘ “ ´ ` ř r α s α s “ Y r , Y r ‰¯ (4.21)In addition, inside G ` A ˘ we also have, for all Z , Z P g and a P A such that a “ ` ` a Z ˘ ¨ ` ` a Z ˘ “ ` ` a ` Z ` Z ˘ ˘ “ ` ` a Z ˘ ¨ ` ` a Z ˘ ` ` α s α s X ˘ “ ´ ` ř r α s α s “ Y r , Y r ‰¯ “ ś r ´ ` α s α s “ Y r , Y r ‰¯ (4.22)As next step, recall that in G p A q also hold relations of the form ` ` η Y ˘ ¨ ` ` η Y ˘ “ ´ ` η η “ Y , Y ‰¯ ¨ ` ` η Y ˘ ¨ ` ` η Y ˘ that we can re-shape as ´ ` η η “ Y , Y ‰¯ “ ´` ` η Y ˘ , ` ` η Y ˘¯ (4.23)where in right-hand side we used standard group-theoretical notation ` a , b ˘ : “ a b a ´ b ´ for thecommutator of any two elements a and b in a given group. Now (4.23) together with (4.22) gives ` ` α s α s X ˘ “ ś r ´ ` α s α s “ Y r , Y r ‰¯ “ ś r ´` ` α s Y r ˘ , ` ` α s Y r ˘¯ (4.24)Eventually, matching (4.24) with (4.20) we get ` ` c X ˘ “ ś s ` ` α s α s X ˘ “ ś s,r ´` ` α s Y r ˘ , ` ` α s Y r ˘¯ P @ G ă ´ p A q D “ G ´ p A q (where the factors in the last product can be taken in any order). Thus the element ` ` c X ˘ belongs to G ´ p A q ; since N r s G p A q is generated by such elements, we get N r s G p A q Ď G ´ p A q , whenceclearly N r s G p A q ¨ G ă ´ p A q Ď G ´ p A q and we are done.Now we go and prove (4.15): like before, it is enough to prove the left-hand side, that is N G p A q “ N G ` A ˘ ¨ G ă ´ p A q , by left-right symmetry.Thanks to Proposition 4.2.3, we can take as generators of N G p A q the elements of the set ! exp ` t j X j ˘ , exp ` η i Y i ˘ “ ` ` η i Y i ˘ ˇˇˇ t j P ˝ A , η i P ˝ A “ A , @ j P J, i P I ) (4.25)where X j ( j P J is any K –basis of g and Y i ( i P I is our fixed, totally ordered K –basis of g .Therefore, our aim is to prove that any n P N G p A q , originally expressed as an unordered productof factors taken from (4.25), can be “re-ordered” so to read as an ordered product of the form n Ñ ś i P I ` ` p η i Y i ˘ , for some n P N G ` A ˘ and p η i P A , which does belong to N G ` A ˘ ¨ G ă ´ p A q .First of all, whenever in our original product n we have two consecutive factors ` ` η s Y i s ˘ and n : “ exp ` t j X j ˘ P N G ` A ˘ ` Ď G ` A ˘ q — that is, we have a “sub-product” of the form ` ` η s Y i s ˘ ¨ n — using relations (b) and (d) in Lemma 4.2.5 we can re-write this subproduct as ` ` η s Y i s ˘ n “ n ´ ` η s Ad ` p n q ´ ˘` Y i s ˘¯ “ n ´ ` η s ř j P I c s,j Y j ¯ “ n Ñ ś j P I ` ` η s c s,j Y j ˘ (4.26)for suitable c s,j P K ( j P I ); note in particular that the rightmost term in the chain of identities(4.26) does belong to N G ` A ˘ ¨ G ă ´ p A q , i.e. it has the form we are looking form. Applying thisprocedure, whenever in our element n — written as a product as above — we have a factor of type n P N G ` A ˘ that occurs on the right of any factor of type ` ` η s Y i s ˘ , we can “move the formerto the left of the latter” in the sense that we apply (4.26). Then, after finitely many repetitions ofthis move we end up with a new factorization of n of the form n “ n ¨ N ś s “ Ñ ś j P I ` ` η s k s,j Y j ˘ (4.27)23or some n P N G ` A ˘ and k s,j P K (where N is the number of factors of type ` ` η s Y i s ˘ occurring in the initial factorization of n . Now, in this last factorization, the right-hand side gives N ś s “ Ñ ś j P I ` ` η s k s,j Y j ˘ P G ´ p A q “ N r s G p A q ¨ G ă ´ p A q thanks to (4.14). This together with (4.27) gives n P N G ` A ˘ ¨ G ´ p A q “ N G ` A ˘ ¨ N r s G p A q ¨ G ă ´ p A q “ N G ` A ˘ ¨ G ă ´ p A q — because N r s G p A q is a subgroup of N G ` A ˘ , by construction — and we are done.Finally, as to (4.16) we prove it via the following chain of identities: G p A q “ G p K q ¨ N G p A q “ G p K q ¨ ` N G ` A ˘ ¨ G ă ´ p A q ˘ ““ ` G p K q ¨ N G ` A ˘˘ ¨ G ă ´ p A q “ G ` A ˘ ¨ G ă ´ p A q “ G p A q ¨ G ă ´ p A q where we first used Boseck’s splitting for G p A q — cf. (4.1) — and then (4.15). (c) We begin with the proof of (4.17), for which again it is enough to prove the left-hand side,that is G ´ p A q “ N r s G p A q ¨ exp ` A b K g ˘ .First of all, the inclusion N r s G p A q ¨ exp ` A b K g ˘ Ď G ´ p A q follows at once from (4.14) togetherwith claim (a) . Moreover, again by claim (a) we have G ´ p A q “ @ exp ` A b K g ˘D so it is enough toprove that the product of any two generators exp ` Y ˘ and exp ` Y ˘ of G ´ p A q “ @ exp ` A b K g ˘D lies in N r s G p A q ¨ exp ` A b K g ˘ . In fact, this follows at once from the identityexp ` Y ˘ exp ` Y ˘ “ exp ´ P p d q ` Y , Y ˘¯ exp ´ Y ` Y ` P p d q ` Y , Y ˘¯ in Lemma 4.2.5 (h) , just because it tells us exactly that exp ´ P p d q ` Y , Y ˘¯ P N r s G p A q andexp ´ Y ` Y ` P p d q ` Y , Y ˘¯ P exp ` A b K g ˘ .The same argument used above also applies to prove (4.18).Finally, (4.19) follows from the chain of identities G p A q “ G p K q ¨ N G p A q “ G p K q ¨ ` N G ` A ˘ ¨ exp ` A b K g ˘˘ ““ ` G p K q ¨ N G ` A ˘˘ ¨ exp ` A b K g ˘ “ G ` A ˘ ¨ exp ` A b K g ˘ “ G p A q ¨ exp ` A b K g ˘ using Boseck’s splitting for G p A q — cf. (4.1) — and then (4.18).The previous proposition provides remarkable factorizations for the A –points of the Lie super-group G and their remarkable subgroups N G and G ´ . Our ultimate goal is to improve such aresult, eventually achieving stronger factorization results: in case of G itself, this will be what is(more or less) known as “global splitting” for Lie supergroups. We still need a technical lemma: Lemma 4.2.9.
Given a Lie supergroup G and A P ( Wsalg ) k , let ζ i P A ( i P I ). Then:(a) if g : “ Ñ ś i P I ` ` ζ i Y i ˘ P G p A q Ş G ă ´ p A q , then ζ i “ for all i P I ;(b) if g : “ exp ` ř i P I ζ i Y i ˘ P G p A q Ş exp ` A b K g ˘ , then ζ i “ for all i P I . roof. (a) Recall that, by definition, we have G p A q : “ š x P| G | Hom ( salg ) K ` O | G | ,x , A ˘ ; therefore, itmakes sense to formally expand the product defining g as g : “ Ñ ś i P I ` ` ζ i Y i ˘ “ ` ř i P I ζ i Y i ` O p q (4.28)where O p q is a short-hand notation for “ additional summands of higher order in the ζ i ’s ”. Let a : “ ` t ζ i u i P I ˘ be the ideal of A generated by the ζ i ’s; then (4.28) yields r g s : “ ` ř i P I r ζ i s Y i (4.29)inside G ` A L a ˘ : “ š x P| G | Hom ( salg ) K ` O | G | ,x , A L a ˘ . On the other hand, the assumption that g P G p A q Ş G ă ´ p A q implies r g s P G ` A L a ˘ Ş G ă ´ ` A L a ˘ as well, hence in particular — thinkingof r g s as an A L a –valued map — we have Im ` r g s ˘ Ď ` A L a ˘ .Now, as Y i ( i P I is a K –basis of g , there exists a local system of coordinates around theunit point 1 G P | G | , say t y i u i P I , such that Y i p y j q “ δ i,j for all i, j P I . Then (4.29) gives r g s p y j q : “ ` ř i P I r ζ i s Y i p y j q “ r ζ j s , in particular r g s p y j q “ r ζ j s P ` A L a ˘ ; this togetherwith Im ` r g s ˘ Ď ` A L a ˘ implies r ζ j s “ r s P A L a , i.e. ζ j P a “ ` t ζ i u i P I ˘ , for all j P I :thus ζ j P a n for n P N , j P I . But a n “ n " ζ j “ j P I , q.e.d. (b) Acting like in (a) , we can expand g : “ exp ` ř i P I ζ i Y i ˘ P G p A q Ş exp ` A b K g ˘ into aformal power series (indeed a finite sum) of the form g : “ exp ` ř i P I ζ i Y i ˘ “ ` ř i P I ζ i Y i ` O p q ,just like in (4.28). Then the same argument applies again, and leads to conclusion.Finally, we can now state and prove the main result of the present subsection: Theorem 4.2.10. (existence of Global Splittings for Lie supergroups)
Let G be a Lie supergroup, and g its tangent Lie superalgebra.(a) The restriction of group multiplication in G provides isomorphisms of (set-valued) functors N r s G ˆ G ă ´ – G ´ , N G ˆ G ă ´ – N G , G ˆ G ă ´ – GG ă ´ ˆ N r s G – G ´ , G ă ´ ˆ N G – N G , G ă ´ ˆ G – GN r s G ˆ exp ` p´q b K g ˘ – G ´ , N G ˆ exp ` p´q b K g ˘ – N G , G ˆ exp ` p´q b K g ˘ – G exp ` p´q b K g ˘ ˆ N r s G – G ´ , exp ` p´q b K g ˘ ˆ N G – N G , exp ` p´q b K g ˘ ˆ G – G with exp ` p´q b K g ˘ the set-valued functor ( Wsalg ) K ÝÑ ( sets ) given by A ÞÑ exp ` A b K g ˘ .(b) Setting notation d : “ dim K ` g ˘ “ | I | , there exist isomorphisms of (set-valued) functors A | d K – G ă ´ and A | d K – exp ` p´q b K g ˘ , given on A –points — for every A P ( Wsalg ) K — by A | d K p A q “ A d ÝÝÑ G ă ´ p A q , ` η i ˘ i P I ÞÑ Ñ ś i P I p ` η i Y i q and by A | d K p A q “ A d ÝÝÑ exp ` A b K g ˘ , ` η i ˘ i P I ÞÑ exp ` ř i P I η i Y i ˘ (c) There exist isomorphisms of (set-valued) functors N r s G ˆ A | d K – G ´ , N G ˆ A | d K – N G , G ˆ A | d K – G A | d K ˆ N r s G – G ´ , A | d K ˆ N G – N G , A | d K ˆ G – G induced in the obvious way by those in (a) and (b). roof. (a) The claim yields a (strong) refinement of the factorization results of Proposition 4.2.8 (b) ,now stated in functorial terms. Just like for that proposition, it is enough to prove half of the results,say those in first and in third line; moreover, we bound ourselves to prove the claim concerning G ,as the others go along similarly.Assume that ˆ g ` ˆ g ´ “ ˇ g ` ˇ g ´ with ˆ g ` , ˇ g ` P G p A q , ˆ g ´ , ˇ g ´ P G ă ´ p A q ; then g : “ ˆ g ´ ˇ g ´ ´ “ ˆ g ´ ` ˇ g ` P G p A q , as G p A q is a subgroup in G p A q . Now ˆ g ´ , ˇ g ´ P G ă ´ p A q have the form ˆ g ´ “ Ñ ś i P I ` ` ˆ η i Y i ˘ and ˇ g ´ “ Ñ ś i P I ` ` ˇ η i Y i ˘ , so that ˇ g ´ ´ “ Ð ś i P I ` ´ ˇ η i Y i ˘ , where once more Ñ ś and Ð ś respectively denote an ordered and a reversely-ordered product. Therefore we have G p A q Q g : “ ˆ g ´ ˇ g ´ ´ “ Ñ ś i P I ` ` ˆ η i Y i ˘ Ð ś i P I ` ´ ˇ η i Y i ˘ (4.30)Let b : “ ` ˇ η i α j ( i,j P I ˘ be the ideal of A generated by the products of the ˇ η i ’s and the α j : “ ` ˆ η j ´ ˇ η j ˘ ’s; also, write π b : A ÝÝ ։ A L b for the quotient map, r a s b : “ π b p a q for a P A ,and G p π b q : G p A q ÝÝÑ G ` A L b ˘ for the associated group morphism, with r y s b : “ G p π b qp y q for y P G p A q . Now applying to (4.30) the commutation relations in Lemma 4.2.5 we get r g s b “ Ñ ś i P I ` ` r ˆ η i s Y i ˘ Ð ś i P I ` ´ r ˇ η i s Y i ˘ “ Ñ ś i P I ` ` r α i s Y i ˘ P G ă ´ ` A L b ˘ Since it is also r g s b P G ` A L b ˘ , we can apply Lemma 4.2.9 (a) , with A L b playing the rˆole of A , thus finding r α i s b “ r s b P A L b , that is α i P b , for all i P I . In turn, this implies at once that α i P a n (for i P I ) for all n P N ` , where a : “ ` ˆ η i ˇ η j ( i,j P I ˘ is the ideal of A generated by all theˆ η i ’s and ˇ η j ’s; but a n “ t u for n " η i ´ ˇ η i “ : α i “ η i “ ˇ η i , for all i P I . This yieldsˆ g ´ “ ˇ g ´ , and from this we get also ˆ g ` “ ˇ g ` , q.e.d.Next we prove the result for G in third line: again, acting pointwise this amounts to provingthat, for any A P ( Wsalg ) k , the multiplication map G p A q ˆ exp ` A b K g ˘ ÝÝÝÑ G p A q —which is surjective by (4.19) in Proposition 4.2.8 (b) — is also injective : that is, we must show that,if ˆ g ` ˆ g e “ ˇ g ` ˇ g e for ˆ g ` , ˇ g ` P G p A q and ˆ g ´ , ˇ g e P exp ` A b K g ˘ , then ˆ g ` “ ˇ g ` and ˆ g e “ ˇ g e .From ˆ g ` ˆ g e “ ˇ g ` ˇ g e we get ˇ g e ˆ g ´ e “ ˇ g ´ ` ˆ g ` P G p A q . Writing ˆ g e and ˇ g e explicitly asˆ g e “ exp ` ř i P I ˆ η i Y i ˘ and ˇ g e “ exp ` ř i P I ˇ η i Y i ˘ — with ˆ η i , ˇ η i P A — we findˇ g e ˆ g ´ e “ exp ` ř i P I ˆ η i Y i ˘ exp ` ´ ř i P I ˇ η i Y i ˘ “ exp ´` ř i P I ˆ η i Y i ˘ ˚ ` ´ ř i P I ˇ η i Y i ˘¯ (4.31)where “ ˚ ” denotes again the formal product given by the Campbell-Baker-Hausdorff formula. Nowusing again Lemma 4.2.5 (h) we getˇ g e ˆ g ´ e “ exp ´ P p d q ´ ř i ˆ η i Y i , ´ ř i ˇ η i Y i ¯¯ ¨ exp ´ ř i ` ˆ η i ´ ˇ η i ˘ Y i ` P p d q ´ ř i ˆ η i Y i , ´ ř i ˇ η i Y i ¯¯ with exp ´ P p d q ´ ř i ˆ η i Y i , ´ ř i ˇ η i Y i ¯¯ P G p A q Q ˇ g e ˆ g ´ e ; this in turn impliesexp ´ ř i P I ` ˆ η i ´ ˇ η i ˘ Y i ` P p d q ´ ř i P I ˆ η i Y i , ´ ř i P I ˇ η i Y i ¯¯ P G p A q as well, hence eventuallyexp ´ ř i P I ` ˆ η i ´ ˇ η i ˘ Y i ` P p d q ´ ř i P I ˆ η i Y i , ´ ř i P I ˇ η i Y i ¯¯ P G p A q č exp ` A b K g ˘ (4.32)Now consider any Lie monomial of degree greater than 1, say M : denoting by M ` ℓ , ℓ ˘ anyarbitrary way of filling it with two Lie variables ℓ and ℓ , we always have M ` ℓ , ℓ ´ ℓ ˘ “ M ` ℓ , ℓ ˘ .As an application, for ˆ Y : “ ř i P I ˆ η i Y i , ˇ Y : “ ř i P I ˇ η i Y i and Y α : “ ř i P I α i Y i “ ˆ Y ´ ˇ Y , this gives26 ` ˆ Y , ´ ˇ Y ˘ “ M ` ˆ Y , Y α ´ ˆ Y ˘ “ M ` ˆ Y , Y α ˘ . By Lemma 4.2.5 (h) we know that P p d q p x, y q is a K –linear combination of Lie monomials of degree greater than 1, hence we can conclude that P p d q ` ˆ Y , ´ ˇ Y ˘ “ P p d q ` ˆ Y , Y α ˘ “ P p d q ´ ř i P I ˆ η i Y i , ř i P I α i Y i ¯ “ ř i P I β i Y i where in the right-hand side expansion of P p d q ´ ř i P I ˆ η i Y i , ř i P I ´ ˇ η i Y i ¯ “ P p d q ` ˆ Y , ´ ˇ Y ˘ P A b g we have β i P b for all i P I , with b : “ ` ˇ η i α j ( i,j P I ˘ the ideal of A given above. Then (4.32) readsexp ´ ř i P I ` α i ` β i ˘ Y i ¯ P G p A q č exp ` A b K g ˘ which by Lemma 4.2.9 (b) implies ` α i ` β i ˘ “ α i “ ´ β i P b Ď a , for all i P I , where a : “ ` ˆ η i ˇ η j ( i,j P I ˘ is the ideal of A introduced above. By construction this implies α i P a n for all i P I and n P N , hence — since a n “ t u for n " η i ´ ˇ η i “ : α i “ η i “ ˇ η i , for all i P I . This means that ˆ g e “ ˇ g e , whence ˆ g ` “ ˇ g ` too, q.e.d. (b) To begin with, by definition of G ă ´ there exists a functor epimorphism Θ ă : A | d K ÝÝÑ G ă ´ which is given on every single A P ( Wsalg ) K byΘ ă A : A | d K p A q : “ A ˆ d ÝÝÑ G ă ´ p A q , ` η i ˘ i P I ÞÑ Θ ă A `` η i ˘ i P I ˘ : “ Ñ ś i P I ` ` η i Y i ˘ We prove now that all these Θ ă A ’s are injective, so that Θ ă is indeed an isomorphism.Let ` ˆ η i ˘ i P I , ` ˇ η i ˘ i P I P A ˆ d be such that Θ ă A `` ˆ η i ˘ i P I ˘ “ Θ ă A `` ˇ η i ˘ i P I ˘ , in other words we have Ñ ś i P I ` ` ˆ η i Y i ˘ “ Ñ ś i P I ` ` ˇ η i Y i ˘ . Then we can replay the proof of the first part of claim (a) , nowwith ˆ g ` : “ “ : ˇ g ` ; the outcome is again ˆ η i “ ˇ η i for all i P I , that is ` ˆ η i ˘ i P I “ ` ˇ η i ˘ i P I , q.e.d.As to the isomorphism A | d K – exp ` p´q b K g ˘ , definitions provide a functor epimorphismΘ e : A | d K ÝÝÑ exp ` p´q b K g ˘ given on each A P ( Wsalg ) K byΘ eA : A | d K p A q : “ A ˆ d ÝÝÑ G ă ´ p A q , ` η i ˘ i P I ÞÑ Θ eA `` η i ˘ i P I ˘ : “ exp ` ř i P I η i Y i ˘ But all these Θ eA ’s are indeed injective, so that overall Θ e is indeed an isomorphism. In fact, let ` ˆ η i ˘ i P I , ` ˇ η i ˘ i P I P A ˆ d give Θ eA `` ˆ η i ˘ i P I ˘ “ Θ eA `` ˇ η i ˘ i P I ˘ , i.e. exp ` ř i P I ˆ η i Y i ˘ “ exp ` ř i P I ˇ η i Y i ˘ .Then we can proceed again as in the proof of the second part of claim (a) , now with ˆ g ` : “ “ : ˇ g ` ;this gives again ˆ η i “ ˇ η i for all i P I , thus ` ˆ η i ˘ i P I “ ` ˇ η i ˘ i P I , q.e.d. (c) It follows at once from claims (a) and (b) together.
Remark 4.2.11.
For every Lie supergroup G , we shall refer to the isomorphisms in claim (a) and/or (c) of Theorem 4.2.10 as to “Global Splittings” of G — or of N G , or of G ´ , respectively. In this section we provide two different functors Ψ that are quasi-inverse to the functor Φof Theorem 3.2.6. In both cases, for any super Harish-Chandra pair P , we define as associatedΨ p P q : “ G P a suitable functor from Weil superalgebras to groups, and then prove that it has the“right properties”. Concretely, we follow the pattern provided by the Global Splitting Theorem G P shouldlook like, in terms of P itself: this leads us, eventually, to provide two different recipes.In both cases, we proceed along the following lines: we define our looked-for G P as a group-valued functor on the category of Weil K –superalgebras, such that each single group G P p A q —with A P ( Wsalg ) K — is given by generators and relations, in a uniform way (with respect to A ).The idea that dictates the desired presentation by generators and relations is somewhat simple:as part of our ultimate goal, we want Φ ` G P ˘ “ P ` “ p G ` , g q ˘ , so we must have ` G P ˘ “ G ` and Lie ` G P ˘ “ g . The former requirement gives us the reduced Lie subgroup; the latter insteadprescribes what the Lie superalgebra of G P must be: then we might think of using this to realizethe “missing part” of G P as “ exp p g q ”. So each G P p A q should be presented as generated by G ` p A q “ G ` p A q and exp ` A b g ˘ , or at least enough of its elements, and suitable relations.To realize all this, we follow the pattern provided by the “Global Splitting(s) Theorem” — cf.Theorem 4.2.10 — which essentially prescribes, in two ways, how G P must be done. In this subsection we construct a first Lie supergroup functor, denoted G ˝ P , along the linesmentioned above. As a matter of notation, hereafter we shall adopt the following: given P “p G ` , g q P ( sHCp ) K , A P ( Wsalg ) K and c P A such that c “ X P g we set ` G ` ` c X ˘ : “ exp ` c X ˘ P G ` p A q (5.1)which is obviously inspired by the formal series expansion of the “ exp ” function; when no confusionis possible we shall drop the subscript G ` and simply write ` ` c X ˘ instead. Similarly, we shallpresently introduce new formal elements of type “ ` ` η Y ˘ “ exp ` η Y ˘ ” with η P A , Y P g . Definition 5.1.1.
Let P : “ ` G ` , g ˘ P ( sHCp ) K be a super Harish-Chandra pair over K . (a) We introduce a functor G ˝ P : ( Wsalg ) K ÝÝÑ ( group ) as follows. For any Weil superalgebra A P ( Wsalg ) K , we define G ˝ P p A q as being the group with generators the elements of the setΓ A : “ g ` , ` ` η Y ˘ ˇˇ g ` P G ` p A q , p η, Y q P A ˆ g ( “ G ` p A q Ť p ` η Y q ( p η ,Y q P A ˆ g and relations (for g , g P G ` p A q , η , η , η P A , Y , Y , Y P g , c P K ) g ¨ g “ g ¨ G ` g , ` ` η Y ˘ ¨ g ` “ g ` ¨ ` ` η Ad ` g ´ ` ˘ p Y q ˘` ` η Y ˘ ¨ ` ` η Y ˘ “ ´ G ` ` η η Y x y ¯ G ` ¨ ` ` ` η ` η ˘ Y ˘` ` η Y ˘ ¨ ` ` η Y ˘ “ ´ G ` ` η η “ Y , Y ‰¯ G ` ¨ ` ` η Y ˘ ¨ ` ` η Y ˘` ` η Y ˘ ¨ ` ` η Y ˘ “ ` ` η ` Y ` Y ˘˘` ` p c η q Y ˘ “ ` ` η p cY q ˘ , ` ` η g ˘ “ , ` ` A Y ˘ “ G ` p A q their product, denoted with“ ¨ ”, inside G ˝ P p A q is the same as in G ` p A q , where it is denoted with “ ¨ G ` ”; moreover, notation like ´ G ` ` η η Y x y ¯ G ` and ´ G ` ` η η “ Y , Y ‰¯ G ` denotes two elements in G ` p A q as in (5.1).This yields the functor G ˝ P on objects, and one then defines it on morphisms in the obvious way.Namely, for any morphism f : A ÝÑ A in ( Wsalg ) K we let G ˝ P p f q : G ˝ P ` A ˘ ÝÑ G ˝ P ` A ˘ be thegroup morphism uniquely defined on generators — for all g P G ` ` A ˘ , η P A , Y P g — by G ˝ P p f q ` g ˘ : “ G ` p f q ` g ˘ , G ˝ P p f q ` ` η Y ˘ : “ ` ` f ` η ˘ Y ˘
28s the defining relations of each G ˝ P p A q are independent of A , such a G ˝ P p f q is well defined indeed. (b) We define a functor G ˝ , ´ P : ( Wsalg ) K ÝÝÑ ( set ) on any object A P ( Wsalg ) K by G ˝ , ´ P p A q : “ ! ś ns “ ` ` η s Y s ˘ ˇˇˇ n P N , p η s , Y s q P A ˆ g @ s P t , . . . , n u ) ` Ď G ˝ P p A q ˘ and on morphism in the obvious way — just like for G P . (c) Let us fix in g a K –basis Y i ( i P I — for some index set I — and a total order in I . Wedefine a functor G ă ´ : ( Wsalg ) K ÝÝÑ ( set ) as follows. For A P ( Wsalg ) K we set G ă ´ p A q : “ " Ñ ś i P I ` ` η i Y i ˘ ˇˇˇˇ η i P A @ i P I * ` Ď G ˝ , ´ P p A q Ď G ˝ P p A q ˘ where Ñ ś i P I denotes an ordered product — with respect to the fixed total order in I . This definesthe functor G ă ´ on objects, and its definition on morphism is the obvious one (like for G ˝ P ) . ♦ Remark 5.1.2.
By their very definition, both G ˝ , ´ P and G ă ´ can be thought of as subfunctorsof G ˝ P . Moreover, every group G ˝ , ´ P p A q is clearly the subgroup of G ˝ P p A q generated by G ă ´ p A q , oreven by p ` η Y q ( p η ,Y qP A ˆ g . In particular, although G ă ´ depends on the choice of Y i ( i P I , thesupergroup subfunctor that it generates (inside G ˝ P ) instead is independent of any such choice.Next result shows that G ˝ P can also be described using a much smaller set of generators: Proposition 5.1.3.
Let P : “ ` G ` , g ˘ P ( sHCp ) K be a super Harish-Chandra pair over K ; also,we fix in g a K –basis Y i ( i P I — for some index set I — and a total order in I .Then for every Weil K –superalgebra A P ( Wsalg ) K the group G ˝ P p A q is generated by the set Γ ˛ A : “ G ` p A q Ť ` ` η i Y i ˘ ˇˇ η i P A , @ i P I ( Proof.
Given A P ( Wsalg ) K , let G ˛ P p A q be the subgroup of G ˝ P p A q generated by Γ ˛ A . We shallprove that every generator of the (larger, a priori ) group G ˝ P p A q of the form p ` η Y q with p η , Y q P A ˆ g also belongs to the subgroup G ˛ P p A q : this then will prove the claim.So let p η , Y q P A ˆ g ; then, in terms of the K –basis Y i ( i P I of g , our Y expands into Y “ ř ks “ c j s Y j s . By repeated applications of relations of the form ` ` η Y ˘ ¨ ` ` η Y ˘ “ ` ` η ` Y ` Y ˘˘ , we find that the generator p ` η Y q in G ˝ P p A q factors as ` ` η Y ˘ “ ´ ` η ř ks “ c j s Y j s ¯ “ ś ks “ ` ` c j s η Y j s ˘ (5.2)where the product can be done in any order, as the factors in it mutually commute. Now theproduct in right-hand side does belong to G ˛ P p A q , hence we are done. G ˝ P . Let P “ ` G ` , g ˘ P ( sHCp ) K be a super Harish-Chandrapair; we present now yet another way of realizing the K –supergroup G ˝ P introduced in Definition5.1.1 (a) . In the following, if K is any group presented by generators and relations, we write K “ @ Γ DM` R ˘ if Γ is a set of free generators (of K ), R is a set of relations among generators and ` R ˘ is the normal subgroup in K generated by R . As a matter of notation, given a presentation K “ @ Γ DM` R ˘ “ @ Γ DM` R Y R ˘ with R “ R Y R , the Double Quotient Theorem gives us K “ @ Γ DM` R ˘ “ @ Γ DM` R Y R ˘ “ @ Γ DM` R ˘O` R Y R ˘M` R ˘ “ @ Γ DM` R ˘ (5.3)where Γ and R respectively denote the images of Γ and of R in the quotient group @ Γ DM` R ˘ .29or any fixed A P ( Wsalg ) K , we denote by G r s` p A q the subgroup of G ` p A q generated by theset p ` c X q ˇˇ c P A r s , X P r g , g s ( — cf. § A r s . Note then that G r s` p A q is normal in G ` p A q , as one easily sees by construction (taking into account that, as P : “ ` G ` , g ˘ is a super Harish-Chandra pair, the “adjoint” action of G ` onto g maps r g , g s into itself).We consider also the three setsΓ ` A : “ G ` p A q , Γ r s A : “ G r s` p A q , Γ ´ A : “ Γ r s A Ť p ` η Y q ( p η ,Y q P A ˆ g and the sets of relations — for all g ` , g , g P Γ ` A , g r s , g s , g s P Γ r s A , η , η , η P A , X Pr g , g s , Y, Y , Y P g , with ¨ G ` and ¨ G r s` being the product in G ` p A q and in G r s` p A q — given by R ` A : g ¨ g “ g ¨ G ` g R ´ A : $’’’’’’’’’’’&’’’’’’’’’’’% g s ¨ g s “ g s ¨ G r s` g s ` ` η Y ˘ ¨ g r s “ g r s ¨ ` ` η Ad ` g ´ r s ˘ p Y q ˘` ` η Y ˘ ¨ ` ` η Y ˘ “ ´ ` η η Y x y ¯ ¨ ` ` ` η ` η ˘ Y ˘` ` η Y ˘ ¨ ` ` η Y ˘ “ ´ ` η η “ Y , Y ‰¯ ¨ ` ` η Y ˘ ¨ ` ` η Y ˘` ` η Y ˘ ¨ ` ` η Y ˘ “ ` ` η ` Y ` Y ˘˘` ` η g ˘ “ , ` ` A Y ˘ “ R ˙ A : g r s ¨ g ` “ g ` ¨ ` g ´ ` ¨ G ` g r s ¨ G ` g ` ˘ , ` ` η Y ˘ ¨ g ` “ g ` ¨ ` ` η Ad ` g ´ ` ˘ p Y q ˘ R r s A : ` g r s ˘ Γ r s A “ ` g r s ˘ Γ ` A R A : “ R ` A Ť R ´ A Ť R ˙ A Ť R r s A (in particular, note that the relations of type R r s A in down-to-earth terms just identify each elementin Γ r s A with its corresponding copy inside Γ ` A ). Then we define a new group, by generators andrelations, namely G ˝ , ´ P p A q : “ @ Γ ´ A DM` R ´ A ˘ .From the very definition of G ˝ P p A q — cf. Definition 5.1.1 — it follows that G ˝ P p A q – @ Γ ` A Ť Γ ´ A DM` R A ˘ “ @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R ˙ A Ť R r s A ¯ (5.4)indeed, here above we are just taking larger sets of generators and of relations (w.r.t. Definition5.1.1), but with enough redundancies as to find a different presentation of the same group.From this we find a neat description of G ˝ P p A q by achieving the presentation (5.4) in a series ofintermediate steps, namely adding only one bunch of relations at a time. As a first step, we have @ Γ ` A Ť Γ ´ A DM` R ` A Ť R ´ A ˘ “ @ Γ ` A DM` R ` A ˘ ˚ @ Γ ´ A DM` R ´ A ˘ – G ` p A q ˚ G ˝ , ´ P p A q (5.5)where G ` p A q – @ Γ ` A DM` R ` A ˘ by construction and ˚ denotes the free product (of two groups).For the next two steps we can follow two different lines of action. On the one hand, one has @ Γ ` A Ť Γ ´ A DM` R ` A Ť R ´ A Ť R ˙ A ˘ – ´ G ` p A q ˚ G ˝ , ´ P p A q ¯N´ R ˙ A ¯ – G ` p A q ˙ G ˝ , ´ P p A q G ` p A q ˙ G ˝ , ´ P p A q is the semidirect product of G ` p A q with G ˝ , ´ P p A q with respect to the obvious (“adjoint”) action of the former on the latter. Then @ Γ ` A Ť Γ ´ A DM` R A ˘ – @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R ˙ A Ť R r s A ¯ –– ´ G ` p A q ˙ G ˝ , ´ P p A q ¯N´ R r s A ¯ – ´ G ` p A q ˙ G ˝ , ´ P p A q ¯N N r s p A q where N r s p A q is the normal subgroup of G ` p A q ˙ G ˝ , ´ P p A q generated by !` g r s , g ´ r s ˘) g r s P Γ r s A .This together with (5.4) eventually yields G ˝ P p A q “ ´ G ` p A q ˙ G ´ p A q ¯N N r s p A q On the other hand, again from (5.3) and (5.5) together we get @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R r s A ¯ – ´ G ` p A q ˚ G ˝ , ´ P p A q ¯N´ R r s A ¯ – G ` p A q ˚ G r s` p A q G ˝ , ´ P p A q where G ` p A q ˚ G r s` p A q G ˝ , ´ P p A q is the amalgamated product of G ` p A q and G ˝ , ´ P p A q over G r s` p A q w.r.t. the natural monomorphisms G r s` p A q ã ÝÝÑ G ` p A q and G r s` p A q ã ÝÝÑ G ˝ , ´ P p A q . Then @ Γ ` A Ť Γ ´ A DM` R A ˘ – @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R r s A Ť R ˙ A ¯ –– ´ G ` p A q ˚ G r s` p A q G ˝ , ´ P p A q ¯N´ R ˙ A ¯ – ´ G ` p A q ˚ G r s` p A q G ˝ , ´ P p A q ¯N N ˙ p A q where N ˙ p A q is the normal subgroup of G ` p A q ˚ G r s` p A q G ˝ , ´ P p A q generated by ! g ` ` ` η Y ˘ g ´ ` ` ` η Ad p g ` qp Y q ˘ ´ ) g ` P G ` p A qp η,Y qP A ˆ g Ť ! g ` g r s g ` ` g ` ¨ G ` g r s ¨ G ` g ` ˘ ´ ) g r s P Γ r s A g ` P G ` p A q All this along with (5.4) eventually gives G ˝ P p A q “ ´ G ` p A q ˚ G r s` p A q G ˝ , ´ P p A q ¯N N ˙ p A q for all A P ( Wsalg ) K . In functorial terms this yields G ˝ P “ ´ G ` ˙ G ˝ , ´ P ¯N N r s and G ˝ P “ ´ G ` ˚ G r s` G ˝ , ´ P ¯N N ˙ , or G ˝ P “ G ` ˙ G r s` G ˝ , ´ P where the last, (hopefully) more suggestive notation G ˝ P “ G ` ˙ G r s` G ˝ , ´ P tells us that G ˝ P is the“amalgamate semidirect product” of G ` and G ˝ , ´ P over their common subgroup G r s` . G ˝ P as a Lie supergroup We aim now to proving that the functor G ˝ P is actually a Lie supergroup. We keep definitionsand notations as before: in particular, recall that for A P ( Wsalg ) K we denote by G r s` p A q thesubgroup of G ` p A q generated by p ` c X q ˇˇ c P A r s , X P r g , g s ( — cf. § A r s .Our first step is the following “factorization result” for G ˝ P :31 roposition 5.2.1. Let P : “ ` G ` , g ˘ P ( sHCp ) K be a super Harish-Chandra pair over K , let Y i ( i P I be a totally ordered K –basis of g (for our fixed order in I ) and A P ( Wsalg ) K . Then:(a) letting @ G ă ´ p A q D be the subgroup of G ˝ P p A q generated by G ă ´ p A q , we have @ G ă ´ p A q D “ G ˝ , ´ P p A q and there exist set-theoretic factorizations (with respect to the group product “ ¨ ”) G ˝ , ´ P p A q “ G r s` p A q ¨ G ă ´ p A q , G ˝ , ´ P p A q “ G ă ´ p A q ¨ G r s` p A q (b) there exist set-theoretic factorizations (with respect to the group product “ ¨ ”) G ˝ P p A q “ G ` p A q ¨ G ă ´ p A q , G ˝ P p A q “ G ă ´ p A q ¨ G ` p A q Proof.
Claim (a) is the exact analogue of Proposition (4.14), and claim (b) the analogue of (4.16),in Proposition 4.2.8 (b) . In both cases the proof (up to trivialities) is identical, so we can skip it. G ˝ P ÝÝÑ GL p V q . When discussing the structure of a Lie supergroup G , the factorization G “ G ¨ G ă ´ was just a intermediate step; Proposition 5.2.1 above givesus the parallel counterpart for G ˝ P . This factorization result for G is improved by the “GlobalSplitting Theorem” — i.e. Theorem 4.2.10 — that, roughly speaking, states that for any g P G p A q the factorization pertaining to G p A q ¨ G ă ´ p A q has uniquely determined factors, and similarlyany element in G ă ´ p A q has a unique factorization into an ordered product of factors of the form ` ` η i Y i ˘ . Both results are proved by showing that two factorizations of the same object necessarilyhave identical factors; in other words, distinct choices of factors always give rise to different elementsin G p A q or in G ă ´ p A q . This last fact was proved using the concrete realization of G p A q as a specialset of maps, namely G p A q : “ Ů x P| G | Hom ( salg ) K ` O | G | ,x , A ˘ ; indeed, this algebra is rich enough to“separate” different elements of G p A q itself just looking at their values as A –valued maps. Whendealing with G ˝ P p A q instead, that is defined abstractly, such a built-in realization is not available:our strategy then is to replace it with a suitable “partial linearization”, namely a representation of G ˝ P p A q that, although not being faithful, is still “rich enough” to (almost) separate elements.Let P “ ` G ` , g ˘ P ( sHCp ) K be given; as before, we fix a K –basis Y i ( i P I of g , where I is anindex set in which we fix some total order, hence the basis itself is totally ordered as well.Recall that the universal enveloping algebra U p g q is given by U p g q : “ T p g q M J where T p g q isthe tensor algebra of g and J is the two-sided ideal in T p g q generated by the set ! x y ´ p´ q | x | | y | y x ´ r x, y s , z ´ z x y ˇˇˇ x, y P g Ť g , z P g ) — where z x y : “ ´ r z, z s , see Definition 2.2.1 (c) . It is known then — see for instance [25], § K –supercoalgebras U p g q – U p g q b K Ź g – Ź g b K U p g q (5.6)In addition, Ź g has K –basis Y i Y i ¨ ¨ ¨ Y i s ˇˇ s ď | I | , i ă i ă ¨ ¨ ¨ ă i s ( ; hereafter, we drop thesign “ ^ ” to denote the product in Ź g .Now let be the (one-dimensional) trivial representation of g . By the standard process of induction from g to g — the former being thought of as a Lie subsuperalgebra of the latter — wecan consider the induced representation V : “ Ind gg p q , that is a g –module. Looking at and V respectively as a module over U p g q and over U p g q , taking (5.6) into account we get V : “ Ind gg p q “ U p g q b U p g q – Ź g b K – Ź g (5.7)The last one above is a natural isomorphism of K –superspaces, uniquely determined once a specificelement b P is fixed to form a K –basis of itself: the isomorphism is ω b b ÞÑ ω for all ω P Ź g .32his representation-theoretical construction and its outcome clearly give rise to similar func-torial counterparts, for the Lie algebra valued K –superfunctors L g and L g , as well as for the K –superfunctors associated with U p g q and U p g q , in the standard way, namely A ÞÑ A b K U p g q and A ÞÑ ` A b K U p g q ˘ “ A b K U p g q À A b K U p g q for all A P ( Wsalg ) K .On the other hand, recall that g “ Lie p G ` q , and clearly is also the trivial representationfor G ` , as a classical Lie group. Then, by construction, the representation of g on the space V alsoinduces a representation of the super Harish-Chandra pair P “ p G ` , g q on the same V , in otherwords V bears also a structure of p G ` , g q –module — in the obvious sense: we have a morphism p r ` , ρ q : p G ` g q ÝÑ ` GL p V q , gl p V q ˘ of super Harish-Chandra pairs. We shall write again ρ for therepresentation map ρ : U p g q ÝÑ End K p V q giving the U p g q –module structure on V .Our key step now is to remark that the above p G ` , g q –module structure on V actually “inte-grates” to a G ˝ P –module structure, in a natural way. Proposition 5.2.3.
Retain notation as above for the p G ` , g q –module V . There exists a uniquestructure of (left) G ˝ P –module onto V which satisfies the following conditions: for every A P ( Wsalg ) K , the representation map r ˝ P ,A : G ˝ P p A q ÝÑ GL p V qp A q is given on generators of G ˝ P p A q — namely, all g ` P G ` p A q and p ` η i Y i q for i P I , η i P A — by r ˝ P ,A p g ` q : “ r ` p g ` q , r ˝ P ,A p ` η i Y i q : “ ρ p ` η i Y i q “ id V ` η i ρ p Y i q or, in other words, g ` .v : “ r ` p g ` qp v q and p ` η i Y i q .v : “ ρ p ` η i Y i qp v q “ v ` η i ρ p Y i qp v q forall v P V p A q . Overall, this yields a morphism a K –supergroup functors r ˝ P : G ˝ P ÝÑ GL p V q .Proof. This is, essentially, a straightforward consequence of the whole construction, and of the verydefinition of G ˝ P . Indeed, by definition of representation for the super Harish-Chandra pair P wesee that the operators r ˝ P ,A p g ` q and r ˝ P ,A p ` η i Y i q on V — associated with the generators of G ˝ P p A q — do satisfy all relations which, by Definition 5.1.1 (a) , are satisfied by the generators themselves.Thus they uniquely provide a well-defined group morphism r ˝ P ,A : G ˝ P p A q ÝÑ GL p V qp A q as required.The construction is clearly functorial in A , whence the claim.The advantage of introducing the representation r P of G ˝ P on V is that it allows us to “separate”,in a sense, the “odd points of G ˝ P p A q from each other and from the even ones”, i.e. we can separatethe points in G ă ´ p A q from each other (in a “very fine” sense) and from those in G ` p A q .We are now ready to state and prove the main result of the present subsection, that is just the“global splitting theorem” for G ˝ P (cf. Theorem 4.2.10): Proposition 5.2.4. (a) The restriction of group multiplication in G ˝ P provides isomorphisms of (set-valued) functors G ` ˆ G ă ´ – G ˝ P , G ă ´ ˆ G ` – G ˝ P , G r s` ˆ G ă ´ – G ˝ , ´ P , G ă ´ ˆ G r s` – G ˝ , ´ P (b) There exists an isomorphism of (set-valued) functors A | d K – G ă ´ , with d : “ | I | “ dim K ` g ˘ , given on A –points — for every A P ( Wsalg ) K — by A | d K p A q “ A d ÝÑ G ă ´ p A q , ` η i ˘ i P I ÞÑ Ñ ś i P I p ` η i Y i q (c) There exist isomorphisms of (set-valued) functors G ` ˆ A | d K – G ˝ P , G r s` ˆ A | d K – G ˝ , ´ P , and A | d K ˆ G ` – G ˝ P , A | d K ˆ G r s` – G ˝ , ´ P given on A –points — for every A P ( Wsalg ) K — respectively by ` g ` , ` η i ˘ i P I ˘ ÞÑ g ` ¨ Ñ ś i P I p ` η i Y i q and `` η i ˘ i P I , g ` ˘ ÞÑ Ñ ś i P I p ` η i Y i q ¨ g ` roof. The proof is quite close to (half of) that of Theorem 4.2.10, with some technical differences,involving the use of the representation V of § (a) It is enough to prove the first identity concerning G ˝ P , as all other are similar. Thus our goalamounts to showing the following: for any A P ( Wsalg ) K , if ˆ g ` ˆ g ´ “ ˇ g ` ˇ g ´ for ˆ g ` , ˇ g ` P G ` p A q and ˆ g ´ , ˇ g ´ P G ă ´ p A q , then ˆ g ` “ ˇ g ` and ˆ g ´ “ ˇ g ´ .The assumption ˆ g ` ˆ g ´ “ ˇ g ` ˇ g ´ implies g : “ ˆ g ´ ˇ g ´ ´ “ ˆ g ´ ` ˇ g ` P G ` p A q , as G ` p A q isa subgroup in G ˝ P p A q . Now ˆ g ´ P G ă ´ p A q has the form ˆ g ´ “ Ñ ś i P I ` ` ˆ η i Y i ˘ and similarlyˇ g ´ “ Ñ ś i P I ` ` ˇ η i Y i ˘ so that ˇ g ´ ´ “ Ð ś i P I ` ´ ˇ η i Y i ˘ ; therefore we have g : “ ˆ g ´ ˇ g ´ ´ “ Ñ ś i P I ` ` ˆ η i Y i ˘ Ð ś i P I ` ´ ˇ η i Y i ˘ P G ` p A q Ď G ˝ P p A q (5.8)Let a : “ ` ˆ η i , ˇ η i ( i P I ˘ be the ideal of A generated by the ˆ η i ’s and the ˇ η i ’s, set A π n ÝÝ ։ A L a n forthe quotient map and r a s n : “ π n p a q for a P A , then G ˝ P p A q G p π n q ÝÝÝÝÑ G ˝ P ` A L a n ˘ for the associatedgroup morphism and r y s n : “ G ˝ P p π n qp y q for every y P G ˝ P p A q . Now, the defining relations for G ă ´ ` A L a ˘ — taking into account that ˆ η h ˇ η k , ˇ η k ˆ η h P a (for all h, k P I ) — yield r g s “ Ñ ś i P I ` ` r ˆ η i s Y i ˘ ¨ Ð ś i P I ` ´ r ˇ η i s Y i ˘ “ Ñ ś i P I ` ` r α i s Y i ˘ P G ă ´ ` A L a ˘ (5.9)Next step then is to let r g s act onto b P V ` A L a ˘ . To avoid confusion, when we describe V as V “ Ź g . b – Ź g , we write the elements of t Y i u i P I of g as ¯ Y i instead of Y i : thus the isomorphism Ź g . b – Ź g is given by p Y i Y i ¨ ¨ ¨ Y i s q . b ÞÑ ¯ Y i ¯ Y i ¨ ¨ ¨ ¯ Y i s — for all i ă i ă ¨ ¨ ¨ ă i s .Taking into account that r α h s r α k s “ r s P A L a (for all h, k P I ) from (5.9) we get thatthe action of r g s onto b P V ` A L a ˘ is given by r g s . b “ Ñ ś i P I ` ` r α i s Y i ˘ . b “ . b ` ÿ i P I r α i s Y i . b “ b ` ÿ i P I r α i s ¯ Y i P V ` A L a ˘ (5.10)On the other hand, we have also r g s . b “ b because r g s P G ` ` A L a ˘ and G ` acts trivially on V . This compared with (5.10), taking into account that b ( Y ¯ Y i ( i P I is part of the chosen basisof V , implies that r α i s “ r s P A L a , i.e. α i P a , for all i P I . But then we can repeat theprevious argument, slightly improved: indeed, again, the defining relations of G ˝ P ` A L a ˘ give r g s “ Ñ ś i P I ` ` r ˆ η i s Y i ˘ Ð ś i P I ` ´ r ˇ η i s Y i ˘ “ Ñ ś i P I ` ` r α i s Y i ˘ P G ˝ P ` A L a ˘ (5.11)— now because the new occurring factors depend on coefficients of the form α h ˇ η k and α h ˇ η k ,that all belong to a . Then we repeat the second step, namely we let r g s act onto b P V ` A L a ˘ ,for which (5.11) gives the analogue of (5.10), namely r g s . b “ b ` ÿ i P I r α i s ¯ Y i P V ` A L a ˘ which in turn implies α i P a , for all i P I . Clearly, we can iterate this process, and find α i P a n for all n P N , i P I ; as a n “ t u for n " a is generated by finitely many odd elements)we end up with α i “ η i “ ˇ η i , for all i P I . This means ˆ g ´ “ ˇ g ´ , whence ˆ g ` “ ˇ g ` as well. (b) By construction there exists a morphism Θ ă : A | d K ÝÝÑ G ă ´ of set-valued functors that isgiven on A –points — for every single A P ( Wsalg ) K — by the mapΘ ă A : A | d K p A q : “ A ˆ d ÝÝÑ G ă ´ p A q , ` η i ˘ i P I ÞÑ Θ ă A `` η i ˘ i P I ˘ : “ Ñ ś i P I ` ` η i Y i ˘ that is actually surjective . We need to prove that all these maps Θ ă A are also injective, so thatoverall Θ ă is indeed an isomorphism. 34et ` ˆ η i ˘ i P I , ` ˇ η i ˘ i P I P A ˆ d be such that Θ ă A `` ˆ η i ˘ i P I ˘ “ Θ ă A `` ˇ η i ˘ i P I ˘ , that is Ñ ś i P I ` ` ˆ η i Y i ˘ “ Ñ ś i P I ` ` ˇ η i Y i ˘ . Then we can replay the proof of claim (a) , now with ˆ g ` : “ “ : ˇ g ` : the outcomewill be again ˆ η i “ ˇ η i for all i P I , i.e. ` ˆ η i ˘ i P I “ ` ˇ η i ˘ i P I . Thus Θ ă A is injective, as desired. (c) This is a direct consequence of (a) and (b) together. G ˝ P . For any given P P ( sHCp ) K and A P ( Wsalg ) K ,consider the group G ˝ P p A q . Thanks to Proposition 5.2.4 (c) , we have a particular bijection φ ˝ A : G ` p A q ˆ A | d K p A q – ã ÝÝÝÝÝ ։ G ˝ P p A q (5.12)whose restriction to G ` p A q , identified with G ` p A q ˆ p q i P I ( Ď G ` p A q ˆ A | d K p A q , is the identity— onto the copy of G ` p A q naturally sitting inside G ˝ P p A q .Now, G ` p A q is by definition an A –manifold (cf. § P : “ p G ` , g q it pertains to; on the other hand, A | d K p A q carriesnatural, canonical structures of A –manifold of any possible type (real smooth or real analytic if K “ R , complex holomorphic if K “ C ), in particular then also of the type of G ` p A q . Thenwe know that there is also a canonical “product structure” of A –manifold — of the same typeof G ` p A q , i.e. of P — onto the Cartesian product G ` p A q ˆ A | d K p A q . Using the bijection φ ˝ A in(5.12) we push-forward this canonical A –manifold structure of G ` p A q ˆ A | d K p A q onto G ˝ P p A q ,which then is turned into an A –manifold on its own, still of the same type as P .Strictly speaking, the structure of A –manifold defined on G ˝ P p A q formally depends on thechoice of G ă ´ , hence of a totally ordered K –basis of g , as this choice enters in the constructionof φ ˝ A in (5.12) above. However, thanks to the special form of the defining relations of G ˝ P p A q it isstraightforward to show that changing such a basis amounts to changing local charts for the same,unique A –manifold structure ; so in the end the structure is actually independent of such a choice.Now, using the above mentioned structure of A –manifold on G ˝ P p A q for each A P ( Wsalg ) K ,given a morphism f : A ÝÑ A in ( Wsalg ) K it is straightforward to check that the correspond-ing group morphism G ˝ P p f q : G ˝ P p A q ÝÑ G ˝ P p A q is a morphism of A –manifolds, hence it is amorphism of A –manifolds (cf. § G ˝ P can also be seen as a functor from Weil K –superalgebras to A –manifolds (real smooth, real analytic or complex holomorphic as P is) .At last, again looking at the commutation relations in G ˝ P p A q , we see that the group multiplica-tion and the inverse map are “regular” (that is to say, “real smooth”, “real analytic” or “complexholomorphic” depending on the type of P ); indeed, this is explicitly proved by calculations likethose needed in the proof of Proposition 5.2.1 (b) — that we skipped, so refer instead to the proofof Proposition 4.2.8 (b) . Thus they are morphisms of A –manifolds, so G ˝ P p A q is a group elementamong A –manifolds, i.e. it is a Lie A –group ; hence (cf. Proposition 2.4.7), overall the functor G ˝ P is a Lie supergroup , of real smooth, real analytic or complex holomorphic type as P is .Eventually, the outcome of this discussion — and core result of the present section — is thefollowing statement, which provides a “backward functor” from sHCp’s to Lie supergroups: Theorem 5.2.6.
The recipe in Definition 5.1.1 provides functors Ψ ˝ : ( sHCp ) R ÝÑ ( Lsgrp ) R , Ψ ˝ : ( sHCp ) ω R ÝÑ ( Lsgrp ) ω R , Ψ ˝ : ( sHCp ) ω C ÝÑ ( Lsgrp ) ω C given on objects by P ÞÑ Ψ ˝ p P q : “ G ˝ P and on morphisms by ´ p φ ` , ϕ q : P ÝÝÑ P ¯ ÞÑ ´ Ψ ˝ ` p φ ` , ϕ q ˘ : Ψ ˝ ` P ˘ : “ G ˝ P ÝÝÑ G ˝ P “ : Ψ ˝ ` P ˘ ¯ here the functor morphism Ψ ˝ ` p φ ` , ϕ q ˘ : Ψ ˝ ` P ˘ : “ G ˝ P ÝÝÑ G ˝ P “ : Ψ ˝ ` P ˘ is defined by Ψ ˝ ` p φ ` , ϕ q ˘ A : g ÞÑ φ ` ` g ˘ , ` ` η Y ˘ ÞÑ ` ` η ϕ ` Y ˘˘ (5.13) for all A P ( Wsalg ) K , g P G p A q , η P A , Y P g , with P “ ` G , g ˘ and P “ ` G , g ˘ .Proof. What is still left to prove is only that the given definition for Ψ ˝ ` p φ ` , ϕ q ˘ actually makessense, as all the rest is already proved by our previous analysis — in particular, by § ˝ ` p φ ` , ϕ q ˘ A ongenerators of Ψ ˝ ` P ˘ p A q : “ G ˝ P p A q : then a straightforward check shows that all defining relations among such generators — inside G ˝ P p A q — are mapped to corresponding (defining) relations in G ˝ P p A q , thus providing a unique, well-defined group morphism as required. However, we mustshow that this is a morphism of A –manifolds too , which needs some extra work.Let Y i ( i P I and Y j ( j P J be K –bases of g and g respectively, both endowed with somefixed total order. Accordingly, both G ˝ P p A q and G ˝ P p A q admit factorizations as in Proposition5.2.4 (a) — say of type G ` ˆ G ă´ . In particular, any given g P G ˝ P p A q uniquely factors into g “ g ¨ ÝÑ ź i P I ` ` η i Y i ˘ ; then Ψ ˝ ` p φ ` , ϕ q ˘ A , being a group morphism, maps g ontoΨ ˝ ` p φ ` , ϕ q ˘ A ` g ˘ “ φ ` ` g ˘ ¨ ÝÑ ź i P I ` ` η i ϕ ` Y i ˘˘ and from this, letting ϕ ` Y i ˘ “ ř j P J c i,j Z j — with c i,j P K — we getΨ ˝ ` p φ ` , ϕ q ˘ A ` g ˘ “ φ ` ` g ˘ ¨ÝÑ ź i P I ´ ` η i ` ř j P J c i,j Z j ˘¯ “ φ ` ` g ˘ ¨ÝÑ ź i P I ś j P J ´ ` η i c i,j Z j ¯ (5.14)where in the second product in the rightmost term the order of factors is irrelevant, as they docommute with each other. Now we must re-order the result according to the factorization of G ˝ P p A q of the form G ` ˆ G ă´ ; in doing this, when we reorder the second factor ÝÑ ź i P I ś j P J ` ` η i c i,j Z j ˘ in (5.14) above we find — via calculations as in the proof of Proposition 5.2.1 (which means likethose for Proposition 4.2.8 (b) ) — an outcome of the form n ś r “ ` ` a r X r ˘ G ¨ ÝÑ ź j P J ` ` α j Z j ˘ where— (a) the X r ’s belong to g ,— (b) the a r ’s are (even) polynomial expressions in the η i ’s,— (c) the α j ’s are (odd) polynomial expressions in the η i ’s,Overall, this implies that the map ÝÑ ź i P I ` ` η i Y i ˘ ÞÑ n ś r “ ` ` a r X r ˘ G ¨ ÝÑ ź j P J ` ` α j Z j ˘ is a map of A –manifolds from ` G ˝ P ˘ ă´ p A q to G ˝ P p A q . But φ ` : G p A q ÝÝÑ G p A q is a map of A –manifolds too, by assumptions; this along with the previous remark and (5.14) above eventuallyimplies that Ψ ˝ ` p φ ` , ϕ q ˘ A is a map of A –manifolds as claimed. In this subsection we construct a second Lie supergroup functor, denoted G e P , which we laterprove is a Lie supergroup: this is in fact a “sibling alternative” to the functor G ˝ P considered in § P “ p G ` , g q P ( sHCp ) K , A P ( Wsalg ) K and X P A b K g there exists a well-defined exp G ` p X q P G ` p A q ; furthermore, if in particular X P A b K g , then the formal series expansion of exp p X q can be actually realized as a finite sum.36hen no confusion is possible we shall drop the subscript G ` and simply write exp p X q instead.Similarly, we shall presently introduce new formal elements of type “ exp p Y q ” with Y P A b K g .Finally, we extend the built-in G ` –action onto g to a (same-name) G ` –action onto A b K g by Ad p g q ` ř ns “ η s Y s ˘ : “ ř ns “ η s Ad p g qp Y s q for all ř ns “ η s Y s : “ ř ns “ η s b Y s P A b K g . Definition 5.3.1.
Let P : “ ` G ` , g ˘ P ( sHCp ) K be a super Harish-Chandra pair over K . (a) We introduce a functor G e P : ( Wsalg ) K ÝÝÑ ( group ) as follows. For any Weil superalgebra A P ( Wsalg ) K , we define G e P p A q as being the group with generators the elements of the set Γ A : “ g ` , exp p Y q ˇˇ g ` P G ` p A q , Y P A b K g ( “ G ` p A q Ť exp p Y q ( Y P A b K g and relations (for g ` , g , g P G ` p A q , Y , Y , Y P A b K g ) g ¨ g “ g ¨ G ` g , exp p q “ , exp ` Y q ¨ g ` “ g ` ¨ exp ` Ad ` g ´ ` ˘ p Y q ˘ exp ` Y ˘ ¨ exp ` Y ˘ “ exp ´ P p d q ` Y , Y ˘¯ ¨ exp ´ Y ` Y ` P p d q ` Y , Y ˘¯ with P p d q and P p d q as given in Lemma 4.2.5 (h) . This yields the functor G e P on objects.To define G e P on morphisms, for any morphism f : A ÝÑ A in ( Wsalg ) K we define the groupmorphism G e P p f q : G e P ` A ˘ ÝÑ G e P ` A ˘ as being the unique one given on generators — for all g P G ` ` A ˘ , η P A , Y P A b K g — by G e P p f q ` g ˘ : “ G ` p f q ` g ˘ , G e P p f q ` exp ` Y q ˘ : “ exp ` f p Y q ˘ where f ` Y ˘ : “ ř ns “ f ` η s ˘ Y s for all Y : “ ř ns “ η s Y s P A b K g . (b) We define a functor G e, ´ P : ( Wsalg ) K ÝÝÑ ( set ) on any object A P ( Wsalg ) K by G e, ´ P p A q : “ A exp ` A b K g ˘E ` Ď G e P p A q ˘ ` Ď G e P p A q ˘ — the subgroup of G e P p A q generated by exp ` A b K g ˘ : “ exp p Y q ( Y P A b K g — and on mor-phisms in the obvious way. By definition, G e, ´ P can be thought of as subfunctor of G e P . ♦ G e P . Given a super Harish-Chandra pair P “ ` G ` , g ˘ P ( sHCp ) K ,we present now another way of realizing the K –supergroup G e P introduced in Definition 5.3.1 (a) :this mimics what we did in § A P ( Wsalg ) K , we denote by G x y` p A q the subgroup of G ` p A q generated by theset exp p X q ˇˇ X P A r s b K r g , g s ( . Then one easily sees that G x y` p A q is normal in G ` p A q We consider also the three sets Γ ` A : “ G ` p A q , Γ x y A : “ G x y` p A q , Γ ´ A : “ Γ x y A Ť exp ` A b K g ˘ and the five sets of relations — for all g ` , g , g P Γ ` A , g x y , g y , g y P Γ x y A , Y , Y , Y P A b K g — given by R ` A : g ¨ g “ g ¨ G ` g R ´ A : $’’’&’’’% g y ¨ g y “ g y ¨ G x y` g y , exp p Y q ¨ g x y “ g x y ¨ exp ` Ad ` g ´ x y ˘ p Y q ˘ , exp p q “ ` Y ˘ exp ` Y ˘ “ exp ´ P p d q ` Y , Y ˘¯ exp ´ Y ` Y ` P p d q ` Y , Y ˘¯ with P p d q and P p d q as given in Lemma 4.2.5 (h) ˙ A : g x y ¨ g ` “ g ` ¨ ` g ´ ` ¨ G ` g x y ¨ G ` g ` ˘ , exp p Y q ¨ g ` “ g ` ¨ exp ` Ad ` g ´ ` ˘ p Y q ˘ R x y A : ` g x y ˘ Γ x y A “ ` g x y ˘ Γ ` A R A : “ R ` A Ť R ´ A Ť R ˙ A Ť R x y A Then we define a new group, by generators and relations, namely G e, ´ P p A q : “ @ Γ ´ A DM` R ´ A ˘ .Directly from Definition 5.3.1 it follows that G e P p A q – @ Γ ` A Ť Γ ´ A DM` R A ˘ “ @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R ˙ A Ť R x y A ¯ (5.15)but we can also achieve the presentation (5.15) in a series of intermediate steps, just adding oneset of relations at a time. As a first step, we have @ Γ ` A Ť Γ ´ A DM` R ` A Ť R ´ A ˘ “ @ Γ ` A DM` R ` A ˘ ˚ @ Γ ´ A DM` R ´ A ˘ – G ` p A q ˚ G e, ´ P p A q (5.16)where G ` p A q – @ Γ ` A DM` R ` A ˘ by construction and ˚ denotes the free product (of two groups).For the next two steps we can follow two different lines of action. The first one gives @ Γ ` A Ť Γ ´ A DM` R ` A Ť R ´ A Ť R ˙ A ˘ – ´ G ` p A q ˚ G e, ´ P p A q ¯N´ R ˙ A ¯ – G ` p A q ˙ G e, ´ P p A q because of (5.3) and (5.16) together, where G ` p A q ˙ G e, ´ P p A q is the semidirect product of G ` p A q with G e, ´ P p A q . Then @ Γ ` A Ť Γ ´ A DM` R A ˘ – @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R ˙ A Ť R x y A ¯ –– ´ G ` p A q ˙ G ˝ , ´ P p A q ¯N´ R x y A ¯ – ´ G ` p A q ˙ G ˝ , ´ P p A q ¯N N x y p A q where N x y p A q is the normal subgroup of G ` p A q ˙ G e, ´ P p A q generated by !` g x y , g ´ x y ˘) g x y P Γ x y A .This together with (5.15) eventually yields G e P p A q “ ´ G ` p A q ˙ G ´ p A q ¯N N x y p A q On the other hand, (5.3) and (5.16) jointly give @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R x y A ¯ – ´ G ` p A q ˚ G e, ´ P p A q ¯N´ R x y A ¯ – G ` p A q ˚ G x y` p A q G e, ´ P p A q with G ` p A q ˚ G x y` p A q G e, ´ P p A q the amalgamated product of G ` p A q and G e, ´ P p A q over G x y` p A q . Then @ Γ ` A Ť Γ ´ A DM` R A ˘ – @ Γ ` A Ť Γ ´ A DN´ R ` A Ť R ´ A Ť R x y A Ť R ˙ A ¯ –– ´ G ` p A q ˚ G x y` p A q G e, ´ P p A q ¯N´ R ˙ A ¯ – ´ G ` p A q ˚ G x y` p A q G e, ´ P p A q ¯N N ˙ p A q where N ˙ p A q is the normal subgroup of G ` p A q ˚ G x y` p A q G e, ´ P p A q generated by ! g ` exp p Y q g ´ ` exp ` Ad p g ` qp Y q ˘ ´ ) g ` P G ` p A q Y P A b K g Ť ! g ` g x y g ` ` g ` ¨ G ` g x y ¨ G ` g ` ˘ ´ ) g x y P Γ x y A g ` P G ` p A q G e P p A q “ ´ G ` p A q ˚ G x y` p A q G e, ´ P p A q ¯N N ˙ p A q for all A P ( Wsalg ) K . In functorial terms this yields G e P “ ´ G ` ˙ G e, ´ P ¯N N x y and G e P “ ´ G ` ˚ G x y` G e, ´ P ¯N N ˙ , or G e P “ G ` ˙ G x y` G e, ´ P so that G e P is the “amalgamate semidirect product” of G ` and G e, ´ P over G x y` . G e P as a Lie supergroup We aim now to proving that the functor G e P is actually a Lie supergroup. Keeping notations asbefore, we follow closely the same line of arguing as in § G e P : Proposition 5.4.1.
Let P : “ ` G ` , g ˘ P ( sHCp ) K be a super Harish-Chandra pair over K . Then:(a) there exist group-theoretic factorizations G e, ´ P p A q “ G x y` p A q ¨ exp ` A b K g ˘ , G e, ´ P p A q “ exp ` A b K g ˘ ¨ G x y` p A q (b) there exist group-theoretic factorizations G e P p A q “ G ` p A q ¨ exp ` A b K g ˘ , G e P p A q “ exp ` A b K g ˘ ¨ G ` p A q Proof.
Claim (a) is the exact analogue of (4.17), and claim (b) the analogue of (4.19), in Proposition4.2.8 (c) . In both cases the proof (up to details) is the same, so we can skip it. G e P ÝÝÑ GL p V q . Let P “ ` G ` , g ˘ P ( sHCp ) K be any given superHarish-Candra pair over K . Just like we did for G ˝ P in § G e P as well a suitablelinear representation V , which we now define along the same lines, keeping the same notation.Let U p g q be the universal enveloping algebra of g , and let V : “ Ind gg p q “ U p g q b U p g q – Ź g b K – Ź g be the g –representation induced from the trivial representation of g (as in § § p r ` , ρ q : p G ` g q ÝÑ ` GL p V q , gl p V q ˘ of super Harish-Chandrapairs that gives to V a structure p G ` , g q –module; by ρ again we denote also the representation map ρ : U p g q ÝÑ End K p V q giving the U p g q –module structure on V , and similarly — in functorial way— for the representation maps of ` A b K g ˘ and ` A b K U p g q ˘ onto ` A b K V ˘ .We will now show that the p G ` , g q –module structure on V can be naturally “integrated” to a G e P –module structure too. 39 roposition 5.4.3. Retain notation as above for the p G ` , g q –module V . There exists a uniquestructure of (left) G e P –module onto V which satisfies the following conditions: for every A P ( Wsalg ) K , the representation map r e P ,A : G e P p A q ÝÑ GL p V qp A q is given on generators of G e P p A q — namely, all g ` P G ` p A q and exp p Y q for i P I , Y P A b K g — by r e P ,A p g ` q : “ r ` p g ` q , r e P ,A ` exp p Y q ˘ : “ ρ ` exp p Y q ˘ “ exp ` ρ p Y q ˘ that is, g ` .v : “ r ` p g ` qp v q and exp p Y q .v : “ exp ` ρ p Y q ˘ p v q — with exp ` ρ p Y q ˘ being a finite sum— for all v P V p A q . Overall, this yields a morphism of K –supergroup functors r e P : G e P ÝÑ GL p V q .Proof. This follows from the whole construction: indeed, by definition of representation for thesuper Harish-Chandra pair P , the operators r e P ,A p g ` q and r e P ,A ` exp p Y q ˘ on V satisfy all relationswhich, by Definition 5.3.1 (a) , are satisfied by the generators of G e P p A q . So they define a groupmorphism r e P ,A : G e P p A q ÝÑ GL p V qp A q , functorial in A by construction, whence the claim.We are now ready to state the “global splitting theorem” for G e P (cf. Theorem 4.2.10): Proposition 5.4.4. (a) The restriction of group multiplication in G e P provides isomorphisms of (set-valued) functors G ` ˆ exp ` p´q b K g ˘ – G e P , exp ` p´q b K g ˘ ˆ G ` – G e P G x y` ˆ exp ` p´q b K g ˘ – G e, ´ P , exp ` p´q b K g ˘ ˆ G x y` – G e, ´ P (b) For any fixed K –basis Y i ( i P I of g , there exists an isomorphism of (set-valued) functors A | d K – exp ` p´q b K g ˘ , with d : “ dim K ` g ˘ “ | I | , given on A –points, for A P ( Wsalg ) K , by A | d K p A q “ A d ÝÝÝÑ exp ` A b K g ˘ , ` η i ˘ i P I ÞÑ exp ` ř i P I η i Y i ˘ (c) There exist isomorphisms of (set-valued) functors G ` ˆ A | d K – G e P , G x y` ˆ A | d K – G e, ´ P , and A | d K ˆ G ` – G e P , A | d K ˆ G x y` – G e, ´ P given on A –points — for every A P ( Wsalg ) K — respectively by ` g ` , ` η i ˘ i P I ˘ ÞÑ g ` ¨ exp ` ř i P I η i Y i ˘ and `` η i ˘ i P I , g ` ˘ ÞÑ exp ` ř i P I η i Y i ˘ ¨ g ` Proof.
Like for Theorem 5.2.4, the proof is very close to (half of) that of Theorem 4.2.10, witha few, technical differences that involve the representation V of § G e P . For any given P P ( sHCp ) K and A P ( Wsalg ) K ,by Proposition 5.4.4 (c) , we have a particular bijection for the corresponding group G e P p A q , that is φ eA : G ` p A q ˆ A | d K p A q – ã ÝÝÝÝÝ ։ G e P p A q (5.17)whose restriction to G ` p A q , identified with G ` p A q ˆ p q i P I ( Ď G ` p A q ˆ A | d K p A q , is the identity— onto the natural copy of G ` p A q inside G e P p A q .Now, G ` p A q is by definition an A –manifold (cf. § P : “ p G ` , g q ; on the other hand, A | d K p A q carries canonical40tructures of A –manifold of any type (real smooth, etc.), then also of the type of G ` p A q . Sothere exists also a canonical “product structure” of A –manifold — of the same type of G ` p A q ,i.e. of P — onto G ` p A q ˆ A | d K p A q . Then we push-forward this canonical A –manifold structureof G ` p A q ˆ A | d K p A q — through the bijection φ eA in (5.17) — onto G e P p A q , which then is turnedinto an A –manifold on its own, still of the same type as P .Using the above mentioned structure of A –manifold on G e P p A q for each A P ( Wsalg ) K , given amorphism f : A ÝÑ A in ( Wsalg ) K one can easily check that the corresponding group morphism G e P p f q : G e P p A q ÝÑ G e P p A q is a morphism of A –manifolds, hence it is a morphism of A –manifolds(cf. § G e P can also be seen as a functor from Weil K –superalgebras to A –manifolds(real smooth, real analytic or complex holomorphic as P is) .Finally, looking again at the commutation relations in G e P p A q one finds that the group mul-tiplication and the inverse map are “regular” (i.e., “real smooth”, “real analytic” or “complexholomorphic”, according to the type of P ); namely, this is proved via calculations like those usedto prove Proposition 5.4.1 (b) — that we skipped — or to prove Proposition 4.2.8 (c) . Thus theyare morphisms of A –manifolds, so G e P p A q is a group element among A –manifolds, i.e. it is a Lie A –group ; therefore (cf. Proposition 2.4.7), overall the functor G e P is a Lie supergroup — of realsmooth, real analytic or complex holomorphic type as P is .Eventually, the outcome of this discussion — the key result of the present section — is thefollowing statement, which provides a second “backward functor” from sHCp’s to Lie supergroups: Theorem 5.4.6.
The recipe in Definition 5.3.1 provides functors Ψ e : ( sHCp ) R ÝÑ ( Lsgrp ) R , Ψ e : ( sHCp ) ω R ÝÑ ( Lsgrp ) ω R , Ψ e : ( sHCp ) ω C ÝÑ ( Lsgrp ) ω C given on objects by P ÞÑ Ψ e p P q : “ G e P and on morphisms by ´ p φ ` , ϕ q : P ÝÝÑ P ¯ ÞÑ ´ Ψ e ` p φ ` , ϕ q ˘ : Ψ e ` P ˘ : “ G e P ÝÝÑ G e P “ : Ψ ˝ ` P ˘ ¯ where the functor morphism Ψ ˝ ` p φ ` , ϕ q ˘ : Ψ ˝ ` P ˘ : “ G e P ÝÝÑ G e P “ : Ψ e ` P ˘ is defined by Ψ e ` p φ ` , ϕ q ˘ A : g ÞÑ φ ` ` g ˘ , exp ` Y ˘ ÞÑ exp ` ϕ ` Y ˘˘ (5.18) for all A P ( Wsalg ) K , g P G p A q , Y P A b K g , with P “ ` G , g ˘ and P “ ` G , g ˘ .Proof. We are only left to prove that the given definition for Ψ e ` p φ ` , ϕ q ˘ really makes sense:indeed, all the rest is proved by our previous analysis (in particular, by § e ` p φ ` , ϕ q ˘ A on generators ofΨ e ` P ˘ p A q : “ G e P p A q : a direct check shows that all defining relations among such generators — in G e P p A q — are mapped to corresponding (defining) relations in G e P p A q , thus yielding a well-definedgroup morphism as required. In particular, this follows from the special properties of the Lie poly-nomials P p d q and P p d q — for d P d : “ dim ` g ˘ , d : “ dim ` g ˘ ( with possibly d “ d —and of their factors/summands T p s q and T p s q mentioned in Lemma 4.2.5 (h) .However, we must still show that each such Ψ e ` p φ ` , ϕ q ˘ A is a morphism of A –manifolds too .Both groups G e P p A q and G e P p A q admit factorizations of type G ` ˆ exp ` A b K g ˘ , as inProposition 5.4.4 (a) : in particular, any g P G e P p A q uniquely factors into g “ g ¨ exp ` Y ˘ ;then Ψ e ` p φ ` , ϕ q ˘ A , being a group morphism, maps g ontoΨ e ` p φ ` , ϕ q ˘ A ` g ˘ “ Ψ e ` p φ ` , ϕ q ˘ A ` g ¨ exp ` Y ˘˘ “ φ ` ` g ˘ ¨ exp ` ϕ A ` Y ˘˘ (5.19)where ϕ A ` Y ˘ stands for the image of Y for the map ϕ A : A b K g ÝÝÑ A b K g obtained byscalar extension from ϕ ˇˇ g : g ÝÝÑ g . As both φ ` and exp ˝ ϕ A ˝ ` exp ˇˇ A b g ˘ ´ are maps of A –manifolds , from (5.19) we deduce that Ψ e ` p φ ` , ϕ q ˘ A is a map of A –manifolds too, q.e.d.41 The new equivalences ( sHCp ) – ( Lsgrp ) . In Section 5 we introduced two functors, denoted Ψ ˝ and Ψ e , from sHCp’s (of any type: realsmooth, real analytic or complex holomorphic) to Lie supergroups (of the same type); in particular,they go the other way round with respect to the “natural” functor Φ considered in Section 3. Weshall now show that both these two functors are quasi-inverse to Φ, so that (together with Φ ) theyprovide equivalences between the categories of super Harish-Chandra pairs and of Lie supergroups. Ψ as a quasi-inverse to Φ : proof of Φ ˝ Ψ – id ( sHCp ) In this subsection we cope with the first half of our task, namely proving that Φ ˝ Ψ – id ( sHCp ) for Ψ P Ψ ˝ , Ψ e ( ; indeed, this is the easy part of our task. Proposition 6.1.1.
Let Φ and Ψ P Ψ ˝ , Ψ e ( be as in Theorems 3.2.6, 5.2.6 and 5.4.6. Then Φ ˝ Ψ ˝ – id ( sHCp ) , where “ ( sHCp ) ” must be read as either ( sHCp ) R , or ( sHCp ) ω R , or ( sHCp ) ω C ,and Φ and Ψ must be taken as working onto the corresponding types of Lie supergroups or sHCp’s.Proof. This follows almost directly from definitions. We begin with the case of Ψ ˝ . Let us considera super Harish-Chandra pair (of either smooth, or analytic, or holomorphic type) P : “ ` G ` , g ˘ ,and let G ˝ P “ Ψ ˝ p P q , so that ` Φ ˝ Ψ ˝ ˘ p P q “ Φ ` G ˝ P ˘ “ `` G ˝ P ˘ , Lie ` G ˝ P ˘˘ . Then, by the veryconstruction of G ˝ P we have ` G ˝ P ˘ “ G ` . In addition, from the definition of Lie p G q in Definition3.2.3 and exploiting the factorization G ˝ P – G ` ˆ G ă´ from Proposition 5.2.4, one finds that Lie ` G ˝ P ˘ “ Lie ` G ` ˆ G ă ´ ˘ “ Lie ` G ` ˘ ‘ T e ` G ă ´ ˘ “ L g ‘ L g “ L g (cf. § L g with g as usual —simply Lie ` G ˝ P ˘ “ g , this being an identification as Lie K –superalgebras. Therefore, in the end ` Φ ˝ Ψ ˝ ˘ p P q “ Φ ` G ˝ P ˘ “ `` G ˝ P ˘ , Lie ` G ˝ P ˘˘ – ` G ` , g ˘ “ P which means that Φ ˝ Ψ ˝ acts on objects — up to natural isomorphisms — as the identity, q.e.d.As to morphisms, let p φ ` , ϕ q : P “ ` G , g ˘ ÝÝÑ ` G , g ˘ “ P be a morphism of sHCp’sand φ : “ Ψ ˝ ` p φ ` , ϕ q ˘ : Ψ ˝ ` P ˘ “ G P ÝÝÑ G P “ Ψ ˝ ` P ˘ the corresponding (via Ψ ˝ ) morphismof supergroups; we aim to prove that ` Φ ˝ Ψ ˝ ˘` p φ ` , ϕ q ˘ “ Φ p φ q does coincide (up to the naturalisomorphisms ` Φ ˝ Ψ ˝ ˘` P ˘ – P and ` Φ ˝ Ψ ˝ ˘` P ˘ – P mentioned above) with p φ ` , ϕ q itself.By definition Φ p φ q : “ ` φ ˇˇ p G P q , dφ ˘ . Now, on the one hand by the very construction of φ we have φ ˇˇ p G P q “ Ψ ˝ ` p φ ` , ϕ q ˘ˇˇ G “ φ ` . On the other hand, like in the proof of Theorem5.2.6 we consider factorizations G P “ G ˆ G , ă ´ and G P “ G ˆ G , ă ´ ; using these and the veryconstructions we find that the action of dφ onto T e ` G P ˘ “ T e ` G ˘ ‘ T e ` G , ă ´ ˘ “ g ‘ g “ g is given by dφ ˇˇ g “ dφ ` “ ϕ ˇˇ g onto g and by dφ ˇˇ g “ ϕ ˇˇ g onto g ; eventually, this gives dφ “ dφ ˇˇ g ‘ dφ ˇˇ g “ ϕ ‘ ϕ “ ϕ as expected, so that Φ p φ q : “ ` φ ˇˇ p G P q , dφ ˘ “ ` φ ` , ϕ ˘ .Finally, the case of Ψ e is dealt with in an entirely similar way. One simply has to exploitthe parallel results for G e P to those for G ˝ P , like Proposition 5.4.4 (instead of Proposition 5.2.4),Theorem 5.4.6 (instead of Theorem 5.2.6), the factorizations G P “ G ˆ exp ` p´q b K g ˘ and G P “ G ˆ exp ` p´q b K g ˘ , and so on and so forth. Details are left to the reader.42 .2 The functor Ψ as a quasi-inverse to Φ : proof of Ψ ˝ Φ – id ( Lsgrp ) We can now add up the last missing piece, proving that Ψ ˝ Φ is isomorphic to the identityfunctor on Lie supergroups, for all Ψ P Ψ ˝ , Ψ e ( . This last step, together with Proposition 6.1.1,will prove that Φ and Ψ are quasi-inverse to each other, in particular they are equivalences betweenLie supergroups an super Harish-Chandra pairs (and viceversa). The result reads as follows: Proposition 6.2.1.
Let Φ and Ψ P Ψ ˝ , Ψ e ( be as in Theorems 3.2.6, 5.2.6 and 5.4.6. Then Ψ ˝ ˝ Φ – id ( Lsgrp ) , where “ ( Lsgrp ) ” must be read as either ( Lsgrp ) R , or ( Lsgrp ) ω R , or ( Lsgrp ) ω C ,and Φ and Ψ must be taken as working onto the corresponding types of Lie supergroups or sHCp’s.Proof. We begin again by looking at the case of Ψ ˝ .Given a Lie supergroup G , set g : “ Lie p G q and P : “ Φ g p G q “ p G , g q . We look at thesupergroup Ψ ˝ ` Φ p G q ˘ “ Ψ ˝ p P q : “ G ˝ P , and prove that the latter is naturally isomorphic to G .For any A P ( salg ) k , by abuse of notation we denote with the same symbol any element g P G p A q as belonging to G p A q — via the embedding of G p A q into G p A q — and as an elementof G ˝ P p A q — actually, one of the distinguished generators of G ˝ P p A q given from scratch.With this convention, it is immediate to see that Lemma 4.2.5 yields the following: there existsa unique group morphism φ A : G ˝ P p A q ÝÝÝÑ G p A q such that φ A p g q “ g for all g P G p A q and φ A ` p ` η Y q ˘ “ p ` η Y q for all η P A , Y P g . Indeed, thanks to that lemma we knowthat the defining relations among generators of G ˝ P p A q are also satisfied by their images in G p A q through φ A under the above prescription.Due to the factorization (4.16) in Proposition 4.2.8, we have also that the morphism φ A isactually surjective . Even more, the Global Splitting Theorem for G (namely, Theorem 4.2.10) andfor G ˝ P (that is, Proposition 5.2.4) together easily imply that the morphism φ A is also injective ,hence it is a group isomorphism . Finally, it is clear that all these φ A ’s are natural in A , thusaltogether they provide an isomorphism between G ˝ P “ Ψ ˝ ` Φ p G q ˘ and G , which ends the proof.The case of Ψ e is dealt with again similarly, just using the parallel results for G e P to thoseapplied for G ˝ P in the above arguments: namely, Lemma 4.2.5 (the last part), formula (4.19) insteadof (4.16), Proposition 5.4.4 instead of Proposition 5.2.4, etc. Details are left to the reader. We shall now briefly discuss the fallout, in representation theory, of the existence of a an equiv-alence between Lie supergroups and sHCp’s, in particular when realized via the functors Ψ of § The very construction of our functors Ψ ˝ and Ψ e gains a much more concrete meaning whenthe super Harish-Chandra pairs (and Lie supergroups) we deal with are linear .Indeed, assume first that the Lie supergroup G is linear, i.e. it embeds into some GL p V q , where V is a suitable superspace (in other words, there exists a faithful G –module V ). Then both G and g : “ Lie p G q embed into End p V q , and the relations linking them (formalized by saying that“ p G , g q is a super Harish-Chandra pair”) are relations among elements of the unital, associativesuperalgebra End p V q . Conversely, one can formally define a “ linear super Harish-Chandra pair” asbeing any sHCp p G , g q such that both G and g embed into some End p V q , and the compatibilityrelations linking G and g hold true as relations inside End p V q itself — cf. [15], Definition 4.2.1 (b) G is linear then its associated super Harish-Chandra pairΦ p G q “ : P is linear too — both being linearized through their faithful representation onto V .On the other hand, let us start with a linear sHCp, say P “ p G ` , g q : so the latter is embedded(in the obvious sense) into the sHCp ` GL p V q , gl p V q ˘ for some representation superspace V . Thusfor A P ( Wsalg ) K both G ` p A q and A b K g are embedded into ` End p V q ˘ p A q , with relations amongthem — inside the algebra ` End p V q ˘ p A q — induced by the very notion of linear sHCp. Now, onecan consider in ` End p V q ˘ p A q all elements of the form exp p η Y q “ p ` η Y q — with η P A , Y P g — that actually belong to ` GL p V q ˘ p A q : and clearly G ` p A q Ď ` GL p V q ˘ p A q too. Therefore, onecan take inside ` GL p V q ˘ p A q the subgroup G ˝ , V P p A q generated by G ` p A q and by all the p ` η Y q ’s .A trivial check shows that the elements from G ` p A q and the p ` η i Y i q ’s enjoy all relations thatenter in the very definition G ˝ P p A q for their parallel counterparts: thus, there exists a (unique) groupmorphism φ ˝ A : G ˝ P p A q ÝÝÝÑ G ˝ , V P p A q such that φ ˝ A p g ` q “ g ` and φ ˝ A ` p ` η Y q ˘ “ p ` η Y q forall g ` P G ` p A q , η P A , Y P g ; in addition, by construction this φ ˝ A is clearly onto .On the other hand, G ˝ , V P p A q acts faithfully on V , by definition: indeed, it is linear, namely“linearized by V ”. A key fact then is that this linearization allows one to show that the like ofthe Global Splitting Theorem — cf. Theorem 4.2.10 and Proposition 5.4.4 — does hold true for G ˝ , V P p A q ; indeed, one can apply the same analysis and arguments used in the proofs of either Theo-rem 4.2.10 or Proposition 5.4.4, but for one single change: the linearization of G ˝ , V P p A q (induced bythe initial linearization of the sHCp P we started with) has to replace the following key ingredients:— in the proof of Theorem 4.2.10, one has that G p A q : “ š x P| G | Hom ( salg ) K ` O | G | ,x , A ˘ ,— in the proof of Proposition 5.4.4 (and of the lemmas before it, mostly), one has that G ˝ P p A q is acting onto V : “ Ind gg p q — see (5.7).In fact, in both cases — of either G p A q or G ˝ P p A q — the group under exam is realized as a groupof maps (linear operators, in the second case), and these are rich enough to “separate (enough)points” so to guarantee the uniqueness of factorization(s) that is the core part of the Global SplittingTheorem. In the case of G ˝ , V P p A q instead, its built-in linearization provides a similar realizationas “group of maps”, and this again allows to separate enough points to get global splitting(s) for G ˝ , V P p A q too. Finally, thanks to the Global Splitting Theorem for G ˝ P p A q and for G ˝ , V P p A q one canapply again the arguments used in the proof of Proposition 6.2.1 and successfully prove that theabove group (epi)morphism φ ˝ A : G ˝ P p A q ÝÝÝÑ G ˝ , V P p A q is also injective, hence it is an isomorphism .By construction all these isomorphisms φ ˝ A are natural in A , hence they give altogether a functorisomorphism φ ˝ : G ˝ P – ÝÝÑ G ˝ , V P . Therefore G ˝ P – G ˝ , V P , which means that we found a different,concrete realization of G ˝ P , that is now constructed explicitly as the linear Lie supergroup G ˝ , V P .In a parallel way, still starting with a linear sHCp P “ p G ` , g q embedded into ` GL p V q , gl p V q ˘ ,one can consider in ` End p V q ˘ p A q all elements of the form exp p Y q with Y P A b K g , andthen take the subgroup G e, V P p A q of ` GL p V q ˘ p A q generated by G ` p A q and by all the exp p Y q ’s .Acting like above we find (by parallel arguments) that there exists a (unique) group epimorphism φ e A : G e P p A q ÝÝÝÑ G e, V P p A q such that φ e A p g ` q “ g ` , φ e A ` exp p Y q ˘ “ exp p Y q , for all g ` P G ` p A q and Y P A b K g . Even more, still by the same method as above (up to minimal changes) weeventually find that all these φ e A ’s are in fact isomorphisms , natural in A , hence they define afunctor isomorphism φ e : G e P – ÝÝÑ G e, V P . This gives yet another concrete realization of G e P , nowexplicitly realized as the linear Lie supergroup G e, V P .44 .2 Representations 1: supergroup modules vs. sHCp modules An important application of the equivalence between Lie supergroups and super Harish-Chandrapairs occurs in representation theory. Indeed, let G and P be a Lie supergroup and a superHarish-Chandra pair that correspond to each other through one of the equivalences Ψ P Ψ ˝ , Ψ e ( presented in §
6, i.e. G “ Ψ p P q and P “ Φ p G q . Let also G –Mod and P –Mod be the category of G –modules and of P –modules (of the correct type: smooth, etc.); in short, we mean that a G –moduleis the datum of a finite dimensional supermodule M with a morphism φ : G ÝÑ GL p M q of Lie su-pergroups (in the proper category), whereas a P –module is the datum of a finite dimensional super-module M with a morphism p φ ` , ϕ q : P ÝÑ ` GL ` M ˘ , gl ` M ˘˘ of super Harish-Chandra pairs.Just to fix notation, we assume to be in the real smooth case, the other cases being entirelysimilar. Let’s assume M is a G –module; applying Φ : ( Lsgrp ) R ÝÑ ( sHCp ) R to the morphism φ : G ÝÑ GL p M q we find a morphism Φ p φ q : Φ p G q ÝÑ Φ ` GL p M q ˘ between the correspondingobjects in ( sHCp ) R . But Φ p G q “ P by assumption and Φ ` GL p M q ˘ “ ` GL p M q , gl p M q ˘ , sowhat we have is a morphism Φ p φ q : P ÝÑ ` GL p M q , gl p M q ˘ making M into a P –module.Conversely, let M be a P –module. Applying the functor Ψ : ( sHCp ) ÝÑ ( Lsgrp ) R to thecorresponding morphism p φ ` , ϕ q : P ÝÑ ` GL p M q , gl p M q ˘ we get a morphism between thecorresponding supergroups, namely Ψ ` p φ ` , ϕ q ˘ : Ψ p P q ÝÑ Ψ `` GL p M q , gl p M q ˘˘ . As Ψ p P q “ G by and Ψ `` GL p M q , gl p M q ˘˘ “ GL p M q , we find a morphism Ψ ` p φ ` , ϕ q ˘ : G ÝÑ GL p M q in ( Lsgrp ) R which makes M into a G –module. In fact, in this way the G –action that one gets on M is just what one obtains by direct application of the recipe in § linear super Harish-Chandra pair p φ ` , ϕ qp P q inside ` GL p M q , gl p M q ˘ that is the image of P through p φ ` , ϕ q .The reader can easily check that the previous discussion — extended to the real analytic caseand to the complex holomorphic case as well — has the following outcome: Theorem 7.2.1.
Let G and P be a Lie supergroup and a super Harish-Chandra pair (of smooth,analytic or holomorphic type) over K corresponding to each other as above. Then:(a) for any fixed finite dimensional K –supermodule M , the above constructions provide twobijections, inverse to each other, between G –module structures and P –module structures on M ;(b) the whole construction above is natural in M , in that the above bijections between G –modulestructures and P –module structures over two finite dimensional K –supermodules x M and Ă M arecompatible with K –supermodule morphisms from x M to Ă M . Thus, all the bijections mentioned in(a), for all different M ’s, do provide equivalences, quasi-inverse to each other, between the categoryof all (finite dimensional) G –modules and the category of all (finite dimensional) P –modules. to G Let G be a supergroup (of any type), with associated classical subsupergroup G . Let V beany G –module: we shall now present an explicit construction of the induced G –module Ind GG p V q .Being a G –module, V is also, automatically, a g –module. Then one does have the induced g –module Ind gg p V q , which can be realized as Ind gg p V q “ Ind U p g q U p g q p V q “ U p g q b U p g q V By construction, it is clear that this bears also a unique structure of G –module which is compatiblewith the g –action and coincides with the original G –action on K b U p g q V – V given from scratch.Indeed, we can describe explicitly this G –action, as follows. First, by construction we have Ind gg p V q “ U p g q b U p g q V “ Ź g b K V
45 because U p g q – Ź g b K U p g q as a K -module, by the PBW theorem for Lie superalgebras, see(5.6) — with the g –action given by x. p y b v q “ ad p x qp y q b v ` y b p x.v q for x P g , y P Ź g , v P V , where by ad we denote the unique g –action on Ź g by algebra derivations induced bythe adjoint g –action on g . Second, this action clearly integrates to a (unique) G –action givenby g . p y b v q : “ Ad p g qp y q b p g .v q for g P G , y P Ź g , v P V , where we write Ad for theunique G –action on Ź g by algebra automorphisms induced by the adjoint G –action on g .The key point is that the above G –action and the built-in g –action on Ind gg p V q are compatible,in that they make Ind gg p V q into a p G , g q –module, i.e. a module for the sHCp P : “ p G , g q “ Φ p G q .Then for Ψ P Ψ ˝ , Ψ e ( , since Ψ ` p G , g q ˘ – G , by § Ind gg p V q bears a uniquestructure of G –module which correspond to the previous P –action — i.e., it yields (by restrictionand “differentiation”) the previously found G –action and g –action. In down-to-earth terms whathappens is the following. The action of P : “ p G , g q “ Φ p G q onto W : “ Ind gg p V q is given bythe G –action (induced by the original action on V ) and a compatible g –action. Then for any A P ( Wsalg ) K we have also that all of A b K g “acts” onto A b K W : thus well-defined operators p ` η Y q and exp p Y q — with p η , Y q P A ˆ g , Y P A b K g — exist in ` GL p W q ˘ p A q .One easily checks that these p ` η Y q ’s, or the exp p Y q ’s, altogether enjoy among themselves andwith the operators given by the G –action the very relations that enter in the definition of eitherΨ ˝ p P q : “ G ˝ P or Ψ e p P q “ G e P , respectively — therefore in both cases we get a well-defined actionof Ψ p P q on W , extending the initial one by G : but Ψ p P q “ G , so we are done.Thus we define as Ind GG p V q the space W : “ Ind gg p V q endowed with this G –action : one checksthat this construction is functorial in V and has the universal property making it adjoint of “re-striction” (from G –modules to G –modules), so it can be correctly called “induction” functor.In addition, if the original G –module V is faithful then the induced G –module Ind GG p V q isfaithful too: in particular, this means that if G is linearizable, then G is linearizable too; moreprecisely, from a linearization of G one can construct (via induction) a linearization of G as well. References [1] A. Alldridge, M. Laubinger,
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