Representations of involutory subalgebras of affine Kac-Moody algebras
Axel Kleinschmidt, Ralf Köhl, Robin Lautenbacher, Hermann Nicolai
aa r X i v : . [ m a t h . R T ] F e b Representations of involutory subalgebrasof affine Kac–Moody algebras
Axel Kleinschmidt , , Ralf K¨ohl , Robin Lautenbacher and Hermann Nicolai Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut)Am M¨uhlenberg 1, DE-14476 Potsdam, Germany International Solvay InstitutesULB-Campus Plaine CP231, BE-1050 Brussels, Belgium Justus-Liebig-Universit¨at, Mathematisches Institut, Arndtstraße 2, 35392 Gießen, Germany
We consider the subalgebras of split real, non-twisted affine Kac–Moody Lie algebrasthat are fixed by the Chevalley involution. These infinite-dimensional Lie algebrasare not of Kac–Moody type and admit finite-dimensional unfaithful representations.We exhibit a formulation of these algebras in terms of N -graded Lie algebras thatallows the construction of a large class of representations using the techniques ofinduced representations. We study how these representations relate to previouslyestablished spinor representations as they arise in the theory of supergravity. ontents k ( e ) k ( E ) ( C ) as the quotient of a GIM-algebra 28B The Hilbert space completion b k is not a Lie algebra 32C so (16) representations 33 Every (split real) Kac–Moody algebra g = g ( A ) for an indecomposable generalised Cartan matrix A has a Chevalley involution ω that defines an involutory (‘maximal compact’) Lie subalgebra k = k ( g ) < g as its subalgebra of fixed points [K]. If g is of finite type, then k is compact, whencereductive; its structure and representation theory play a key role in studies of symmetric spacesand automorphic representations. For Kac–Moody algebras g , generators and relations for k were given in the early work [B], but, curiously, very little is known about the representationtheory of k in the infinite-dimensional case. One reason for this is that these algebras do not fitinto more standard frameworks, as they do not exhibit a graded, but rather a filtered structure;in particular, they are not Kac–Moody algebras [KN1], unlike the algebras from which theydescend. This means also that the customary tools of representation theory (lowest and highestweight representations, character formulas, etc. ) are not applicable. This being said, we pointout (see appendix A) that the complexification of k is a quotient of a GIM algebra as definedby Slodowy [S]; therefore, the representation theory of the latter, once developed, should helpwith understanding the (real and complex) representations of k .A remarkable property of k is that it inherits an invariant bilinear form from g . For splitreal g the bilinear form on g is indefinite but its restriction to k is (negative) definite and forthis reason k is sometimes referred to as maximal compact. One may thus consider the Hilbertspace completion b k of k with respect to the norm defined by the invariant bilinear form. One ofthe results of our paper is that, for untwisted affine g , the Lie bracket does not close on b k , sothat b k is not a Hilbert Lie algebra with respect to the standard invariant bilinear form; in fact,it is not even a Lie algebra (see appendix B).The Lie algebra k and their associated groups are potentially important in physics appli-cations, namely as infinite-dimensional generalisations of the R symmetries which govern the1ermionic sectors of certain supersymmetric unified models of the fundamental interactions.Perhaps the most surprising result to come out of these physics motivated studies is thatthe infinite-dimensional algebras k , unlike the Kac–Moody algebras they are embedded into,admit non-trivial finite -dimensional, hence unfaithful, representations, in particular fermionic(double-valued) ones, which cannot be obtained by decomposing representations of the an-cestor Kac–Moody algebra g under its subalgebra k . These representations were originallyfound by analysing the action of the infinite-dimensional duality groups arising in the dimen-sional reduction of (super-)gravities to space-time dimensions D ≤ beyond supergravity were identified in [KN3, KN5], and their structure was further anal-ysed in [KNV], where surprising features were discovered, such as the generic non-compactnessof the quotient algebras (and quotient groups) obtained by dividing the original algebra bythe annihilating ideal of the given representation. These representations were further studiedin [HKL, LK] where in particular the action of the corresponding (covering) group representa-tion of ˜ K ( G ) [GHKW] was clarified; moreover, in complete analogy to the finite-dimensionalsituation it turns out that in the simple-laced case this two-fold covering group ˜ K ( G ) is simplyconnected with respect to its natural topology [HK] (induced from the Kac–Peterson topologyon the corresponding Kac–Moody group). All these studies of Lie algebra representations andcorresponding covering group representations so far, while applicable to large classes of gener-alised Cartan matrices (simply-laced; often even any symmetrisable generalised Cartan matrix)have been limited to a small number of explicitly known examples of concrete representations, asall efforts to find a larger class of examples of representations and to understand their underlyingstructure and representation theory in a more systematic way have failed until now.In this paper, we study the simplest case, corresponding to untwisted affine Kac–Moodyalgebras over K = R or K = C . We show that there do exist infinitely many such unfaithful rep-resentations of ever increasing dimensions. Our construction finally provides a systematic raisond’ˆetre for such representations. Here, we aim for an ab initio construction of k representations(in particular of spinorial nature) and not for representations that are obtained by branchingrepresentations of the affine g . The main method, which is inspired by the supergravity reali-sation on fermions [NS], is to map the filtered structure to a graded one, by replacing Laurentpolynomials K [ t, t − ] in a variable t by power series K [[ u ]] in another variable u that is relatedto t by (3.3). We shall refer to the graded structure as the parabolic model of k . In the parabolicmodel based on u , representations can be constructed easily by means of the Poincar´e–Birkhoff–2itt (PBW) basis of the enveloping algebra of the underlying Lie algebra graded by powers of u . We will present several explicit examples (related to maximal supergravity) to illustrate theconstruction, which puts in evidence the rapid growth of the associated representations. Ourresults can be viewed as a prelude to the construction of similar representations for involutorysubalgebras of hyperbolic Kac–Moody algebras, that will be required for a better understandingof the fermionic sector of unified models, and perhaps pave the way for an embedding of theStandard Model fermions into a unified framework [MN1, KN4, MN2].In a more global view, the results of this paper afford not only a completely new perspectiveon the representation theory of such algebras, but even more importantly, open new avenuestowards exploring the structure of the associated groups . This would be especially relevant forinfinite-dimensional Kac–Moody algebras of indefinite type for which, however, a representationtheory extending the present results remains to be developed. While the loop group approachhas proven very useful for affine Kac–Moody groups [PS] there is no comparable tool availablefor studying indefinite Kac–Moody groups, where often one has to resort to ‘local-to-global’methods. Namely, many mathematical observations concerning (two-spherical) Kac–Moodygroups rely upon ‘gluing’ the SL(2, R ) subgroups associated to the simple real roots of theunderlying Kac–Moody algebra g and on exploiting Tits’ theory of buildings and extensionsthereof [T,KP,AM,CFF,M]. However, in this way one does not gain a truly ‘global’ perspectiveon the associated groups. The same statement applies to an analogous construction of thegroups K ( G ) associated to involutory subalgebras k ⊂ g for indefinite g , which would proceedby similarly ‘gluing’ compact SO(2) ⊂ SL(2, R ) subgroups associated to the simple real rootsof g [GHKW] (which, contrary to the Kac–Moody group G , in fact also works in the non-two-spherical case). By contrast, we here propose a fundamentally different approach which would bebased on the construction of infinite sequences of larger and larger, but still finite-dimensional,quotient groups that, while remaining infinitely degenerate at each step of the sequence, capture‘more and more’ of the infinite-dimensional group (and in particular the information containedin the root spaces associated to imaginary roots), and in such a way that the group action isfully under control at each step of the construction — in more mathematical terms, we proposeto investigate whether the involutory Lie subalgebra k and its corresponding group K ( G ) mightbe residually finite-dimensional. That this is true for k in the affine case is part of proposition 20.The paper is organised as follows: In section 2 we fix our notation for the involutory sub-algebra k of an untwisted affine Kac–Moody algebra. We construct the parabolic analogues N ( K [[ u ]]) and N ( P N ) of k over different rings in section 3 and show that there exist homomor-phisms from k to them. Furthermore we establish that the homomorphism from k to N ( K [[ u ]])is injective. In section 4 we take a slight detour in first constructing induced representations of N ( K [ u ]) from representations of its classical subalgebra ˚ k . Although there is no homomorphismfrom k to N ( K [ u ]) we will show that these induced representations can be truncated to providefinite-dimensional representations of N ( P N ) and k . We will conclude section 4 by proving thatthe inverse limit of these representations provides a fatihful infinite-dimensional representationof k . We specialise our results to the case k ( E ) in section 5 where we also describe the problemof analysing the finite-dimensional representations’ structure in more detail.3 cknowledgments: The work of H.N. has received funding from the European ResearchCouncil (ERC) under the European Union’s Horison 2020 research and innovation programme(grant agreement No 740209). The work of R.K. and R.L. has received funding from the DeutscheForschungsgemeinschaft (DFG) via the grant KO 4323/13-2. The work of R.L. has receivedfunding from the Studienstiftung des deutschen Volkes. Furthermore, R.K. and R.L. gratefullyacknowledge the hospitality of the Max-Planck-Gesellschaft during several extended visits at theMax-Planck-Institute for Gravitational Physics at Golm, Potsdam.
Let A be an indecomposable generalised Cartan matrix of untwisted affine type and let K denote R or C , see [K, §
4] for a complete list of the associated diagrams D ( A ). The Kac–Moody algebra g := g ( A ) ( K ) can be constructed explicitly as an extension of the loop algebra L (˚ g ), where ˚ g denotes the classical Lie-subalgebra ˚ g < g ( A ) ( K ) that one obtains by deleting the affine node inthe generalised Dynkin diagram D ( A ). Denote by ω the Chevalley involution on g ( A ) ( K ) andset k := k ( A ) ( K ) = Fix ω ( g ( A ) ( K )) . (2.1) k ( A ) ( K ) is called the maximal compact subalgebra of g ( A ) ( K ) since the restriction of the stan-dard invariant bilinear form on g to k is (negative) definite. Let us start with a description of k that is adapted to the presentation of g in terms of the loop algebra L (˚ g ). In section 5, we willcomplement this by a collection of correspondences to the basis described in [KNP, §
4] for theparticular case g = e , the affine extension of e .Denote by ˚∆ the root system of ˚ g and its Chevalley involution by ˚ ω . The ± ω provide the Cartan decomposition ˚ g = ˚ k ⊕ ˚ p . In terms of a Cartan basis (cid:8) E γ | γ ∈ ˚∆ (cid:9) ∪ (cid:8) h , . . . , h d ∈ ˚ h (cid:9) , this decomposition is realised as˚ k = span n E γ − E − γ | γ ∈ ˚∆ o , ˚ p = span (cid:16)n E γ + E − γ | γ ∈ ˚∆ o ∪ ˚ h (cid:17) , (2.2)where ˚ h denotes the Cartan subalgebra of ˚ g . Recall that ˚ p is a ˚ k -module but not a Lie-algebrasince [˚ p , ˚ p ] = ˚ k . Denote by L the ring of Laurent polynomials over K . Then the loop algebra of L (˚ g ) is given by the tensor product L ⊗ ˚ g with the commutator given by the bilinear extensionof [ P ⊗ x, Q ⊗ y ] = ( P Q ) ⊗ [ x, y ] , (2.3)where [ · , · ] on the right-hand side denotes the bracket on ˚ g . As t n , t − n (for n ∈ N = { , , . . . } )span L , the Lie algebra L (˚ g ) is spanned by n t ± n ⊗ E γ | γ ∈ ˚∆ o ∪ n t ± n ⊗ h | h ∈ ˚ h o for n ∈ N .Now (see [K, §
7] or [GO]) g = L (˚ g ) ⊕ K · K ⊕ K · d (2.4) Note that the abstract Chevalley involution of ˚ g = g ( ˚ A ) as a Kac–Moody algebra of finite type agrees withthe restriction of ω : g → g to ˚ g < g because the Dynkin diagram of ˚ g is a sub-diagram of D ( A ) due to ourassumption that A is untwisted. K, d ∈ h so that k ⊂ L (˚ g ) because ω ( h ) = − h ∀ h ∈ h . Thus, in order to describe k = k ( A ) ( K ) we only need to study the loop algebra L (˚ g ) and in particular do not need toconsider the central extension of L (˚ g ) by a 2-cocycle, giving rise to K . The Chevalley involutionrestricted to L (˚ g ) is given by the linear extension of ω : P ( t ) ⊗ x P (cid:0) t − (cid:1) ⊗ ˚ ω ( x ) ∀ P ∈ L . (2.5)Denote by L ± the ± L under the involution η : t t − . Observe that evaluationof Laurent polynomials at t = ± η which iswhy we refer to t = ± η . The fixed point set of ω can be constructedfrom L ± , ˚ k and ˚ p : ω ( P ⊗ x ) = P ⊗ x ∀ P ∈ L + , x ∈ ˚ k , ω ( Q ⊗ y ) = Q ⊗ y ∀ Q ∈ L − , y ∈ ˚ p From this we arrive at the following explicit realisation of k : ⇒ k ( A ) ( K ) = (cid:16) L + ⊗ ˚ k (cid:17) ⊕ (cid:0) L − ⊗ ˚ p (cid:1) . (2.6) Remark . (i) The Laurent polynomials in L ± are spanned by t n ± t − n for n ∈ N . The productof two such basis Laurent polynomials is for example( t m + t − m )( t n − t − n ) = (cid:16) t m + n − t − ( m + n ) (cid:17) − sgn( m − n ) (cid:16) t | m − n | − t −| m − n | (cid:17) . (2.7)We shall refer to this as a filtered structure on L that by (2.6) induces a filtered structureon k .(ii) The construction in (2.6) can be generalised to the case where L denotes a finitely generatedcommutative K -algebra with involution and L ± denote the ± g is a generalised current algebra. However, our interest lies in studying differentmodels of k . In particular, we will explore the relation of (2.6) to the cases L = K [[ u ]], K [ u ] and K [ u ] (cid:30) I N , where I N is the ideal generated by the monomial u N +1 .(iii) It is well known that one can form the semi-direct sum of any affine Lie algebra with theVirasoro algebra of centrally extended infinitesimal diffeomorphisms of the circle [K, GO],where both central extensions are identified. This structure descends to k and a ‘maximalcompact’ subalgebra of the Virasoro algebra [JN]. As this will play no role in our generalanalysis, we defer its discussion to section 5.(iv) The Lie algebra k comes with a definite and invariant bilinear form that is inherited from g . In appendix B, we show that the Hilbert space completion with respect to this norm isnot compatible with the Lie algebra structure to form a Hilbert Lie algebra.5 Reformulation in terms of parabolic algebras
In this section, we construct a Lie algebra monomorphism from k to a larger Lie algebra byreplacing Laurent polynomials with formal power series. This corresponds to an expansionaround the fixed points t = ± η instead of 0 as described in [NS, KNP].Denote by P := K [[ u ]] the ring of formal power series together with the involutive ringautomorphism η : u k ( − k u k . We refer to a formal power series in P either as P ∞ k =0 a k u k orjust as ( a k ) k ≥ with a k ∈ K . Denote the ± P with respect to η by P ± , i.e., theformally even/odd power series in u . One has P + := (cid:8) ( a k ) k ∈ N | a k ∈ K , a k +1 = 0 ∀ k ∈ N (cid:9) , P − := (cid:8) ( a k ) k ∈ N | a k +1 ∈ K , a k = 0 ∀ k ∈ N (cid:9) . We mimic the loop algebra construction by setting N ( K [[ u ]]) := (cid:16) P + ⊗ ˚ k (cid:17) ⊕ (cid:16) P − ⊗ ˚ p (cid:17) , [ P ⊗ x, Q ⊗ y ] = ( P · Q ) ⊗ [ x, y ] , (3.1)where the bracket on the right-hand side again denotes the ˚ g -bracket. Recall that multiplicationin the ring of formal power series is given by convolution, i.e., for P = P ∞ n =0 a n u n and Q = P ∞ n =0 b n u n one has P · Q = ∞ X n =0 c n u n , c n = n X k =0 a k b n − k . (3.2) Remark . One could also consider replacing P by the ring of polynomials K [ u ]. In contrast to L ± which have a filtered structure, K [ u ] has a gradation given by the degree of polynomials. Thisextends to a gradation on (cid:0) K [ u ] + ⊗ ˚ k (cid:1) ⊕ ( K [ u ] − ⊗ ˚ p ). If we were to construct a monomorphismfrom k using only K [ u ] one would expect to be able to pull this gradation back to k ( A ) ( K ).Since we doubt this to be possible we work over K [[ u ]] which arises as the formal completion of K [ u ]. There we do not have a gradation by degree of polynomials any more in the proper sensebecause for this any element in K [[ u ]] would have to decompose into the sum of finitely manyhomogeneous elements. This is true for polynomials but not for formal power series. Thus, wesacrifice a graded structure in order to achieve injectivity. However, in section 4 we will considerthis case as a preliminary step.As the power series K [[ u ]] are associated with the expansion around the fixed point t = ± t = 0, we construct a homomorphism by relating the expansion via a (M¨obius-type)transformation u = 1 ∓ t ± t ⇔ t = ± − u u (3.3)through the Taylor series ( n ≥ t n + t − n = ( ± n X k ≥ a ( n )2 k u k , t n − t − n = ( ± n X k ≥ a ( n )2 k +1 u k +1 . (3.4)The filtered multiplication of the Laurent polynomials in L ± is then captured by the followinglemma. 6 emma 3. For each n ∈ N the sequences (cid:16) a ( n )2 k (cid:17) k ∈ N and (cid:16) a ( n )2 k +1 (cid:17) k ∈ N given by given by a ( n )2 k = 2 n X ℓ =0 n ℓ ! k − ℓ + n − k − ℓ ! , a ( n )2 k +1 = − n − X ℓ =0 n ℓ + 1 ! k − ℓ + n − k − ℓ ! (3.5) satisfy k X ℓ =0 a ( m )2 ℓ a ( n )2( k − ℓ ) = a ( m + n )2 k + a ( | m − n | )2 k (3.6a) k X ℓ =0 a ( m )2 ℓ a ( n )2( k − ℓ )+1 = a ( m + n )2 k +1 + sgn( n − m ) a ( | m − n | )2 k +1 (3.6b) k − X ℓ =0 a ( m )2 ℓ +1 a ( n )2( k − ℓ ) − = a ( m + n )2 k − a ( | m − n | )2 k . (3.6c) The coefficients ( − n a ( n )2 k and ( − n a ( n )2 k +1 also satisfy eqs. (3.6a), (3.6b) and (3.6c). Further-more, for fixed n ∈ N ∗ = { , , . . . } , the values of a ( n )2 k and a ( n )2 k +1 are given by polynomials in k of degree n − for all k ∈ N ∗ .Remark . The transformations (3.3) have the property that they interchange the points 0 and ∞ (that are exchanged by the involution t ↔ t − ) with the fixed points ± η . The maps t u ( t ) in (3.3) are fixed uniquely by the requirement that (0 , ∞ , +1 , − (+1 , − , , ∞ )and (0 , ∞ , +1 , − (+1 , − , ∞ , Proof.
We first compute the Taylor series for the M¨obius transformation (3.3) for n ∈ N t n + t − n = ( ± n − u ) n n X ℓ =0 (cid:18) n ℓ (cid:19) u ℓ = 2( ± n X k ≥ n X ℓ =0 (cid:18) n ℓ (cid:19)(cid:18) n + k − k (cid:19) u k + ℓ ) , from which the first formula in (3.5) follows. The second identity in (3.5) is deduced similarlyfrom the expansion of t n − t − n . The order of the polynomial follows from the binomial coefficient (cid:0) n + k − ℓ − k − ℓ (cid:1) .The convolution properties (3.6) follow from multiplying out (cid:0) t m + t − m (cid:1) (cid:0) t n + t − n (cid:1) = t m + n + t − ( m + n ) + t | m − n | + t −| m − n | for (3.6a) for the first one upon using (3.2). The identities (3.6b) and (3.6c) follow in the sameway by changing the appropriate signs.We collect some formulae concerning the coefficients a ( n )2 k and a ( n )2 k +1 . One has a ( n )0 = 2 forall n ∈ N . For k > a ( n )2 k are given by the following polynomials in k (note that these expressions are not valid for k = 0): a (0)2 k = 0 , a (1)2 k = 4 , a (2)2 k = 16 k, a (3)2 k = 32 k + 4 , a (4)2 k = 1283 k + 643 k, (3.7)7hile the first few coefficients a ( n )2 k +1 are given by polynomials in k as well: a (1)2 k +1 = − , a (2)2 k +1 = − k − , a (3)2 k +1 = − k − k − ,a (4)2 k +1 = − k − k − k − . (3.8)With lemma 3 above one can now construct a Lie algebra homomorphism from k to N ( K [[ u ]]),defined in (3.1), that is constructed using (3.4). Proposition 5.
The linear map ρ ± : k ( A ) ( K ) → N ( K [[ u ]]) defined by (cid:0) t n + t − n (cid:1) ⊗ x ( ± n ∞ X k =0 a ( n )2 k u k ⊗ x ∀ x ∈ ˚ k (3.9a) (cid:0) t n − t − n (cid:1) ⊗ y ( ± n ∞ X k =0 a ( n )2 k +1 u k +1 ⊗ y ∀ y ∈ ˚ p (3.9b) with a ( n )2 k , a ( n )2 k +1 as in (3.5) extends to a homomorphism of Lie algebras.Proof. This can be checked case by case for pairs ( a, b ) ∈ (cid:0) L + ⊗ ˚ k (cid:1) × (cid:0) L + ⊗ ˚ k (cid:1) , (cid:0) L + ⊗ ˚ k (cid:1) × ( L − ⊗ ˚ p ),( L − ⊗ ˚ p ) × ( L − ⊗ ˚ p ) and extending the result by (bi-)linearity. We demonstrate this for the firstpair, the other pairs work analogously.First, set x ( n ) := ( t n + t − n ) ⊗ x for all x ∈ ˚ k . Then by definition h x ( m )1 , x ( n )2 i = (cid:16) t m + n + t − ( m + n ) (cid:17) ⊗ [ x , x ] + (cid:16) t | m − n | + t −| m − n | (cid:17) ⊗ [ x , x ] , so that from (3.9a) ρ ± (cid:16)h x ( m )1 , x ( n )2 i(cid:17) = ( ± m + n ∞ X k =0 (cid:16) a ( m + n )2 k + a ( | m − n | )2 k (cid:17) u k ⊗ [ x , x ] . On the other hand h ρ (cid:16) x ( m )1 (cid:17) , ρ (cid:16) x ( n )2 (cid:17)i = ( ± m + n ∞ X k =0 k X ℓ =0 a ( m )2( k − ℓ ) a ( n )2 ℓ u k ⊗ [ x , x ]= ( ± m + n ∞ X k =0 (cid:16) a ( m + n )2 k + a ( | m − n | )2 k (cid:17) u k ⊗ [ x , x ] , by (3.6a), proving the claim. For the other pairs one proceeds similarly, using in particular a ( n )0 =2 for all n , and this shows that the map ρ ± extends to a homomorphism of Lie algebras.It is now possible to introduce a cutoff in (3.9) in order to make all expressions finite sums.Denote by I N := (cid:0) u N +1 (cid:1) (3.10)8he ideal in K [[ u ]] that is generated by the element u N +1 and set P N := K [[ u ]] (cid:30) I N . Furthermore,define the even and odd parts as P + N := span K (cid:26) u k + I N | k = 0 , . . . , (cid:22) N (cid:23)(cid:27) , P − N := span K (cid:26) u k +1 + I N | k = (cid:22) N − (cid:23)(cid:27) and consider the Lie algebra N ( P N ) := (cid:16) P + N ⊗ ˚ k (cid:17) ⊕ (cid:0) P − N ⊗ ˚ p (cid:1) , [ P ⊗ x, Q ⊗ y ] = ( P · Q ) ⊗ [ x, y ] (3.11)where the bracket on the right-hand side still denotes the ˚ g -bracket. Remark . Note that the ring of formal power series P is isomorphic to the inverse limit of theinverse system ( P N ) N ∈ N with the obvious bonding maps [R]. As a consequence N ( K [[ u ]]) is theinverse limit of the N ( P N ): N ( K [[ u ]]) = lim ←− N →∞ N ( P N ) . (3.12)In the following we will construct homomorphisms ρ ( N ) ± : k ( A ) ( K ) → N ( P N ) which are thenshown to be surjective. As ρ ( N ) ± is constructed via ρ ± and the natural projection Pr N from N ( K [[ u ]]) to N ( P N ) the ρ ( N ) ± are compatible with the natural projections from N ( P N + M ) to N ( P N ). Assuming we had started with the ρ ( N ) ± then by the universal property of inverse limitswe would have obtained ρ ± as the unique homomorphism ρ ± : k ( A ) ( K ) → N ( K [[ u ]]) such that ρ ( N ) ± = ρ ± ◦ Pr N . Corollary 7.
The homomorphisms of Lie algebras ρ ± : k ( A ) ( K ) → N ( K [[ u ]]) induce homo-morphisms ρ ( N ) ± : k ( A ) ( K ) → N ( P N ) which are given explicitly by (cid:0) t n + t − n (cid:1) ⊗ x ( ± n ⌊ N/ ⌋ X k =0 a ( n )2 k (cid:16) u k + I N (cid:17) ⊗ x ∀ x ∈ ˚ k (3.13) (cid:0) t n − t − n (cid:1) ⊗ y ( ± n ⌊ ( N − / ⌋ X k =0 a ( n )2 k +1 (cid:16) u k +1 + I N (cid:17) ⊗ y ∀ y ∈ ˚ p . (3.14) Proof. N ( P N ) is a quotient of N ( K [[ u ]]) because P N = K [[ u ]] (cid:30) I N is a quotient of P .Next, we want to show that the homomorphisms ρ ± are injective but not surjective. Towardsthis we will need the following fact about matrices whose entries are obtained from evaluationof polynomials: Lemma 8.
Let = p , . . . , p n ∈ K [ t ] be linearly independent polynomials. Then there exist N , . . . , N n ∈ N such that the matrix M ( N , . . . , N n ) := ( p i ( N j )) ni,j =1 is regular.Proof. This can be achieved by induction on n and an expansion of the determinant which yieldsa linear combination of linearly independent polynomials. As a nonzero polynomial p ( t ) is equalto 0 for only finitely many t ∈ K this can be used to show regularity of M ( N , . . . , N n ) forsuitable N , . . . , N n . 9 roposition 9. The homomorphism of Lie algebras ρ ± : k ( A ) ( K ) → N ( K [[ u ]]) from proposi-tion 5 is injective. Furthermore its image does not contain elements in N ( K [[ u ]]) = ( P + ⊗ k ) ⊕ ( P − ⊗ p ) whose formal power series contain only finitely many nonzero coefficients. In particu-lar, the elements u k +2 ⊗ x for x ∈ ˚ k and u k +1 ⊗ y for y ∈ ˚ p and k ≥ are not contained in theimage of k ( A ) ( K ) in N ( K [[ u ]]) .Remark . Observe that there is a certain asymmetry. It is possible to map elements from k ( A ) ( K ) to N ( K [[ u ]]) by allowing formal power series but in the reverse direction it is notpossible to define such a map for elements such as u k +2 ⊗ x because the Laurent polynomialsdo not have a completion that behaves well under the involution η . One can only complete inone direction, i.e., for an element P n ∈ Z c n u n ⊗ x in some completion of k ( A ) ( K ), c n = 0 is onlypossible for either n > N or n < N but not both at the same time. But this does not agreewith the demand c − n = ± c n unless c n = 0 for only finitely many n . Proof.
For a generic element in k ( A ) ( K ) one can split the analysis into two pieces becauseelements from L + ⊗ ˚ k are mapped to elements which only involve even powers u k while theones from L − ⊗ ˚ p are mapped to series involving only odd powers u k +1 . Therefore consider theimage of χ := K X i =1 (cid:0) t n i + t − n i (cid:1) ⊗ x i under ρ : ρ ( χ ) = K X i =1 X k ≥ a ( n i )2 k u k ⊗ x i = X k ≥ K X i =1 a ( n i )2 k u k ⊗ x i = 0 , so that we need P Ki =1 a ( n i )2 k x i = 0 for all k ≥
0. For a basis e , . . . , e d of ˚ k and x i = P dj =1 c ji e j this yields K X i =1 d X j =1 a ( n i )2 k c ji e j = 0 ∀ k ≥ ⇔ K X i =1 a ( n i )2 k c ji = 0 ∀ k ≥ , ∀ j = 1 , . . . , d. This way one sees that ρ ( χ ) = 0 admits nontrivial solutions if and only if the infinite linearsystem of equations K X i =1 a ( n i )2 k z i = 0 ∀ k ≥ a ( n )2 N is given by the evaluation at N of a nontrivial polynomial p n ∈ K [ x ]of degree n − p n , . . . , p n K are linearlyindependent if the n , . . . , n K are pairwise distinct because then they are of different degree.Consider a subsystem of linear equations of (3.15) given by K X i =1 a ( n i )2 k z i = 0 ∀ k ∈ { N , . . . , N K } ⇔ K X i =1 p n i ( k ) z i = 0 ∀ k ∈ { N , . . . , N K } .
10y lemma 8 it is possible to choose N , . . . , N K such that the matrix ( p n i ( N j )) Ki,j =1 is regular.Thus, this subsystem of linear equations admits only the trivial solution and therefore so does(3.15). Another way to put this result is that for a basis { e , . . . , e d } of ˚ k the elements of the set (cid:8) ρ (cid:0)(cid:0) t n + t − n (cid:1) ⊗ e i (cid:1) | n ≥ , i = 1 , . . . , d (cid:9) ∈ N ( K ) (3.16)are linearly independent. Since a ( n )2 N +1 is also given by a polynomial the same argument worksfor (cid:8) ρ (cid:0)(cid:0) t n − t − n (cid:1) ⊗ f i (cid:1) | n ≥ , i = 1 , . . . , D (cid:9) ∈ N ( K ) , (3.17)where { f , . . . , f D } is a basis of ˚ p . This shows that ρ ± is injective.We next consider the claim that elements P k ≥ b k u k ⊗ x k with only finitely many b k = 0are not contained in the image of k ( A ) ( K ) in N ( K [[ u ]]). Any element in the image of k ( A ) ( K )in N ( K [[ u ]]) can be written as a linear combination of elements in (3.16) and (3.17). It sufficesto focus on one set as they are split into even and odd coefficients. For elements of type (3.16)this implies that there exist b , . . . , b N ∈ K \ { } such that N X j =1 b j a ( n j )2 k = 0 ∀ k > ⇔ N X j =1 b j p ( n j ) ( k ) = 0 ∀ k > . The polynomials p ( n j ) are linearly independent and so their sum is a polynomial of fixed degreegreater than 0. Then the above equation is a contradiction to the fact that a nonzero polynomialcan be equal to 0 only at finitely many points. Proposition 11.
The homomorphisms ρ ( N ) ± : k ( A ) ( K ) → N ( P N ) are surjective.Proof. Set ˚ k (2 k ) := span K n(cid:16) u k + I N (cid:17) ⊗ x | x ∈ ˚ k o ⊂ N ( P N ) , (3.18a)˚ p (2 k +1) := span K n(cid:16) u k +1 + I N (cid:17) ⊗ y | y ∈ ˚ p o ⊂ N ( P N ) (3.18b)and note that N ( P N ) decomposes into vector spaces as N ( P N ) ∼ = ⌊ N/ ⌋ M k =0 ˚ k (2 k ) ⊕ ⌊ ( N − / ⌋ M k =0 ˚ p (2 k +1) . (3.19)The image of ρ ( N ) ± in N ( P N ) is spanned by elements of the form ⌊ N/ ⌋ X k =0 a ( n )2 k (cid:16) u k + I N (cid:17) ⊗ x ∀ n ∈ N , x ∈ ˚ k , ⌊ ( N − / ⌋ X k =0 a ( n )2 k +1 (cid:16) u k +1 + I N (cid:17) ⊗ y ∀ n ∈ N ∗ , y ∈ ˚ p . Since a (0)2 k = 2 δ k, one already has ˚ k (0) = n ⊗ x | x ∈ ˚ k o ⊂ im ρ ( N ) ± . With this it is possible toremove the ˚ k (0) -part from other elements: ρ ( N ) ± (cid:18) x ( m ) −
12 ( ± m a ( m )0 x (0) (cid:19) = ⌊ N/ ⌋ X k =1 a ( n )2 k (cid:16) u k + I N (cid:17) ⊗ x ∀ n ∈ N , x ∈ ˚ k .
11y the properties of the Cartan decomposition (cf. [HN, prop. 13.1.10]) one has that if ˚ g issimple and non-compact then ˚ k = [˚ p , ˚ p ] and that ˚ p is a simple ˚ k -module. Therefore any element x ∈ ˚ k can be written as an iterated commutator (cid:2) x (1) , (cid:2) x (2) , . . . , x ( k ) (cid:3)(cid:3) for x (1) , . . . , x ( k ) ∈ ˚ k or ˚ p .Choose levels n , . . . , n k such that n + · · · + n k = ⌊ N/ ⌋ and set e x ( i )( n i ) := x ( i )( n i ) −
12 ( ± n i a ( n i )0 x ( i )(0) . Then ρ ( N ) ± (cid:16)he x (1)( n ) , h . . . , e x ( k )( n k ) ii(cid:17) = (cid:16) u ⌊ N/ ⌋ + I N (cid:17) ⊗ x which shows that n(cid:16) u ⌊ N/ ⌋ + I N (cid:17) ⊗ x | x ∈ ˚ k o is contained in the image of ρ ( N ) ± . The same procedure works for (cid:0) u k +1 + I N (cid:1) ⊗ ˚ p because ˚ p is a simple ˚ k -module and therefore there exist iterated commutators here as well. Repeat thisprocess for the lower levels as now the level (cid:0) u ⌊ N/ ⌋ + I N (cid:1) can be removed. This shows byinduction that each homogeneous space in (3.19) lies in the image of ρ ( N ) ± which concludes theproof.We next consider the structure of the Lie algebras N ( P N ). Proposition 12.
Let ˚ k (2 k ) and ˚ p (2 k +1) be as in (3.18a) and (3.18b) and denote by z (cid:0) ˚ k (0) (cid:1) thecenter of ˚ k (0) , then J ( N ) := z (cid:0) ˚ k (0) (cid:1) ⊕ ⌊ N/ ⌋ M k =1 ˚ k (2 k ) ⊕ ⌊ ( N − / ⌋ M k =0 ˚ p (2 k +1) ⊂ N ( P N ) is the radical of N ( P N ) , i.e., the unique maximal solvable ideal in N ( P N ) . Hence, the Levidecomposition of N ( P N ) is given by N ( P N ) ∼ = h ˚ k (0) , ˚ k (0) i ⋉ J ( N ) . Proof.
The N -graded structure of N ( P N ) is given by its decomposition into vector spaces (3.19).By the gradation one deduces (cid:2) ˚ p (2 k − , ˚ p (2 ℓ − (cid:3) ⊆ ˚ k (2 k +2 ℓ − , h ˚ k (2 k ) , ˚ p (2 ℓ − i ⊆ ˚ p (2 k +2 ℓ − , h ˚ k (2 k ) , ˚ k (2 ℓ ) i ⊆ ˚ k (2 k +2 ℓ ) , where it is understood that ˚ k (2 k ) = { } = ˚ p (2 ℓ +1) for k > ⌊ N/ ⌋ and ℓ > ⌊ ( N − / ⌋ . From thisit follows that J ( N ) is an ideal as in particular h ˚ k (0) , ˚ k (2 ℓ ) i ⊆ ˚ k (2 ℓ ) and h ˚ k (0) , ˚ p (2 ℓ − i ⊆ ˚ p (2 ℓ − . Note that for a generalised Dynkin diagram A of untwisted affine type, the only cases when z (˚ k (0) ) is nontrivialare A = C (1) l and A = A (1)1 . For ˚ k (0) = k ( C l ) one has k ( C l ) ∼ = u l which contains a nontrivial center, whereas for k ( A ) ∼ = R the center is already all of ˚ k (0) . h z (cid:0) ˚ k (0) (cid:1) , z (cid:0) ˚ k (0) (cid:1)i = { } the lowest degree in (cid:2) J ( N ) , J ( N ) (cid:3) is 1 and therefore J ( N ) is solvablebecause ˚ k (2 k ) = { } = ˚ p (2 l +1) for k > ⌊ N/ ⌋ and ℓ > ⌊ ( N − / ⌋ . Consider the ideal generatedby J ( N ) + x for 0 = x ∈ h ˚ k (0) , ˚ k (0) i . As h ˚ k (0) , ˚ k (0) i is semi-simple so is the ideal j in h ˚ k (0) , ˚ k (0) i generated by x . Since ideals of semisimple Lie algebras are semisimple, j is also perfect. Thus,the upper derived series j ( n +1)0 := h j ( n )0 , j ( n )0 i becomes constant at j := [ j , j ] and therefore theupper derived series J ( n +1)( N ) := h J ( n )( N ) , J ( n )( N ) i will always contain j . Thus, J ( N ) is a maximalsolvable ideal and therefore by definition the radical of N ( P N ). Remark . We call N ( K [[ u ]]) a parabolic Lie algebra since it is the inverse limit of the parabolicLie algebra N ( P N ). By proposition 5 and corollary 7, we can construct representations of theinvolutory subalgebra k by considering representations of N ( K [[ u ]]) or N ( P N ), respectively. Remark . By a consequence of Lie’s theorem one has that every simple representation of N ( P N ) over a complex vector space is given by the tensor product of a simple representationof N ( P N ) (cid:30) rad ( N ( P N )) ∼ = [˚ k , ˚ k ] with a one-dimensional representation of N ( P N ). Therefore,the simple representations of k ( A ) ( K ) that factor through N ( P N ) are essentially the simplerepresentations of ˚ k ( K ). These correspond to truncation at N = 0.The kernels ker ρ ( N ) ± are described by linear systems of equations as the following propositionshows: Proposition 15.
Let ker ρ ( N ) ± be the kernel of the homomorphism described in cor. 7. Thenwith x ( m ) := (cid:0) t m + t − m (cid:1) ⊗ x ∀ x ∈ ˚ k , y ( m ) := (cid:0) t m − t − m (cid:1) ⊗ y ∀ y ∈ ˚ p one has ker ρ ( N ) ± = span ( M X i =1 ( ± m i b i x ( m i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X i =1 ( ± m i b i a ( m i )2 k = 0 ∀ k =0 , . . ., ⌊ N/ ⌋ , ∀ x ∈ ˚ k ) ⊕ span ( M X i =1 ( ± m i b i y ( m i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X i =1 ( ± m i b i a ( m i )2 k +1 = 0 ∀ k =0 , . . ., ⌊ ( N − / ⌋ , ∀ y ∈ ˚ p ) ker ρ ( N ) ± ⊃ ker ρ ( N +1) ± . Proof.
In general one has for x ∈ ˚ k , y ∈ ˚ p that ρ ( N ) ± (cid:0) x ( m ) (cid:1) = ( ± m ⌊ N/ ⌋ X k =0 a ( m )2 k u k ⊗ x, ρ ( N ) ± (cid:0) y ( m ) (cid:1) = ( ± m ⌊ ( N − / ⌋ X k =0 a ( m )2 k +1 u k +1 ⊗ y and one computes for P Mi =1 ( ± m i b i x ( m i ) that ρ ( N ) ± M X i =1 ( ± m i b i x ( m i ) ! = M X i =1 ( ± m i b i ⌊ N/ ⌋ X k =0 a ( m i )2 k u k ⊗ x = ⌊ N/ ⌋ X k =0 M X i =1 ( ± m i b i a ( m i )2 k ! u k ⊗ x = 013f and only if M X i =1 ( ± m i b i a ( m i )2 k = 0 ∀ k = 0 , . . . , ⌊ N/ ⌋ . Similarly one deduces that ρ ( N ) ± M X i =1 ( ± m i b i y ( m i ) ! = 0 ⇔ M X i =1 ( ± m i b i a ( m i )2 k +1 = 0 ∀ k = 0 , . . . , ⌊ ( N − / ⌋ . These equations remain unaltered by changing N N + N , N ∈ N , there only appearadditional equations to be satisfied. This shows thatker ρ ( N ) ± ⊃ ker ρ ( N +1) ± . Remark . Specialised to A = E the ideals ker ρ (0) ± in k ( E ) ( K ) coincide with the Dirac idealsof [KNP]. Furthermore, these ideals can be shown to be principal ideals. In this section, we consider representations of k that are constructed using induced representa-tions of N ( K [ u ]), the model of k over the polynomial ring which will be defined below. Sincerepresentations of N ( K [ u ]) in general do not provide representations of k , we will then describehow to recover representations of k .Construct the Lie algebra N ( K [ u ]) in the same way as N ( K [[ u ]]) in (3.1) but replace K [[ u ]]by the ring of polynomials K [ u ]. As mentioned in remark 2, N ( K [ u ]) is N -graded and its gradeddecomposition is given by N ( K [ u ]) = ∞ M k =0 N ( k ) , (4.1)where N ( k ) = ( span K n u n ⊗ ˚ k o , k = 2 n even , span K (cid:8) u n +1 ⊗ ˚ p (cid:9) , k = 2 n + 1 odd . We believe that there does not exist any non-trivial Lie algebra homomorphism from k into N ( K [ u ]). Such homomorphisms, similar to those of proposition 11, only exist when we quotientby I N defined in (3.10) which can also be thought of as an ideal of K [ u ]. The structure of N ( K [ u ]) is analogous to that of N ( P N ) described in proposition 12. In particular, the (now nolonger solvable) ideal N + is given by N + = z (˚ k ) ⊕ M k> N ( k ) , z (˚ k ) denotes the center of ˚ k . The ideal N + inherits the N -grading from N ( K [ u ]). Weshall in the following assume that z (˚ k ) = 0 for simplicity. All statements can be generalisedstraight-forwardly.The universal enveloping algebras U ( N + ) and U ( N ( K [ u ])) also inherit the N -gradation fromthe Lie algebra and the Poincar´e–Birkhoff–Witt theorem provides a basis of them. One has U ( N + ) = ∞ M ℓ =0 U ℓ , (4.2)where U ℓ denotes the (ordered) words in the tensor algebra of degree ℓ with respect to the N -grading, where x ≤ y if deg( x ) ≤ deg( y ). For the first few levels, this means U = 1 , U = N (1) , U = Sym (cid:16) N (1) (cid:17) ⊕ N (2) , U = Sym (cid:16) N (1) (cid:17) ⊕ N (1) ⊗ N (2) ⊕ N (3) . The N ( k ) are ˚ k -modules because the adjoint action of ˚ k preserves the degree. We will make theseexpressions more explicit in the example in section 5. For the full universal enveloping algebraone has U ( N ( K [ u ])) ∼ = U (˚ k ) ⊗ K U ( N + ) (4.3)as a tensor product of K -vector spaces. In terms of the multiplication · in U ( N ( K [ u ])) the abovedecomposition is better written as U ( N ( K [ u ])) = U (˚ k ) · U ( N + ) (4.4)This fact is due to the PBW-theorem applied to a suitably chosen order that is such that elementsof degree 0 appear to the left in the basis that is provided by the PBW-theorem. Recall that allelements of ˚ k = N (0) ⊂ N ( K [ u ]) have degree 0.Now consider a finite-dimensional ˚ k -module V which we view as a left U (cid:0) ˚ k (cid:1) -module. As U (cid:0) N ( K [ u ]) (cid:1) allows the structure of a right U (cid:0) ˚ k (cid:1) -module we build the induced N ( K [ u ])-modulevia the tensor product V := U ( N ( K [ u ])) ⊗ U (˚ k ) V , (4.5)where the tensor product is defined with U ( N ( K [ u ])) as a right U (cid:0) ˚ k (cid:1) -module. Explicitly onehas a ⊗ ( x · v ) = ax ⊗ v for v ∈ V , x ∈ U (cid:0) ˚ k (cid:1) , a ∈ U ( N ( K [ u ])). V , however, is viewed as a left U ( N ( K [ u ]))-module, i.e.for a, b ∈ U ( N ( K [ u ])), v ∈ V one has a · ( b ⊗ v ) = ( a · b ) ⊗ v. emma 17. The induced module V inherits the N -grading V = ∞ M i =0 V i (4.6) from U ( N ( K [ u ])) and is an infinite-dimensional representation of N ( K [ u ]) . V is generated bythe action of U ( N + ) on ⊗ V . Furthermore, a K -basis of V is given by the Kronecker productof a PBW-basis of U ( N + ) with a basis of V .Proof. The inherited gradation on V is given by assigning deg( u ⊗ v ) = deg( u ). It is infinite-dimensional because U ( N + ) is. In order to determine a K -basis of (4.5), consider it as a˚ k -moduleand show that it is isomorphic to the K -tensor product of U ( N + ) and V . Recall that ˚ k actson N + via the adjoint action from N ( K [ u ]). Due to the above decomposition (4.4) one has for x ∈ U (cid:0) ˚ k (cid:1) , y ∈ U ( N + ) and v ∈ V that xy ⊗ v = ( yx + [ x, y ]) ⊗ v = y ⊗ v ′ + y ′ ⊗ v (4.7)with y ′ = [ x, y ] ∈ U ( N + ) and v ′ = x.v ∈ V . We want to view (4.5) as a K -tensor productmodulo an equivalence relation. The equivalence relation on U (cid:0) ˚ k (cid:1) ⊗ K U ( N + ) ⊗ K V induced by ⊗ U ( ˚ k ) now is the multilinear extension of x ⊗ K y ⊗ K v ∼ ⊗ K y ⊗ K v ′ + 1 ⊗ K y ′ ⊗ K v, (4.8)where y ′ = [ x, y ] ∈ U ( N + ), v ′ = x.v ∈ V and U ( N ( K [ u ])) ⊗ U ( ˚ k ) V ∼ = U (cid:0) ˚ k (cid:1) ⊗ K U ( N + ) ⊗ K V (cid:30) ∼ as K -vector spaces. The original equivalence relation in U ( N ( K [ u ])) ⊗ K V is yx ⊗ K v ∼ y ⊗ K xv ∀ x ∈ U (cid:0) ˚ k (cid:1) but decomposing yx = xy − [ x, y ] according to (4.4) leads to the formulation (4.8) of ∼ . Eq.(4.8) shows that each element x ⊗ K y ⊗ K v is ∼ -equivalent to an element of 1 ⊗ K U ( N + ) ⊗ K V ∼ = U ( N + ) ⊗ K V where the isomorphism is as K -vector spaces. Since1 ⊗ K U ( N + ) ⊗ K V ⊂ U (cid:0) ˚ k (cid:1) ⊗ K U ( N + ) ⊗ K V one deduces that the elements of U (cid:0) ˚ k (cid:1) ⊗ K U ( N + ) ⊗ K V (cid:30) ∼ are in 1-1-correspondence with theelements of U ( N + ) ⊗ K V and hence, U ( N ( K [ u ])) ⊗ U ( ˚ k ) V ∼ = U ( N + ) ⊗ K V (4.9)as K -vector spaces. Now U (cid:0) ˚ k (cid:1) acts on U ( N + ) ⊗ U ( ˚ k ) V via left-multiplication but as a result of(4.7) one has that xy ⊗ v = [ x, y ] ⊗ v + y ⊗ xv. U (cid:0) ˚ k (cid:1) on the K -tensor product of ˚ k -modules U ( N + ) and V . Thus, one finds U ( N ( K [ u ])) ⊗ U ( ˚ k ) V ∼ = U ( N + ) ⊗ K V as˚ k -modules. The Kronecker basis of this tensor product also provides a basis for the U (cid:0) N (cid:0) K [ u ] (cid:1)(cid:1) -module V because U ( N + ) ∩ U (cid:0) ˚ k (cid:1) = K · N -graded structure on N ( K [ u ]), V has many invariant subspaces. In particular,any V ( N ) = ∞ M i = N V i (4.10)for N > V . The quotient representation V / V ( N ) (4.11)is by construction a finite-dimensional representation of N ( K [ u ]) for any fixed choice N . Theimportant point now is Proposition 18.
The quotient V / V ( N ) defined in (4.11) is a finite-dimensional module of N ( P N ) and therefore, by propositions 5 and 11, a representation of k .Proof. This statement follows from the fact that I N ⊗ (˚ k ⊕ ˚ p ) ∩ N ( K [ u ]) acts trivially on V / V ( N ) by the N -grading and therefore V / V ( N ) is a representation of the corresponding quotient. Sincethe quotients K [ u ] (cid:30) I N and K [[ u ]] (cid:30) I N are isomorphic, we deduce that V / V ( N ) is a finite-dimensional representation of N ( P N ) defined in (3.11) and therefore can be pulled back to arepresentation of k by proposition 11. Remark . The quotient algebra acts non-trivially on V / V ( k ) and is given by all k generatorsof degree at most k . While all other elements of the parabolic model N ( K [ u ]) act trivially, wededuce from proposition 5 that infinitely many generators of k act non-trivially.Other invariant subspaces of V can be considered by taking any vector w ∈ V and consideringthe subrepresentation W ⊂ V that it generates under the action of N ( K [ u ]). A natural choicewould be to select some irreducible ˚ k representation W k within one of the V k . Clearly, thesubmodule W generated by W k is a subspace of V ( k ) . In general, it can be of arbitrary co-dimension in it. The representation of N ( K [ u ]) we are interested in then is the quotient V / W . (4.12)If this quotient is finite-dimensional it can again be pulled back to a representation of k sinceit is a quotient of one of the spaces V / V ( N ) described above. Analysing (4.12) in general ismore complicated than in the case (4.11). To illustrate this point, we shall analyse an explicitexample for e in the next section 5. 17 1 2 3 4 5 6 78Figure 1: Dynkin diagram of E with nodes labelled. Proposition 20.
Recall the graded decomposition (4.6) of V and consider its formal completion V := { ( v i ) | v i ∈ V i } . (4.13) On V there exists a natural, faithful N ( K [[ u ]]) -action. Furthermore, V is the inverse limit ofthe modules of proposition 18 and it is spinorial as a k -representation if the ˚ k -representation V that induces V is. Also, faithfulness shows that k is residually finite-dimensional, that is, foreach non-trivial element x ∈ k there exists a finite-dimensional representation on which x actsnon-trivially. Proof.
The natural action is given by v x · v with ( x · v ) i = i X j =0 x j · v i − j for i ∈ N (4.14)for x = ( x i ) ∈ N ( K [[ u ]]) and v = ( v i ) ∈ V . Since only x i and v N − i enter at degree N thisaction commutes with the projection to V (cid:30) V ( i ) , where V ( i ) denotes the formal completion of V ( i ) . As V (cid:30) V ( i ) ∼ = V (cid:30) V ( i ) , one can check that V is the inverse limit of the finite-dimensionalrepresentations V (cid:30) V ( i ) by also establishing that the elements in V and the inverse limit of the V (cid:30) V ( i ) are in 1-1-correspondence. The representation is faithful , because for ( x i ) ∈ N ( K [[ u ]])one can pick ( v , , , . . . ) ∈ V with 0 = v ∈ V . Since N ( k ) is part of the PBW-basis of U ( N ( K [ u ])), there always exists a k > = v such that x k .v = 0 ∈ V according tolemma 17. The only exception is when ( x i ) ∈ N ( K [[ u ]]) is such that only x = 0. Even if V isthe trivial module, the action of this element is nontrivial because U ( N + ) is a faithful ˚ k -module.Spinoriality of V implies that of V because ˚ k maps to N (0) ⊂ N ( K [[ u ]]). k ( e ) In this section, we illustrate the general considerations from the previous sections in the case of k ( e ) where g = e ≡ e (1)8 denotes the non-twisted affine extension of the split real exceptionalLie algebra ˚ g = e (also denoted as e ). The algebra k ( e ) enters in the fermionic sector ofmaximal supergravity in D = 2 dimensions where unfaithful representations have been foundpreviously [NS] and is therefore of particular interest in physics. We believe this property to be known already for affine g ( A ), although we were unable to find a reference.Hence, our proposition just reproves residual finiteness of k as a byproduct. .1 The affine algebra e The Cartan decomposition of e is e = ˚ k ⊕ ˚ p with ˚ k ∼ = so (16) (5.1)and ˚ p is a 128-dimensional irreducible spinor representation of so (16). To explicitly parametrisethe symmetric space decomposition we introduce the following adapted basis of generators so (16) = (cid:10) X IJ (cid:12)(cid:12) X IJ = − X JI , I, J = 1 , . . . (cid:11) , ˚ p = (cid:10) Y A (cid:12)(cid:12) A = 1 , . . . , (cid:11) , (5.2)so that the X IJ are the exterior square of the defining 16-dimensional representation of so (16)and the Y A are in an irreducible spinor representation. The conjugate spinor representation willbe denoted with dotted indices ˙ A and will play a role in the construction of k ( e ) representationsbelow.In order to write the e commutation relations in the basis (5.2), we utilise so (16) gammamatrices Γ IA ˙ A where A ˙ A denote the matrix components of degree one elements in the Cliffordalgebra of 16-dimensional Euclidean space. These are real 128-by-128 matrices satisfying theClifford multiplication law Γ IA ˙ A Γ JB ˙ A + Γ JA ˙ A Γ IB ˙ A = 2 δ IJ δ AB (5.3)where repeated indices are summed over (a convention we shall employ throughout this section)and δ IJ and δ AB are the Euclidean so (16)-invariant bilinear forms in the representation spaces. k -fold antisymmetric products of gamma matrices are written as Γ I ...I k normalised with a 1 /k !times the signed sum over permutations of I , . . . , I k , e.g.Γ I I AB = 12! (cid:16) Γ I A ˙ A Γ I B ˙ A − Γ I A ˙ A Γ I B ˙ A (cid:17) , Γ I I ˙ A ˙ B = 12! (cid:16) Γ I A ˙ A Γ I A ˙ B − Γ I A ˙ A Γ I A ˙ B (cid:17) , Γ I I I A ˙ B = 13! (cid:16) Γ I A ˙ A Γ I B ˙ A Γ I B ˙ B ± (cid:17) . (5.4)For more details on the use of these gamma matrices see [NS, KNP].The explicit commutations relations of (5.1) in the basis (5.2) are then given by (cid:2) X IJ , X KL (cid:3) = δ JK X IL − δ IK X JL − δ JL X IK + δ IL X JK , (5.5a) (cid:2) X IJ , Y A (cid:3) = −
12 Γ
IJAB Y B , (5.5b) (cid:2) Y A , Y B (cid:3) = 14 Γ IJAB X IJ . (5.5c)The first line is just the ˚ k ∼ = so (16) Lie algebra while the others are manifestations of the Cartandecomposition’s properties (cid:2) ˚ k , ˚ p (cid:3) ⊂ ˚ p and (cid:2) ˚ p , ˚ p (cid:3) = ˚ k . In particular, the last relation can beviewed as a projection from Alt (˚ p ) to the factor that is isomorphic to the adjoint module of ˚ k .19he affine algebra is then obtained according to (2.4) by introducing the loop generators X IJm = t m ⊗ X IJ , Y Am = t m ⊗ Y A (5.6)for m ∈ Z . The commutation relations, including the central element K and derivation d , arethen given by (cid:2) X IJm , X
KLn (cid:3) = δ JK X ILm + n − δ IK X JLm + n − δ JL X IKm + n + δ IL X JKm + n − m (cid:0) δ IK δ JL − δ JK δ IL (cid:1) δ m, − n K , (5.7) (cid:2) X IJm , Y An (cid:3) = −
12 Γ
IJAB Y Bm + n , (5.8) (cid:2) Y Am , Y Bn (cid:3) = 14 Γ IJAB X IJm + n + mδ AB δ m, − n K , (5.9) (cid:2) d, X
IJm (cid:3) = − mX IJm , (cid:2) d, Y Am (cid:3) = − mY Am , (5.10)and K commutes with everything.As is well known, there is an action of the Virasoro algebra on loop algebras [K, GO]. TheVirasoro algebra is a central extension of the Witt algebra generated by the operators L m = − t m +1 ddt ( m ∈ Z ) (5.11)acting on Laurent polynomials and the non-trivial commutators with the e loop generators are (cid:2) L m , X IJn (cid:3) = − nX IJm + n , (cid:2) L m , Y An (cid:3) = − nY Am + n . (5.12) L therefore acts like d . The commutators among the Virasoro generators is[ L m , L n ] = ( m − n ) L m + n + c Vir m ( m − δ m, − n K (5.13)with the same central element K as in the affine algebra and c Vir a free coefficient. In anygiven highest or lowest weight representation of the affine algebra, one can find a realisation ofthe Virasoro algebra in the universal enveloping algebra of the loop algebra via the Sugawaraconstruction and this fixes c Vir , see [GO] for a review. k ( e ) The involutory subalgebra k ≡ k ( e ) was defined in (2.6). We shall give explicit forms of thefiltered structure and the parabolic model discussed in sections 2 and 3, respectively. We recall that the central element K and the derivation d are not invariant under the involution ω and therefore not part of k ( e ). The generators of k ( e ) can be expressed in terms of (5.6)20ccording to X IJm = 12 ( t m + t − m ) ⊗ X IJ ≡ (cid:0) X IJm + X IJ − m (cid:1) , (5.14a) Y Am = 12 ( t m − t − m ) ⊗ Y A ≡ (cid:0) Y Am − Y A − m (cid:1) . (5.14b)where t is the spectral parameter and m ∈ N . Note that Y A ≡
0. Here, we have introduced afactor of 1 / X IJ = X IJ is also an so (16) Lie algebra with the same normalisation.The complete k ( e ) Lie algebra in the basis (5.14) reads (cid:2) X IJm , X KLn (cid:3) = 12 δ JK (cid:16) X ILm + n + X IL | m − n | (cid:17) − δ IK (cid:16) X JLm + n + X JL | m − n | (cid:17) − δ JL (cid:16) X IKm + n + X IK | m − n | (cid:17) + 12 δ IL (cid:16) X JKm + n + X JK | m − n | (cid:17) , (5.15a) (cid:2) X IJm , Y An (cid:3) = −
14 Γ
IJAB (cid:16) Y Bm + n − sgn( m − n ) Y B | m − n | (cid:17) , (5.15b) (cid:2) Y Am , Y Bn (cid:3) = 18 Γ IJAB (cid:16) X IJm + n − X IJ | m − n | (cid:17) , (5.15c)where sgn( k ) is the sign function with sgn(0) = 0.In this so (16)-covariant formulation, k ( e ) can be described as the algebra generated by the X IJ , X IJ and Y A with the property that the X IJ form an so (16) algebra and that X IJ and Y A transform correctly under this algebra according to (5.15). The X IJ , X IJ and Y A obey7 Γ IJAB (cid:2) Y A , Y B (cid:3) − (cid:2) X IK , X KJ (cid:3) = − X IJ , (5.16a)Γ I ...I AB (cid:2) Y A , Y B (cid:3) = 0 . (5.16b)These are the so (16)-covariant forms of the Berman relation [ x , [ x , x ]] = − x in a Chevalley–Serre basis x i := e i − f i (5.17)and where we use the convention that 0 is the affine node of the e Dynkin diagram thatattaches to the adjoint node, labelled 1, of the e Dynkin diagram, see figure 1. To write out the‘affine’ Berman generator x explicitly in terms of the basis (5.14), we need to make use of theSO(8) decompositions (A.9) in Appendix A of [KNP]; using the notation and transformationsof Appendix B of that reference there we have x = − γ α ˙ β (cid:0) X α ˙ β + Y α ˙ β (cid:1) (5.18)with the SO(8) gamma matrices γ iα ˙ β (for i = 1 , ..., x i = 14 (cid:16) γ i,i +1 αβ X αβ + γ i,i +1˙ α ˙ β X ˙ α ˙ β (cid:17) (for i = 1 , ..., , x = − γ α ˙ β X α ˙ β . (5.19)21he Virasoro algebra likewise can be restricted to an involutory subalgebra [JN]. The ‘max-imal compact’ subalgebra of the Virasoro algebra is generated by K m := L m − L − m ≡ − (cid:0) t m +1 − t − m +1 (cid:1) ddt for m ≥
1. (5.20)The relevant commutation relations when acting on K ( e ) in the basis (5.14) read (cid:2) K m , X IJn (cid:3) = − n (cid:0) X IJm + n − X IJ | m − n | (cid:1)(cid:2) K m , Y An (cid:3) = − n (cid:0) Y Am + n − sgn( m − n ) Y A | m − n | (cid:1) (5.21)and [ K m , K n ] = ( m − n ) K m + n − sign( m − n )( m + n ) K | m − n | (5.22)Note that the central term drops out here as well. As discussed in sections 3 and 4, it is very useful for constructing representations of k ( e ) toconsider the parabolic algebras N ( K [[ u ]]) and N ( K [ u ]) defined in (3.1) and (4.1), respectively.Since polynomials have a graded product, N ( K [ u ]) is a graded Lie algebra. We write the variableof the polynomial as u as in section 3 and it is related to the variable t in the filtered basis by u = − t t so that expansions around u = 0 are expansions around t = 1 and vice versa.The definition of the basis generators of N ( K [ u ]) is then explicitly A IJ m := u m ⊗ X IJ , S A m +1 := u m +1 ⊗ Y A for m ∈ N . (5.23)The Lie algebra of these generators is graded and given by (cid:2) A IJ m , A KL n (cid:3) = 12 δ JK A IL m + n ) − δ IK A JL m + n ) − δ JL A IK m + n ) + 12 δ IL A JK m + n ) , (5.24a) (cid:2) A IJ m , S KL n +1 (cid:3) = −
14 Γ
IJAB S B m + n )+1 , (5.24b) (cid:2) S A m +1 , S B n +1 (cid:3) = 18 Γ IJAB A IJ m + n +1) . (5.24c)The generators of N ( K [[ u ]]) also include infinite linear combinations of (5.23) since N ( K [[ u ]])is constructed using power series rather than polynomials.The maps (3.9a) and (3.9b) from the filtered to the parabolic bases now read ρ ± ( X IJn ) = ( ± n X k ≥ a ( n )2 k A IJ k , ρ ± ( Y An ) = ( ± n X k ≥ a ( n )2 k +1 S A k +1 , (5.25)22here the factors of are due to our definition (5.14). More specifically, the images of the firstfew generators of k ( e ) according to proposition 5 are given in the above basis by ρ + (cid:0) X IJ (cid:1) = A IJ ,ρ + (cid:0) X IJ (cid:1) = A IJ + 2 X k ≥ A IJ k ,ρ + (cid:0) X IJ (cid:1) = A IJ + 8 X k ≥ k A IJ k ,ρ + (cid:0) X IJ (cid:1) = A IJ + 2 X k ≥ (1 + 8 k ) A IJ k ρ + (cid:0) X IJ (cid:1) = A IJ + 323 X k ≥ ( k + 2 k ) A IJ k ρ + (cid:0) X IJ (cid:1) = A IJ + 23 X k ≥ (3 + 40 k + 32 k ) A IJ k (5.26)for the X IJm . For the Y Am the relations read ρ + (cid:0) Y A (cid:1) = − X k ≥ S A k +1 ,ρ + (cid:0) Y A (cid:1) = − X k ≥ (2 k + 1) S A k +1 ,ρ + (cid:0) Y A (cid:1) = − X k ≥ (8 k + 8 k + 3) S A k +1 ,ρ + (cid:0) Y A (cid:1) = − X k ≥ (3 + 10 k + 12 k + 8 k ) S A k +1 ,ρ + (cid:0) Y A (cid:1) = − X k ≥ (15 + 56 k + 88 k + 64 k + 32 k ) S A k +1 . (5.27)We note that, unlike (5.16), there is no known presentation of N ( K [[ u ]]) as a finitely gen-erated algebra, say by A IJ and S A , with a finite number of Berman-like relations. Using thealgebra (5.24) we find the following Berman-type relations for all k ≥
17 Γ
IJAB X k , k ≥ k + k = k − (cid:2) S A k +1 , S B k +1 (cid:3) − X k ,k ≥ k + k = k (cid:2) A IK k , A KJ k (cid:3) = 448 A IJ k , (5.28a)Γ I ...I AB X k , k ≥ k + k = k − (cid:2) S A k +1 , S B k +1 (cid:3) = 0 , (5.28b)where we have evaluated the commutators involving A in the first line.The maximal compact Virasoro generators K m introduced in (5.20) can also be expressedusing the variable u . In particular, K = L − L − = (cid:0) − t ) ddt = 2 u ddu (5.29)23enerates an SO(1,1) group, and acts as a counting operator on the basis of the parabolicmodel (5.23): (cid:2) K , A IJ k (cid:3) = 4 k A IJ k , (cid:2) K , S A k +1 (cid:3) = 2(2 k + 1) S A k +1 (5.30)More generally the operators K m admit the following realisation as differential operators K m = 12 1(1 − u ) m − (cid:2) (1 + u ) m − (1 − u ) m (cid:3) ddu (5.31)Proceeding as before we find for instance[ K , A IJ m ] = 8 m (cid:0) A IJ m + 2 A IJ m +2 + 2 A IJ m +4 + · · · (cid:1) (5.32)so for m ≥ Representations of k ( e ) can be constructed via the technique described in section 4. This meansthat we construct a basis of the universal enveloping algebra of N + ⊂ N (given by all generatorsin (5.23) with degree greater than 0) and let this act on a ˚ k ∼ = so (16) representation V as theinitial vector space. We shall exploit that everything is so (16) covariant and graded. The basis (4.3) of U ( N + ) at the first few levels becomes ℓ = 0 : 1 (5.33a) ℓ = 1 : S A (5.33b) ℓ = 2 : S ( A S B )1 , A α (5.33c) ℓ = 3 : S ( A S B S C )1 , S A A α , S A (5.33d) ℓ = 4 : S ( A S B S C S D )1 , S ( A S B )1 A α , S B S A , A ( α A β )2 , A α (5.33e)where we now use α ≡ [ IJ ] for I < J to denote an adjoint index of so (16). The symmetrisa-tions ( · · · ) for similar generators on the same level are necessary to implement the ordering inaccordance with the PBW theorem.In terms of so (16) representations the first levels U ℓ of U ( N + ) are ℓ = 0 : (5.34a) ℓ = 1 : s (5.34b) ℓ = 2 : (cid:16) ⊕ ⊕ + (cid:17) ⊕ (5.34c) ℓ = 3 : (cid:16) s ⊕ s ⊕ s ⊕ s (cid:17) ⊕ (cid:16) s ⊕ c ⊕ s (cid:17) ⊕ s (5.34d) The parentheses are used to group the representations according to the different words in the induced repre-sentation module list after decomposing into so (16). so (16) representations by their real dimensions. A translation tohighest weight labels can be found in appendix C.The module V as in (4.5) built from an ˚ k ∼ = so (16) representation V is graded as V = ∞ M ℓ =0 V ℓ , V ℓ := U ℓ ⊗ V and each level decomposes further as a so (16)-representation. For instance, in the case of the V = we obtain V = , (5.35a) V = c ⊕ s , (5.35b) V = (cid:16) ⊕ ⊕ ⊕ ⊕ ⊕ + (cid:17) ⊕ (cid:16) ⊕ ⊕ (cid:17) , (5.35c) V = (cid:16) c ⊕ × s ⊕ c ⊕ × s ⊕ s ⊕ c ⊕ s ⊕ c ⊕ s ⊕ s (cid:17) ⊕ (cid:16) × c ⊕ × s ⊕ c ⊕ c ⊕ s ⊕ s (cid:17) ⊕ (cid:16) c ⊕ s (cid:17) . (5.35d) As is clear from (5.35), the module grows very rapidly and it is desirable to find smaller k ( e )representations by identifying invariant subspaces and taking quotients.The simplest quotient example is to consider W s =1 / = V (1) in the notation (4.10) to begiven by all spaces of degree greater than zero, then we obtain as k ( e ) representation simply V / W s =1 / ∼ = V ∼ = which is nothing but the irreducible spin-1/2 representation appearingin supergravity [NS].A non-trivial example can be obtained by looking at the construction (4.12) that uses theinvariant subspaces W generated by an so (16) representation W k sitting at a given level. Asthe example we take W = s within (5.35). In order to describe this, we shall need a moreexplicit parametrisation of the elements of the module’s homogeneous parts V ℓ for 0 ≤ ℓ ≤ ℓ = 0 we need an element of the 16-dimensional defining representation of so (16) thatwe write as ϕ I .The elements of V ∼ = s ⊗ are of the form S A ϕ I and decompose into a conjugatespinor c and a traceless vector-spinor s S A ϕ I = Γ IA ˙ A χ ˙ A + χ IA according to V = c ⊕ s . Note that the occurrence of c in the above tensor productis tied to the existence of a suitable Γ-matrix Γ IA ˙ A . The condition that projects onto the s is χ IA = S A ϕ I −
116 (Γ I Γ J ) AB S B ϕ J V / W satisfy S B ϕ I = 116 (Γ I Γ J ) BC S C ϕ J (5.36)and any relations obtained from it by acting with k ( e ).To find what this imposes on V as given in (5.35c) we parametrise all its elements as S A S B ϕ I = δ AB ϕ I + Γ J J AB ϕ J J ; I + Γ J ...J AB ϕ J ...J ; I + Γ J ...J AB ϕ J ...J ; I , (5.37)This formula follows from the so (16) tensor product s ⊗ s that is relevant for the word S A S B and that we decompose into its symmetric and anti-symmetric partsSym ( s ) = ⊕ ⊕ + , Alt ( s ) = ⊕ . The first line consists of a scalar, a four-form and a self-dual eight-form of so (16), while thesecond line represents a two-form and a six-form. These intertwiners from s ⊗ s to p -forms are given by the Γ-matrices Γ I ...I p AB . As the anti-symmetric product S [ A S B ]1 is proportionalto the commutator (5.24c) that does not contain a six-form, the ansatz (5.37) does not containa term in Γ I ...I AB . The final so (16) tensor product on the left-hand side of (5.37) is then tomultiply s ⊗ s (written as p -forms) with which is the representation of ϕ I . Thesetensor products are written using a semi-colon, so that for instance ϕ J J ; I ∈ ⊗ = ⊕ ⊕ . Acting on (5.36) with S A then leads to the relation δ AB ϕ I + Γ J J AB ϕ J J ; I + Γ J ...J AB ϕ J ...J ; I + Γ J ...J AB ϕ J ...J ; I = 116 (Γ I Γ K ) BC (cid:18) δ AC ϕ K − Γ J J CA ϕ J J ; K + Γ J ...J CA ϕ J ...J ; K + Γ J ...J CA ϕ J ...J ; K (cid:19) = 116 δ AB ϕ I −
116 Γ
IJAB ϕ J + 116 Γ J J AB ϕ J J ; I −
116 Γ IJ J J AB ϕ J J ; J + 18 Γ J J AB ϕ IJ ; J −
18 Γ
IJAB ϕ JK ; K + 18 δ AB ϕ IJ ; J + 116 Γ J ...J AB ϕ J ...J ; I −
116 Γ IJ ...J AB ϕ J ...J ; J −
14 Γ IJ ...J AB ϕ J ...J K ; K + 14 Γ J ...J AB ϕ IJ ...J ; J + 34 Γ J J AB ϕ IJ J K ; K + 116 Γ J ...J AB ϕ J ...J ; I −
116 Γ IJ ...J AB ϕ J ...J ; J −
12 Γ IJ ...J AB ϕ J ...J K ; K −
12 Γ J ...J AB ϕ IJ ...J ; J + 72 Γ J ...J AB ϕ IJ ...J K ; K . (5.38)Projecting this onto the various irreducible pieces in (5.35c) leads to the conditions ϕ IJ ; J = 152 ϕ I (relation between the two ) ϕ I I I J ; J = 1312 ϕ [ I I ; I ]2 (relation between the two ) (5.39)26nd the fact that all other irreducible components must vanish. Therefore, at level ℓ = 2, thequotient is given by only V / W ∼ = ⊕ , (5.40)a comparatively small subspace of (5.35c). Already at the next level the above computationbecomes almost unfeasible. If one formally substracts the so (16)-decompositions of U ℓ +1 ⊗ V and U ℓ ⊗ W the result indicates that only c survives at level three, and that there are onlytwo s at level four, after which the procedure terminates. In summary, the above computationshows V / W ∼ = V ∼ = , V / W ∼ = c , V / W ∼ = ⊕ , (5.41)and we conjecture V / W ∼ = c , V / W ∼ = 2 × , V ℓ / W ℓ ∼ = 0 ∀ ℓ ≥ . (5.42)This is related to the analogue of the spin- representation studied in [KN5]. The spin-3/2representation of supergravity [NS, KNP] can also be obtained from this construction by takinga further quotient. More precisely, one quotients by all V ℓ with ℓ > representation in (5.40). The remaining so (16) representations are ⊕ ⊕ that form onechiral half of the supergravity spin-3/2 fields. Remark . As is evident from the analysis above, determining the quotient V / W can becomeintricate quickly since the precise structure of the submodule W is hard to analyse. In the caseof complex simple Lie algebras a similar problem arises when constructing irreducible highestweight representations as quotients of Verma modules by the maximal proper submodule. Inthat case, there is a description of the quotient in terms of the Weyl character formula. A similartechnique for representations of k is not known to the best of our knowledge.27 k ( E ) ( C ) as the quotient of a GIM-algebra Generalised intersection algebras, GIM-algebras in short for Generalised Intersection Matrix,are constructed similarly to Kac–Moody algebras. One starts from a so-called generalised in-tersection matrix A where one replaces the condition A ij ≤ i = j by A ij ≤ ⇔ A ji ≤ A ij > ⇔ A ji >
0. GIMs can be visualised quite neatly by drawing solid lines for A ij < A ij >
0. In [B], Berman explores a connection between his involutorysubalgebras and GIM-algebras. As it turns out due to the work of Slodowy, GIM-algebras fallinto two classes. The first class consists of those GIM-algebras which are in fact isomorphic to aKac–Moody-algebra and the second class are those which are isomorphic to an involutory sub-algebra of a Kac–Moody-algebra according to Berman’s construction. This isomorphism relieson doubling the number of vertices of the respective GIM-diagram and the involution that isused involves a diagram automorphism of the new diagram which is of Kac–Moody-type. Soeven though our k is an involutory subalgebra it is not of the type that is directly isomorphic toa GIM-algebra. For k ( E ) ( C ) it turns out that it is a quotient of a GIM-algebra, although wedo not know the precise structure of the defining ideal. In particular we do not know whetherit is generated by (A.15)–(A.19) or whether one needs additional relations. We will collect themost essential definitions and results here (see the work of Slodowy [S] for more details) andstate our result in proposition 27. Definition 22.
Let I be a finite index set and h a C -vector space of dimension r . Let ∆ ∨ := { h i | i ∈ I } ⊂ h and ∆ := { α i | i ∈ I } ⊂ h ∗ . Then ( h , ∆ ∨ , ∆) is called a C -root basis, itsreductive rank is defined to be equal to r , whereas its semi-simple rank is defined to be | I | .Associate a matrix A to ( h , ∆ ∨ , ∆), called its structure matrix, by setting A ij := α j ( h i ) ∀ i, j ∈ I. (A.1)A root basis is called free if both ∆ ∨ and ∆ are linearly independent. Definition 23.
Let A ∈ Z ℓ × ℓ such that( i ) A ii = 2 ∀ i = 1 , . . . , ℓ ( ii ) A ij < ⇔ A ji < ∀ i = j ( iii ) A ij > ⇔ A ji > ∀ i = j, then A is called a generalised intersection matrix (GIM). A GIM A is called symmetrisable ifthere exist D, B ∈ Q ℓ × ℓ such that D is diagonal and B is symmetric and it holds A = DB .A root basis ( h , ∆ ∨ , ∆) whose structure matrix is a generalised intersection matrix is called aGIM-root basis. Definition 24.
Let ( h , ∆ ∨ , ∆) be a GIM-root basis with structure matrix A . Then ∆( A ) is acoloured, weighted graph with vertices ∆ with:1. Two vertices i and j are connected by a dotted edge if A ij = α j ( h i ) > i and j are connected by a solid edge if A ij = α j ( h i ) < i and j if A ij = 0.4. The edges ( i, j ) are weighted by the weight m ij according to the following table α i ( h j ) · α j ( h i ) 0 1 2 3 ≥ m ij ∞ The weights m ij = 2 , m ij = 4 , j if | A ji | = | α i ( h j ) | > | α j ( h i ) | = | A ij | . Definition 25.
Let ( h , ∆ ∨ , ∆) be a GIM-root basis with structure matrix A and let f be thefree Lie algebra over C generated by h and elements e α , e − α for α ∈ ∆. Let I be the ideal in f generated by the relations (identify h − α ≡ − h α for α ∈ − ∆) (cid:2) h, h ′ (cid:3) = 0 , [ h, e α ] = α ( h ) e α ∀ h, h ′ ∈ h , α ∈ ± ∆ , [ e α , e − α ] = h α , ad ( e α ) max(1 , − β ( h α )) ( e β ) = 0 ∀ α, β ∈ ± ∆ and α = − β. Set g := f (cid:30) I then g is called the GIM-Lie algebra to ( h , ∆ ∨ , ∆). Note that every GIM A hasa free realisation ( h , ∆ ∨ , ∆) that is unique up to isomorphism. In this sense one can associate aGIM-Lie algebra gim ( A ) to a structure matrix A or equivalently a GIM-diagram ∆( A ).If one spells out the above relations for A ij <
0, one obtains with e i = e α i , f i = e − α i thefamiliar Serre-relationsad ( e i ) − A ij ( e j ) = 0 = ad ( f i ) − A ij ( f j ) , [ e i , f j ] = 0 = [ e j , f i ]but for A ij > e i ) A ij ( f j ) = 0 = ad ( f i ) A ij ( e j ) , [ e i , e j ] = 0 = [ f i , f j ] . One knows on abstract grounds that the complexification k ( A ) ( C ) of the canonical subalgebra k ( A ) < k ( E ) is isomorphic to B ( C ) ∼ = so (9 , C ). Let us spell out the relationship between thedescription of k ( A ) ( C ) via Berman generators x i = e i − f i (A.2)and the usual description of g ( B ) ( C ) in terms of a Chevalley basis. Let i , i , . . . , i k ∈ { , . . . , } and set x α i + ··· + α ik := (cid:2) x i , (cid:2) x i , (cid:2) . . . , (cid:2) x i k − , x i k (cid:3)(cid:3)(cid:3)(cid:3) . (A.3)Note that the order in the sum α i + · · · + α i k matters. For i < j define roots β (1) i,j , . . . , β (4) i,j ∈ ∆ ( A ) ⊂ ∆ ( E ) by β (1) i,j = α i + · · · + α j − , β (2) i,j = α i + · · · + α j − (A.4) β (3) i,j = α i − + · · · + α j − , β (4) i,j = α i − + · · · + α j − . (A.5)29ow set e ε L i + ε L j := i · (cid:18) x β (1) i,j − iε x β (2) i,j − iε x β (3) i,j − ε ε x β (4) i,j (cid:19) ∀ i < j ∈ { , , , } , (A.6) H j := − ix j − , for j = 1 , . . . , , (A.7) e ± L j := i · (cid:0) x α j + ··· + α ∓ ix α j − + ··· + α j (cid:1) . (A.8) Proposition 26.
Consider the abelian subalgebra h B := span C { H , . . . , H } together with thelinear functionals L i : h ∗ B → C defined via L i ( H j ) = δ ij . Then with the above definitions (A.6),(A.7) and (A.8) one has (cid:2) h, e ε L i + ε L j (cid:3) = ( ε L i + ε L j ) ( h ) e ε L i + ε L j , (cid:2) h, e ± L j (cid:3) = ± L j ( h ) e ± L j ∀ h ∈ h B Thus, (A.6) and (A.8) provide a root space decomposition of k ( A ) ( C ) ∼ = B ( C ) with respect tothe Cartan subalgebra h B spanned by (A.7). A corresponding Chevalley basis is given by e i := e L i − L i +1 , f i := e − L i + L i +1 , h i := H i − H i +1 ∀ i = 1 , , ,e = e + L , f := e − L , h := 2 H . Proof.
One verifies that (A.6) and (A.8) are eigenvectors unde the adjoint action of h B with thecorrect eigenvalues by direct computation. This suffices for a root space decomposition becauseone knows abstractly that this exhausts k ( A ) ( C ) as its isomorphism type is known. Checkingthe Chevalley basis is a matter of fixing suitable prefactors.Now set x ± := i ( x ∓ i [ x , x ]) (A.9)then one has [ x + , x − ] = 2 H , [ H , x ± ] = ± x ± , [ H i , x ± ] = 0 ∀ i = 2 , (A.10)and the Slodowy-type relations [ x + , y ] = 0 ∀ y ∈ { f , e } ∪ { e , e , f , f } , (A.11)ad ( x + ) ( y ) = 0 = ad ( y ) ( x + ) ∀ y ∈ { e , f } ∪ { e , e , f , f } , (A.12)[ x − , y ] = 0 ∀ y ∈ { e , f } ∪ { e , e , f , f } , (A.13)ad ( x − ) ( y ) = 0 = ad ( y ) ( x − ) ∀ y ∈ { f , e } ∪ { e , e , f , f } , (A.14)as well as additional relations that hold in k ( E ) ( C )[ x + , e εL − L ] = x α + α + α − iεx α + α + α + α = [ x − , e εL + L ] (A.15)ad ( x + ) ( e εL − L ) = 2 e εL + L , ad ( x − ) ( e εL + L ) = 2 e εL − L (A.16)[ x + , e + L + εL ] = 0 = [ x − , e − L + εL ] (A.17)[ x + , e − L + εL ] = − εx α + α + α − ix α + α + α + α = − [ x − , e L + εL ] (A.18)30 2 3 4 B B ⋄ Figure 2:
The diagrams associated to B and B ⋄ with labelling of nodes. ad ( x + ) ( e − L + εL ) = − e L + εL , ad ( x − ) ( e L + εL ) = − e − L + εL . (A.19)Consider the Cartan matrix of B and a GIM which we call B ⋄ that extends it: B = − − − − −
10 0 − , B ⋄ = − − − − − − −
10 0 0 − The nontrivial Serre relations (for the e i only) of B spell out to bead ( e i ) ( e i +1 ) = 0 = ad ( e i +1 ) ( e i ) ∀ i = 1 , e ) ( e ) = 0 = ad ( e ) ( e ) . Denote the associated GIM-algebra to B ⋄ over C by gim ( B ⋄ ) ( C ). The diagrams of B and B ⋄ are given in figure 2. Proposition 27.
Denote the Chevalley generators of gim ( B ⋄ ) ( C ) by E , . . . , E , F , . . . , F and H γ , . . . , H γ . There exists a surjective homomorphism of Lie algebras φ : gim ( B ⋄ ) ( C ) → k ( E ) ( C ) that is given on the Level of generators via φ ( E ) = x + , φ ( F ) = x − , φ ( H γ ) = 2 H = 2 h + 2 h + h φ ( E i ) = e i , φ ( F i ) = f i , φ ( H γ i ) = h i ∀ i = 1 , , , . Proof.
One verifies that the defining relations between the generators from definition (25) aresatisfied. The B -relations are unproblematic and towards the relationsad ( E ) ( E ) = 0 = ad ( E ) ( E ) , ad ( E ) ( F ) = 0 = ad ( F ) ( E )we refer to equations (A.11)–(A.14). One also has to check that H γ H satisfies all necessaryidentities which is the case. Thus, φ is a homomorphism of Lie-algebras. Surjectivity followsfrom the fact that all Berman generators of k ( E ) ( C ) can be recovered from the image of thegenerators of gim ( B ⋄ ) ( C ). For x , . . . , x this is a basis transformation within B ( C ) and for x one notes that (A.9) implies x + + x − = 2 ix . E , [ E , E ]] − E L + L with E L + L ∝ [ E , [ E , [ E , [ E , E ]]]]. This element is nonzero because the two summands liein different root spaces and are nonzero themselves. But because of (A.16) it is equal to 0 inthe image. This implies that k ( E ) ( C ) is a (nontrivial) quotient of gim ( B ⋄ ) ( C ).In conclusion, representations of gim ( B ⋄ ) ( C ) could potentially be useful to find represen-tations of k ( E ) ( C ) if it is possible to check whether or not a given representation factorsthrough the projection of proposition 27. Conversely, our results from sections 3 and 4 providerepresentations of gim ( B ⋄ ) ( C ) both of finite and infinite dimension. B The Hilbert space completion b k is not a Lie algebra Because the restriction of the standard bilinear form is positive definite we can complete k toa Hilbert space b k . Here we show by means of a simple explicit example that this Hilbert spacecompletion is not compatible with the Lie algebra structure. Define J N ( ω ) := 12 ω IJ N X n =1 n / ε X IJn (B.1)The positive definite bilinear form can be normalised such that (for m, n ≥ (cid:10) X IJm |X KLn (cid:11) = δ mn δ IJKL (B.2)The induced norm is distinguished by its invariance, which implies that the right-hand side isindependent of m and n . Consequently, || J N || = C N X n =1 n ε (B.3)where C = C ( ω ) is an irrelevant strictly positive constant. For ε > J ∞ ( ω ) ≡ lim N →∞ J N ( ω ) belongs to the Hilbert space b k . We next computethe commutator h J N ( ω ) , J N ( ω ) i = 12 [ ω , ω ] IJ N X n =1 f n X IJn (B.4)with f n = n − X m =1 m / ε n − m ) / ε + N X m =1 m / ε n + m ) / ε (B.5)Estimating the first sum on the right-hand side as ( n > n − X m =1 m / ε n − m ) / ε > n − n ε (B.6)it is easy to see that f n > C n ε (B.7)32ith another irrelevant strictly positive constant C . For ε < the sum P ∞ n =1 | f n | diverges,whence (B.4) does not converge in the limit N → ∞ . In other words, although J ∞ ( ω ) and J ∞ ( ω ) separately do belong to b k , their commutator does not exist as an element of b k . Hence b k is not even a Lie algebra, and a fortiori also not a Hilbert Lie algebra, which would in additionrequire || [ x, y ] || < C || x |||| y || for all x, y ∈ b k .We stress that the failure of b k to be a Lie algebra depends on the norm used for the completionwhich in the analysis above was the standard invariant bilinear form. Other norms are possibleand the corresponding completions of k can be Lie algebras, and even Hilbert Lie algebras.However, in those cases the norm is not invariant.Although b k is not a Lie algebra, an interesting open question is whether one can still makesense of the commutator as a distribution , by considering the commutator of two elementsbelonging to a dense subspace of b k . This would fit with earlier observations in [KNP]. C so (16) representations The translation of dimensions of so (16) representations to the labels of the highest weight in theconventions of the LiE software [vLCL] are ↔ [0,0,0,0,0,0,0,0] ↔ [1,0,0,0,0,0,0,0] ↔ [0,1,0,0,0,0,0,0] s ↔ [0,0,0,0,0,0,0,1] c ↔ [0,0,0,0,0,0,1,0] ↔ [0,0,1,0,0,0,0,0] ↔ [1,1,0,0,0,0,0,0] ↔ [0,0,0,1,0,0,0,0] s ↔ [1,0,0,0,0,0,0,1] ↔ [0,0,0,0,1,0,0,0] + ↔ [0,0,0,0,0,0,0,2] ↔ [1,0,1,0,0,0,0,0] ↔ [0,0,0,0,0,1,0,0] ↔ [0,0,0,0,0,0,1,1] s ↔ [0,1,0,0,0,0,0,1] c ↔ [0,1,0,0,0,0,1,0] c ↔ [2,0,0,0,0,0,1,0] ↔ [1,0,0,1,0,0,0,0] s ↔ [0,0,1,0,0,0,0,1] ↔ [1,0,0,0,1,0,0,0] + ↔ [1,0,0,0,0,0,0,2] ↔ [1,0,0,0,0,1,0,0] ↔ [0,1,0,1,0,0,0,0] s ↔ [1,1,0,0,0,0,0,1] s ↔ [0,0,0,1,0,0,0,1] c ↔ [0,0,0,1,0,0,1,0] ↔ [1,0,0,0,0,0,1,1] s ↔ [0,0,0,0,0,0,0,3] s ↔ [0,0,0,0,1,0,0,1] c ↔ [0,0,0,0,0,0,1,2] c ↔ [0,0,0,0,0,1,1,0] + ↔ [0,1,0,0,0,0,0,2] s ↔ [1,0,1,0,0,0,0,1] s ↔ [1,0,0,1,0,0,0,1] s ↔ [1,0,0,0,0,0,0,3] s ↔ [1,0,0,0,0,1,0,1] References [AM] P. Abramenko and B. 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