Universal central extensions of direct limits of Lie superalgebras
aa r X i v : . [ m a t h . R A ] A p r UNIVERSAL CENTRAL EXTENSIONS OF DIRECT LIMITS OF LIESUPERALGEBRAS
ERHARD NEHER AND JIE SUN
Abstract.
We show that the universal central extension of a direct limit of perfect Liesuperalgebras L i is (isomorphic to) the direct limit of the universal central extensions of L i . As an application we describe the universal central extensions of some infinite rankLie superalgebras. Introduction
Central extensions appear naturally in the theory of infinite dimensional Lie algebras.For example, they are fundamental for the theory of affine Kac-Moody Lie algebras andextended affine Lie algebras. Centrally extended Lie algebras often have a more interestingrepresentation theory than the original Lie algebra, which makes central extension an in-teresting topic for applications, e.g., in physics. A convenient way to find “all” of them, isto determine the universal central extension of a given Lie algebra, which exists for perfectLie algebras (well-known) and superalgebras [N2].Direct limit of Lie superalgebras are an important way to construct infinite dimensionalLie superalgebras. Examples include various types of locally finite Lie (super)algebras [BB,DP, P, PS], locally extended affine Lie algebras [MY, Nee, N3] and Lie superalgebras gradedby locally finite root systems [N1, GN]. These types of Lie algebras and Lie superalgebrashave been intensively studied by many authors, many more than we have quoted, yet nogeneral results seem to be known about their universal central extensions, besides the paper[S] in which the author studied a rather special case, described in Remark 1.7.In this paper, we consider the universal central extensions of general direct limits of Liesuperalgebras over an arbitrary base superring. We show in Theorem 1.6 that the univer-sal central extension of a direct limit lim −→ L i of perfect Lie superalgebras L i is canonicallyisomorphic to the direct limit of the universal central extensions of L i . This result is neweven for the case of Lie algebras. Crucial for its proof is the fact ([N2]) that one has aendo-functor uce on the category of all Lie superalgebras which gives the universal centralextension for perfect Lie superalgebras.As an application, we describe in § sl ( I ; A ) for | I | ≥ A an associative superalgebra(Proposition 2.2, Corollary 2.4), osp ( I ; A ) for A commutative associative (Example 2.6),locally finite Lie superalgebras (Example 2.7) and Lie algebras graded by locally finite rootsystems (Example 2.10). These applications are possible since one knows the universalcentral extension of the Lie superalgebras over which we take the direct limit. Acknowledgements.
The authors thank I. Dimitrov who asked one of us a questionabout the universal central extensions of certain locally finite Lie algebras, now answered
Mathematics Subject Classification. in Example 2.9 and pointed out the reference [S]. The first author gratefully acknowledgeshelpful discussions with N. Lam on the topic of the paper. We also thank V. Serganova forvery useful comments on an earlier version of the paper.1.
Universal central extensions of direct limits of Lie superalgebras:General results
Throughout thissection we consider Lie superalgebras L over a commutative superring S as defined in [N2].Thus S is an associative, unital Z / Z -graded ring which is commutative in the sense that s s = ( − | s || s | s s holds for all homogeneous s i ∈ S . Here and in the following | s | denotes the degree of a homogeneous element. Formulas involving the degree function aresupposed to be valid for homogeneous elements – a condition that we will not mentionexplicitly in the following.We first describe some facts on central extensions which are needed in the following.Proofs can be found in [N2]. A central extension of L is an epimorphism f : K → L of Lie superalgebras with the property that Ker f ⊂ z ( K ), the centre of K . A centralextension f : K → L is called universal if for any other central extension f ′ : K ′ → L thereexists a unique Lie superalgebra morphism g : K → K ′ such that f = f ′ ◦ g . A universalcentral extension of L exists and is then unique up to a unique isomorphism if and onlyif L is perfect. To describe a model of a universal central extension of L one can use thefollowing construction of a Lie superalgebra which is valid for any, not necessarily perfectLie superalgebra L .Let B = B L be the S -submodule of the S -supermodule L ⊗ S L spanned by all elementsof type x ⊗ y + ( − | x || y | y ⊗ x, x ¯0 ⊗ x ¯0 for x ¯0 ∈ L ¯0 , ( − | x || z | x ⊗ [ y, z ] + ( − | y || x | y ⊗ [ z, x ] + ( − | z || y | z ⊗ [ x, y ] , and put uce ( L ) = ( L ⊗ S L ) / B and h x, y i = x ⊗ y + B ∈ uce ( L ) . The supermodule uce ( L ) becomes a Lie superalgebra over S with respect to the product (cid:2) h l , l i , h l , l i (cid:3) = (cid:10) [ l , l ] , [ l , l ] (cid:11) for l i ∈ L . The map(1.1) u = u L : uce ( L ) → L : h x, y i 7→ [ x, y ]is a Lie superalgebra morphism with kernel Ker u ⊂ z ( uce ( L )). If L is perfect, then u : uce ( L ) → L is a universal central extension of L . A morphism of Lie superalgebras f : L → M gives rise to a morphism of Lie superalgebras uce ( f ) : uce ( L ) → uce ( M ) : h l , l i 7→ h f ( l ) , f ( l ) i . The assignments L uce ( L ) and f uce ( f ) define a covariant endo-functor on the category Lie S of Lie S -superalgebras.Similar to Lie algebras, a central extension of a Lie S -superalgebra L can be constructedby using a 2-cocycle τ : L × L → C . Here C is a S -supermodule, τ is S -bilinear ofdegree 0 whence τ ( L α , L β ) ⊂ C α + β for α, β ∈ Z / Z , alternating in the sense that τ ( x, y ) +( − | x || y | τ ( y, x ) = 0 = τ ( x ¯0 , x ¯0 ) for x ¯0 ∈ L ¯0 , and satisfies( − | x || z | τ ( x, [ y, z ]) + ( − | y || x | τ ( y, [ z, x ]) + ( − | z || y | τ ( z, [ x, y ]) = 0 . NIVERSAL CENTRAL EXTENSIONS OF DIRECT LIMITS OF LIE SUPERALGEBRAS 3
Equivalently, a 2-cocycle is a map τ : L × L → C such that L ⊕ C is a Lie superalgebra withrespect to the grading ( L ⊕ C ) α = L α ⊕ C α and product [ l ⊕ c , l ⊕ c ] = [ l , l ] L ⊕ τ ( x , x )where [ ., . ] L is the product of L . In this case, the canonical projection L ⊕ C → L is acentral extension. We recall some notions regarding direct limits. Let ( I, ≤ )be a directed set, which will be fixed throughout this section. A directed system is a family( L i : i ∈ I ) in Lie S together with Lie superalgebra morphisms f ji : L i → L j for every pair( i, j ) with i ≤ j such that f ii = Id L i and f ki = f kj ◦ f ji for i ≤ j ≤ k . A direct limit of thedirected system ( L i , f ji ) is a Lie superalgebra L together with Lie superalgebra morphisms ϕ i : L i → L satisfying ϕ i = ϕ j ◦ f ji , and for any other such pair ( Y, ψ i ), i.e., ψ i = ψ j ◦ f ji for i ≤ j , there exists a unique morphism ϕ : L → Y such that the following diagramcommutes. L i f ji / / ϕ i (cid:31) (cid:31) @@@@@@@@ ψ i (cid:28) (cid:28) L jϕ j (cid:127) (cid:127) ~~~~~~~ ψ j (cid:2) (cid:2) L ϕ (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) Y The usual construction of a direct limit of modules shows that a direct limit of Lie super-algebras exists in
Lie S and is unique, up to a unique isomorphism. We can therefore speakof “the” direct limit, and follow the usual abuse of notation and denote a direct limit of( L i , f ji ) by lim −→ L i . We will call ϕ i the canonical maps .Let ( K i , g ji ) and ( L i , f ji ) be two directed systems of Lie superalgebras, both indexed bythe directed set I . A morphism from ( K i , g ji ) to ( L i , f ji ) is a family ( h i : i ∈ I ) of Liesuperalgebra morphisms h i : K i → L i such that for all pairs ( i, j ) with i ≤ j the diagram K i g ji / / h i (cid:15) (cid:15) K jh j (cid:15) (cid:15) L i f ji / / L j commutes. A morphism from ( K i , g ji ) to ( L i , f ji ) gives rise to a unique Lie superalgebramorphism h = lim −→ h i : lim −→ K i −→ lim −→ L i such that h ◦ ϕ i = ψ i ◦ h i for all i ∈ I , where ϕ i : K i → lim −→ K i and ψ i : L i → lim −→ L i are thecanonical maps. Since direct limits preserve exact sequences [Bo, II, § h is injective (respectively surjective) if all h i are injective (respectively surjective). Let ( L i , f ji ) be a directed system of Lie superalgebras in Lie S and let lim −→ L i be itsdirect limit with canonical maps ϕ i : L i → lim −→ L i . Since uce is a covariant functor, itis immediate that ( uce ( L i ) , uce ( f ji )) is also a directed system of Lie superalgebras. Weabbreviate uce ( f ji ) by b f ji , and let e ϕ i : uce ( L i ) → lim −→ uce ( L i ) be the canonical maps into the ERHARD NEHER AND JIE SUN direct limit of ( uce ( L i ) , b f ji ).(1.2) L i f ji / / ϕ i " " DDDDDDDD L jϕ j | | zzzzzzzz lim −→ L i uce ( L i ) b f ji / / e ϕ i & & LLLLLLLLLL uce ( L j ) e ϕ j x x rrrrrrrrrr lim −→ uce ( L i )Let u i : uce ( L i ) → L i be the Lie superalgebra morphism of (1.1). By construction of themaps b f ji , we have a commutative diagram(1.3) uce ( L i ) b f ji / / u i (cid:15) (cid:15) uce ( L j ) u j (cid:15) (cid:15) L i f ji / / L j for i ≤ j . In other words, the family ( u i , i ∈ I ) is a morphism from the directed system( uce ( L i ) , b f ji ) to the directed system ( L i , f ji ), and therefore gives rise to a morphism(1.4) lim −→ u i : lim −→ uce ( L i ) → lim −→ L i . Lemma 1.4.
In the setting of , the map (1.4) has central kernel, and is a centralextension if all L i are perfect.Proof. To prove that v := lim −→ u i has central kernel, let x ∈ Ker v . Thus x = e ϕ j ( x j ) for some x j ∈ uce ( L j ) and 0 = v ( x ) = ϕ j ( u j ( x j )) in L = lim −→ L i . Hence there exists k ≥ j such that f kj ( u j ( x j )) = 0 ∈ L k . Note ϕ k ( f kj ( u j ( x j ))) = 0 ∈ L . For any y ∈ L , we have to show that[ x, y ] = 0 in L . We have y = e ϕ p ( y p ) for some y p ∈ uce ( L p ). For the above k, p ∈ I thereexists q ∈ I such that q ≥ k ≥ j and q ≥ p . Thus f qj ( u j ( x j )) = ( f qk ◦ f kj )( u j ( x j )) = 0 ∈ L q .The commutative diagram (1.3) for j ≤ q now implies b f qj ( x j ) ∈ Ker u q ⊂ z ( uce ( L q )). So wehave [ b f qj ( x j ) , b f qp ( y p )] uce ( L q ) = 0 ∈ uce ( L q ) and hence[ x, y ] lim −→ uce ( L i ) = [ e ϕ j ( x j ) , e ϕ p ( y p )] lim −→ uce ( L i ) = e ϕ q ([ b f qj ( x j ) , b f qp ( y p )] uce ( L q ) ) = 0 . Thus Ker v ⊂ z (lim −→ uce ( L i )). If all L i are perfect, every u i is surjective, and hence so is v ,proving that v is a central extension. (cid:3) We continue with the setting of 1.3, but assume that every L i is perfect. Then L =lim −→ L i is perfect too and therefore has a universal central extension u : uce ( L ) → L . Ourgoal is to prove that the central extension (1.4) is a universal central extension of L . Bythe construction of L , the canonical maps ϕ i : L i → L are Lie superalgebra morphisms.We therefore get a unique Lie superalgebra morphism b ϕ i : uce ( L i ) → uce ( L ) such that thefollowing diagram commutes(1.5) uce ( L i ) b ϕ i / / u i (cid:15) (cid:15) uce ( L ) u (cid:15) (cid:15) L i ϕ i / / L where u i and u are universal central extensions of L i and L respectively. Applying thecovariant functor uce to the left commutative diagram in (1.2) shows that ˆ ϕ i = uce ( ϕ i ) = NIVERSAL CENTRAL EXTENSIONS OF DIRECT LIMITS OF LIE SUPERALGEBRAS 5 uce ( ϕ j ◦ f ji ) = uce ( ϕ j ) ◦ uce ( f ji ) = ˆ ϕ j ◦ ˆ f ji . Thus the outer triangle in the diagram be-low commutes. Hence, by the universal property of lim −→ uce ( L i ), there exists a unique Liesuperalgebra morphism ϕ : lim −→ uce ( L i ) → uce ( L ) such that all triangles commute.(1.6) uce ( L i ) b f ji / / e ϕ i & & LLLLLLLLLL b ϕ i ! ! uce ( L j ) e ϕ j x x rrrrrrrrrr b ϕ j } } lim −→ uce ( L i ) ϕ (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) uce ( L )For the next theorem we define H ( L ) for a perfect Lie superalgebra L as the kernel of u : uce ( L ) → L . In case L is a perfect Lie algebra over a ring S it is known that H ( L ) isthe second homology group of L with trivial coefficients. Theorem 1.6.
Assume that all Lie superalgebras L i are perfect. Then the map ϕ : lim −→ uce ( L i ) → uce (lim −→ L i ) of (1.6) is an isomorphism of Lie superalgebras, and hence lim −→ u i : lim −→ uce ( L i ) → lim −→ L i isa universal central extension. In particular, lim −→ u i induces an isomorphism (1.7) lim −→ H ( L i ) ∼ = H (lim −→ L i ) . Proof.
We have already noted that L is perfect and therefore has a universal central exten-sion u : uce ( L ) → L . By Lemma 1.4 we know that v = lim −→ u i : lim −→ uce ( L i ) → lim −→ L i is acentral extension. Thus the universal property of uce ( L ) implies that there exists a uniqueLie superalgebra morphism ψ : uce ( L ) → lim −→ uce ( L i ) such that the following diagram com-mutes. uce ( L ) ψ / / u " " EEEEEEEEE lim −→ uce ( L i ) v z z uuuuuuuuuu L We claim that ψ ◦ ϕ = Id lim −→ uce (L i ) and ϕ ◦ ψ = Id uce (L) . For the proof of these two equations,the following diagram may be helpful. uce ( L i ) e ϕ i / / u i (cid:15) (cid:15) b ϕ i ) ) lim −→ uce ( L i ) ϕ / / v & & LLLLLLLLLLLL uce ( L ) ψ o o u (cid:15) (cid:15) L i ϕ i / / L By the universal property of lim −→ uce ( L i ), in order to show ψ ◦ ϕ = Id lim −→ uce (L i ) , we only needto check ( ψ ◦ ϕ ) ◦ e ϕ i = e ϕ i . Since ϕ ◦ e ϕ i = b ϕ i , we are left to check ψ ◦ b ϕ i = e ϕ i and this istrue by the observation v ◦ ψ ◦ b ϕ i = u ◦ b ϕ i = ϕ i ◦ u i = v ◦ e ϕ i and the uniqueness in [N2,Prop. 1.13]. For the proof of ϕ ◦ ψ = Id uce (L) it is in view of the universal property of uce ( L )enough to verify u ◦ ( ϕ ◦ ψ ) = u . Since u = v ◦ ψ , we are left to check u ◦ ϕ = v and thisfollows from u ◦ ϕ ◦ e ϕ i = u ◦ b ϕ i = ϕ i ◦ u i = v ◦ e ϕ i . ERHARD NEHER AND JIE SUN
For the proof of (1.7) it suffices to note that (Ker u i , b f ji | Ker u i ) is a directed system andthat 0 → Ker u i → uce ( L i ) → L i → i ∈ I . The claim then follows from the fact that direct limits preserveexact sequences. (cid:3) Remark 1.7.
Theorem 1.6 is proven in [S, App.] for the case I = N with the natural orderand a directed system of Lie algebras over algebraically closed fields of characteristic zero L f −→ L f −→ · · · satisfying the condition that all f i are monomorphisms with f i (cid:0) z ( L i ) (cid:1) ⊂ z ( L i +1 ). Remark 1.8.
We note that for Lie algebras the formula (1.7) is not new. Indeed, by [Wei,Cor. 7.3.6] the homology groups of a Lie algebra can be interpreted as torsion groups forthe universal enveloping algebra and it is known that torsion commutes with direct limits,see e.g., [Wei, Cor. 2.6.17]. The proof presented here is more direct and works in the supersetting as well.For the next corollary we recall that a perfect Lie superalgebra is called centrally closed if u : uce ( L ) → L is an isomorphism. Corollary 1.9. If ( L i , f ji ) is a directed system of perfect and centrally closed Lie superal-gebras, then lim −→ L i is perfect and centrally closed. Examples: Universal central extensions of some infinite rank Liesuperalgebras
In this section we will consider some examples of universal central extensions of directlimits Lie superalgebras, mainly those which are direct limits of some of the classical Liesuperalgebras. In order to use the known results on their universal central extensions,we will in this section assume that all Lie superalgebras are defined over a commutative,associative, unital ring k , rather than an arbitrary base superring as in Section 1. Example 2.1 (Special linear Lie superalgebra sl ( I ; A ) for A an associative superalgebra) . Let I = I ¯0 ∪ I ¯1 be a superset, i.e., a partitioned set. Let A be a unital associative, butnot necessarily commutative k -superalgebra. We denote by Mat( I ; A ) the associative k -superalgebra whose underlying module consists of | I |×| I | -finitary matrices with entries from A (only finitely many non-zero entries) and Z -grading given by | E ij ( a ) | = | i | + | j | + | a | .Here E ij ( a ) ∈ Mat( I ; A ) has entry a at the position ( ij ) and 0 elsewhere. The productof Mat( I ; A ) is the usual matrix multiplication. Clearly Mat( I ; A ) only depends on thecardinality of I (and of course on A ). For a finite I we putMat( m, n ; A ) := Mat( I ; A ) if | I ¯0 | = m and | I ¯1 | = n .A matrix x ∈ Mat( m, n ; A ) written as(2.1) x = (cid:20) x x x x (cid:21) mnm n is then even (resp. odd) if x and x are matrices with even entries (resp. odd) entries and x and x are matrices with odd (resp. even) entries. NIVERSAL CENTRAL EXTENSIONS OF DIRECT LIMITS OF LIE SUPERALGEBRAS 7
We let gl ( I ; A ) be the Lie superalgebra associated to the associative superalgebra Mat( I ; A ).Its product is [ x, y ] = xy − ( − | x || y | yx . We assume | I | ≥ sl ( I ; A ) := [ gl ( I ; A ) , gl ( I ; A )]is perfect and satisfies sl ( I ; A ) = { x ∈ Mat( I ; A ) : str( x ) ∈ [ A, A ] } . Here the (super)tracestr of a matrix x = ( x ij ) ∈ Mat( I ; A ) is given by str( x ) = P i ∈ I ¯0 x ii − P i ∈ I ¯1 x ii , where[ A, A ] is the span of all commutators [ a , a ] = a a − ( − | a || a | a a , a i ∈ A . Observe thatthe Lie superalgebra sl ( I ; A ) is generated by matrices E ij ( a ), i = j ∈ I , a ∈ A , and thesegenerators satisfy the relations a E ij ( a ) is k -linear;(2.2) [ E ij ( a ) , E pq ( b )] = δ jp E iq ( ab ) − ( − | E ij ( a ) || E pq ( b ) | δ iq E pj ( ba )(2.3)For I as above we define the (linear) Steinberg Lie superalgebra st ( I ; A ) as the Lie k -superalgebra presented by generators e ij ( a ) with i, j ∈ I, i = j , a ∈ A and relations (2.2)–(2.3) with E ij ( a ) replaced by e ij ( a ). We then have a canonical Lie superalgebra epimorphism v I : st ( I ; A ) → sl ( I ; A ) , e ij ( a ) E ij ( a ) . If | I ¯0 | = m and | I ¯1 | = n , we put st ( m, n ; A ) = st ( I ; A ), sl ( m, n ; A ) = sl ( I ; A ) and v mn : st ( m, n ; A ) → sl ( m, n ; A ) for v I .We also need the first cyclic homology group HC ( A ). To define it, we use ≪ A, A ≫ = ( A ⊗ k A ) / H where H is the span of all elements of type a ⊗ b + ( − | a || b | b ⊗ a, a ¯0 ⊗ a ¯0 for a ¯0 ∈ A ¯0 , ( − | a || c | a ⊗ bc + ( − | b || a | b ⊗ ca + ( − | c || b | c ⊗ ab for a, b, c ∈ A . We abbreviate ≪ a, b ≫ = a ⊗ b + H . Observe that there is a well-definedcommutator map c : ≪ A, A ≫ → A, ≪ a, b ≫ 7→ [ a, b ] . We put HC ( A ) = Ker c = { P i ≪ a i , b i ≫ : P i [ a i , b i ] = 0 } We will use the following assumption:(2.4) v F for ≤ | F | < ∞ is a universal central extension with Ker v F ∼ = HC ( A ) . The assumption (2.4) is true in any one of the following situations:(a) n = 0, A an algebra [KL] or a superalgebra [CG],(b) A an algebra [MP1, IK1].The references [CG] and [IK1] assume that k is a commutative ring containing and thatthe underlying module of A is free with a basis containing the identity element of A . Wenote that (2.4) is not true for m + n ≤
4, see the papers [CG, G, GS, SCG] which deal withthe case 3 ≤ m + n ≤ Proposition 2.2.
Assume (2.4) holds and I is a (possibly infinite) set with | I | ≥ . Then v I : st ( I ; A ) → sl ( I ; A ) is a universal central extension with kernel isomorphic to HC ( A ) . ERHARD NEHER AND JIE SUN
Proof.
This can be proven by adapting the proof of (2.4) to our setting. Instead we preferto give a proof based on Theorem 1.6. This is possible since, denoting by F the set of finitesubsets of I ordered by inclusion, the Lie superalgebra sl ( I ; A ) is indeed a direct limit: sl ( I ; A ) = S F ∈F sl ( F ; A ) ∼ = lim −→ F ∈F sl ( F ; A ). Hence uce ( sl ( I ; A )) ∼ = lim −→ F ∈F uce ( sl ( F ; A )) ∼ =lim −→ F ∈F st ( F ; A ). Thus we need to show that lim −→ F ∈F st ( F ; A ) ∼ = st ( I ; A ). This followsfrom the diagram below, where ψ F is given by sending a generator e ij ( a ) ∈ st ( F ; A ) to e ij ( a ) ∈ st ( I ; A ). The existence of ϕ then follows from the definition of a direct limit,applied to ( ψ F ; F ∈ F ). The families ( e ij ( a ) ∈ st ( F ; A ) : F ∈ F ) give rise to elements e ij ( a ) ∈ lim −→ F ∈F st ( F ; A ) satisfying the relations (2.2)–(2.3), whence the existence of themap ψ sending e ij ( a ) ∈ st ( I ; A ) to e ij ( a ). st ( F ; A ) f F ′ F / / ϕ F & & NNNNNNNNNN ψ F " " st ( F ′ ; A ) ϕ F ′ x x ppppppppppp ψ F ′ | | lim −→ st ( F ; A ) ϕ (cid:15) (cid:15) st ( I ; A ) ψ O O It is immediate that ϕ and ψ are inverses of each other, and that Ker v I ∼ = HC ( A ). (cid:3) Example 2.3 ( sl ( I ; A ) for A an associative commutative superalgebra) . Let A be a unitalassociative and commutative k -superalgebra, thus [ A, A ] = 0. Therefore the descriptions of sl ( I ; A ) and HC ( A ) simplify to sl ( I ; A ) = { x ∈ gl ( I ; A ) : str( x ) = 0 } ∼ = sl ( I ; k ) ⊗ k A, HC ( A ) = ≪ A, A ≫ . Moreover, the universal central extension st ( I ; A ) can be described via a 2-cocycle as follows.The Lie superalgebra sl ( m, n ; A ) has a central 2-cocycle τ mn with values in HC ( A ): τ mn ( x, y ) = P ≤ i ≤ m, ≤ j ≤ m + n ≪ x ij , y ji ≫ − P m +1 ≤ i ≤ m + n, ≤ j ≤ m + n ≪ x ij , y ji ≫ for x = ( x ij ), y = ( y ij ) ∈ sl ( m, n ; A ). We let sl ( m, n, A ) ⊕ HC ( A ) be the corresponding Liesuperalgebra, and view it as a central extension of sl ( m, n ; A ) by projecting onto the firstfactor. From now on we suppose m + n ≥ h mn : uce ( sl ( m, n ; A )) → sl ( m, n ; A ) ⊕ HC ( A ) , h mn h x, y i = [ x, y ] ⊕ τ mn ( x, y )is an isomorphism of central extensions:(2.5) uce (cid:0) sl ( m, n ; A ) (cid:1) ∼ = sl ( m, n ; A ) ⊕ HC ( A ) as central extensions. The assumption (2.5) is true in any one of the following situations:(a) n = 0, A an algebra [KL] or a superalgebra [CG],(b) A an algebra [MP1, IK1].Let now ( sl ( m i , n i , A ) , f ji ) be a directed system of Lie superalgebras with m i + n i ≥ f ji : sl ( m i , n i , A ) → sl ( m j , n j , A ) lift uniquely to Lie superalgebramorphisms b f ji : uce ( sl ( m i , n i , A )) → uce ( sl ( m j , n j , A )). Hence, we get a directed system( sl ( m i , n i , A ) ⊕ HC ( A )) i ∈ I with transition maps h m j n j ◦ b f ji ◦ h − m i n i = f ji ⊕ g ji where NIVERSAL CENTRAL EXTENSIONS OF DIRECT LIMITS OF LIE SUPERALGEBRAS 9 the map g ji : HC ( A ) → HC ( A ) is given by g ji ( τ m i n i ( x, y )) = τ m j n j ( f ji ( x ) , f ji ( y )) for x, y ∈ sl ( m i , n i , A ). We now define a central 2-cocycle τ I for the direct limit Lie superalgebra sl ( I ; A ) = lim −→ sl ( m i , n i , A )with values in HC ( A ). Let x, y ∈ sl ( I ; A ). Thus x = ϕ p ( x p ) and y = ϕ q ( y q ) for some p, q ∈ I , x p ∈ sl ( m p , n p , A ) and y q ∈ sl ( m q , n q , A ). Here ϕ p , ϕ q are the canonical maps for sl ( I ; A ). There exists k ∈ I such that k ≥ p, k ≥ q and f kp ( x p ) , f kq ( y q ) ∈ sl ( m k , n k , A ).Then the cocycle τ I for sl ( I ; A ) is given by(2.6) τ I ( x, y ) = τ m k n k ( f kp ( x p ) , f kq ( y q )) . We now get from Proposition 2.2 that st ( I ; A ) ∼ = uce (cid:0) sl ( I ; A ) (cid:1) ∼ = sl ( I ; A ) ⊕ HC ( A ) , wherethe 2-cocycle τ I is given explicitly by (2.6). Summarizing the above, we have proven thefollowing. Corollary 2.4.
Let A be a unital associative commutative superalgebra over a commu-tative ring k . Let ( I, ≤ ) be an arbitrary directed set and let ( sl ( m i , n i , A ) , f ji ) be a di-rected system of Lie superalgebras with m i + n i ≥ . We suppose (2.5) and denote by sl ( I ; A ) := lim −→ sl ( m i , n i , A ) the corresponding direct limit, which is a perfect Lie superalge-bra of possibly infinite rank. Then (2.7) uce (cid:0) sl ( I ; A ) (cid:1) ∼ = sl ( I ; A ) ⊕ HC ( A ) as central extensions, where the Lie superalgebra structure on the right is given by the -cocycle τ I of (2.6) . Example 2.5 ( sl J ( A ) for A an associative algebra) . Let A be an associative unital k -algebra over a commutative ring k containing , and let J be an arbitrary, possible infiniteset with | J | ≥
5. We denote by sl J ( A ) the Lie algebra of finitary matrices over A (onlyfinitely many non-zero entries) and with trace in [ A, A ]. Since sl J ( A ) is the direct limit ofthe Lie algebras sl F ( A ) where F runs through the finite subsets of J , Corollary 2.4 impliesthat uce ( sl J ( A )) ∼ = sl J ( A ) ⊕ HC ( A ). This is proven in [KL] for J finite or countable andin [Wel] for arbitrary J , using the theory of root graded Lie algebras. Example 2.6 (Ortho-symplectic Lie superalgebra osp ( I ; A ) for A an associative commu-tative superalgebra) . The ortho-symplectic Lie superalgebra osp ( m, n ; A ) can be definedin the usual way, see for example [IK1, IK2, MP2]. Since osp ( m, n ; A ) is a subalgebraof sl ( m, n ; A ), the restriction of the 2-cocycle τ mn of Example 2.3 defines a 2-cocycle of osp ( m, n ; A ) with values in HC ( A ) and thus gives rise to a central extension. We supposethat the map uce ( osp ( m, n ; A )) → osp ( m, n ; A ) ⊕ HC ( A ), given by h x, y i 7→ [ x, y ] ⊕ τ mn ( x, y )is an isomorphism:(2.8) uce (cid:0) osp ( m, n ; A ) (cid:1) ∼ = osp ( m, n ; A ) ⊕ HC ( A ) as central extensions. Our assumption (2.8) is fulfilled in any one of the following situations:(a) k a field of characteristic 0 [IK2],(b) A a commutative algebra [IK1, MP2].The reference [MP2] assumes that m ≥ , n ≥ osp ( m i , n i ; A ) , f ji ) be a directed system of Lie superalgebras. One shows as inExample 2.3 that there exists a well-defined 2-cocycle τ ′ I for the direct limit Lie superalgebra osp ( I ; A ) := lim −→ osp ( m i , n i ; A ) with values in HC ( A ). From Theorem 1.6 we then get as in Corollary 2.4 uce (cid:0) osp ( I ; A ) (cid:1) ∼ = lim −→ uce (cid:0) osp ( m i , n i ; A ) (cid:1) ∼ = lim −→ (cid:0) osp ( m i , n i ; A ) ⊕ HC ( A ) (cid:1) ∼ = osp ( I ; A ) ⊕ HC ( A ) . (2.9) Example 2.7 (Locally finite Lie superalgebras) . Classically semisimple locally finite Liesuperalgebras over algebraically closed fields of characteristic 0 were introduced and studiedin [P], including a classification of the simple infinite dimensional ones which admit a localsystem of root injections of classical finite dimensional Lie superalgebras. They are alldirect limits L = lim −→ i L i of classical simple Lie superalgebras L i , i ∈ N with f i : L i → L i +1 being the natural inclusions. Referring the reader to [P] for details, we simple present theclassification list. We abbreviate sl ( m, n ) = sl ( m, n ; k ) and osp ( m, n ) = osp ( m, n ; k ) inthe notation of 2.3 and 2.6 respectively. The Lie superalgebra SP( m ) is the subalgebra of sl ( m, m ) which leaves invariant an odd nondegenerate super-antisymmetric bilinear form,and sq ( m ) is the subalgebra of sl ( m, m ) consisting of matrices of the form (2.1) with x = x , x = x and tr( x ) = 0. With these notations, an infinite dimensional simple Liesuperalgebra, which admits a local system of root injections of classical finite dimensionalLie superalgebras L i , is isomorphic to a Lie superalgebra L in the following table ( i ≥ , n ≥ , k ≥ , r ≥ , m ≥ L L i L L i sl ( ∞ , n ) sl ( i, n ) sl ( ∞ , ∞ ) sl ( i, i )B( ∞ , k ) osp (2 i + 1 , k ) B( ∞ , ∞ ) osp (2 i + 1 , i )B(2 r + 1 , ∞ ) osp (2 r + 1 , i ) C( ∞ ) osp (2 , i )D( ∞ , k ) osp (2 i, k ) D( ∞ , ∞ ) osp (2 i, i )D(2 m, ∞ ) osp (2 m, i ) SP( ∞ ) SP( i ) sq ( ∞ ) sq ( i ) Corollary 2.8.
The Lie superalgebras L listed in table (2.10) are all centrally closed.Proof. The Lie superalgebras L i in table (2.10) are all perfect. In view of Theorem 1.6 ittherefore remains to show that they are centrally closed for large i . For L i of type sl ( m, n )or osp ( m, n ) this follows from (2.5) and (2.8) since HC ( k ) = { } . For the remaining twotypes this follows from [IK2, Th. 5.10]. (cid:3) Example 2.9 (Locally finite Lie algebras) . Corollary 2.8 applies in particular to the simplelocally finite Lie algebras sl ( ∞ ) = sl ( ∞ , o ( ∞ ) = B( ∞ ,
0) = D( ∞ , sp ( ∞ ) = D(0 , ∞ )(the only infinite dimensional simple root reductive Lie algebras), studied for example in[BB], [DP] and [PS]. Example 2.10 (Root-graded Lie algebras) . (a) Let k be a ring in which 2 and 3 areinvertible, and let L be a Lie algebra graded by a locally finite reduced root system R asdefined in [N1], see also [N3, 5.1]. Thus, L is graded by Q ( R ) = Span Z ( R ) with supp Q ( R ) L = R , i.e., L = L α ∈ R L α , [ L α , L β ] ⊂ L α + β , satisfies L = P = α ∈ R [ L α , L − α ] and has the property that for α = 0 there exists an sl -triple ( e α , h α , f α ) ∈ L α × L × L − α such that [ h α , x β ] = h β, α ∨ i x β holds for every β ∈ R and x β ∈ L β . By [LN, 3.15], R is a direct limit of finite root systems, say R = lim −→ R i where i runs through a directed set ( I, ≤ ). The subalgebra(2.11) L i = (cid:0) L = α ∈ R i L α (cid:1) ⊕ P = α ∈ R i [ L α , L − α ]is graded by the root system R i , and it is immediate that L = lim −→ L i . NIVERSAL CENTRAL EXTENSIONS OF DIRECT LIMITS OF LIE SUPERALGEBRAS 11
Any root-graded Lie algebra is perfect, whence uce ( L ) ∼ = lim −→ uce ( L i ) by Theorem 1.6.In fact, something more precise is true. One knows that the universal central extensionof a root-graded Lie algebra is again graded by the same locally finite root system, [N3,Prop. 5.4]. Thus the root-graded Lie algebra uce ( L ) is a direct limit of root-graded Liealgebras, uce ( L ) ∼ = lim −→ uce ( L ) i , where uce ( L ) i is defined in the same way as L i .(b) Suppose in the following that k is a field of characteristic zero. If K is a centrelessLie algebra graded by a finite irreducible reduced root system, the group H ( K ) is knownto be the full skew-dihedral homology group HF( a ), where a is the coordinate algebra of K ,[ABG, Th. 4.13].Let now L be a centreless Lie algebra graded by a locally finite irreducible reduced rootsystem R of rank ≥
9. Then R = lim −→ R i where the R i are finite, irreducible, reduced, haverank ≥ R , [LN, 8.3]. It is moreover no harm to assumethat R ⊆ R i for some fixed 0 ∈ I . It then follows that the root-graded Lie algebras L i of(2.11) all have the same coordinate algebra a (this has also been noted by M. Yousofzadeh).Notice that if f ji ( z ( L i )) ⊂ z ( L j ), then lim −→ L i ∼ = lim −→ L i / z ( L i ). Hence uce ( L ) ∼ = uce (cid:0) lim −→ L i / z ( L i ) (cid:1) ∼ = lim −→ uce ( L i / z ( L i )) ∼ = lim −→ L i / z ( L i ) ⊕ HF( a ) ∼ = L ⊕ HF( a ) . (2.12)Special cases of root-graded Lie algebras are the so-called Lie tori, which occur as cores andcentreless cores of locally extended affine Lie algebras [MY, Nee]. More generally, the coresof affine reflection Lie algebras are root-graded Lie algebras of possibly infinite rank [N3,6.4, 6.5]. References [ABG] B. Allison, G. Benkart and Y. Gao, Central extensions of Lie algebras graded by finite root systems,Math. Ann. (2000), 499–527.[BB] Y. Bahturin and G. Benkart, Some constructions in the theory of locally finite simple Lie algebras,J. Lie Theory (2004), 243–270.[BE] G. Benkart and A. Elduque, Lie superalgebras graded by the root system A ( m, n ), J. Lie Theory (2003), 387–400.[Bo] N. Bourbaki, Alg`ebre. Chapitres 1 `a 3, Hermann, Paris 1970, xiii+635.[CG] H. Chen and N. Guay, Central extensions of matrix Lie superalgebras over Z / Z -graded algebras,preprint, December 2010.[DP] I. Dimitrov and I. Penkov, Locally semisimple and maximal subalgebras of the finitary Lie algebrasgl( ∞ ), sl( ∞ ), so( ∞ ), and sp( ∞ ), J. Algebra (2009), 2069–2081.[G] Y. Gao, On the Steinberg Lie algebras st ( R ), Comm. Algebra (10) (1993), 3691–3706.[GS] Y. Gao and S. Shang, Universal coverings of Steinberg Lie algebras of small characteristic, J. ofAlgebra (2007), 216–230.[GN] E. Garc´ıa and E. Neher, Tits-Kantor-Koecher superalgebras of Jordan superpairs covered by grids,Comm. Algebra (7) (2003), 3335–3375.[IK1] K. Iohara and Y. Koga, Central extensions of Lie superalgebras, Comment. Math. Helv. (2001),110–154.[IK2] , Second homology of Lie superalgebras, Math. Nachr. (9) (2005), 1041–1053.[KL] C. Kassel and J.-L. Loday, Extensions centrales d’alg`ebres de Lie, Ann. Inst. Fourier (Grenoble) (4) (1982), 119–142.[LN] O. Loos and E. Neher, Locally finite root systems, Mem. Amer. Math. Soc. (811) (2004), x+214.[MP1] A. V. Mikhalev and I. A. Pinchuk, Universal central extensions of the matrix Lie superalgebras sl ( m, n, A ), Contemporary Mathematics (2000), 111–125.[MP2] , Universal central extensions of Lie superalgebras, Journal of Mathematical Sciences (4)(2003), 1547–1560. [MY] J. Morita and Y. Yoshii, Locally extended affine Lie algebras, J. Algebra (2006), 59–81.[Nee] K.-H. Neeb, Unitary highest weight modules of locally affine Lie algebras, in Quantum affine algebras,extended affine Lie algebras, and their applications, Contemp. Math., Vol. 506, Amer. Math. Soc.,Providence, RI, 2010, 227–262.[N1] E. Neher, Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid, Amer.J. Math (1996), 439–491.[N2] , An introduction to universal central extensions of Lie superalgebras, Proceedings of the“Groups, rings, Lie and Hopf algebras” conference (St. John’s, NF, 2001), Math. Appl., Vol. 555,Kluwer Acad. Publ. Dordrecht, 2003, 141–166.[N3] , Extended affine Lie algebras and other generalizations – a survey, in “Trends and devel-opments in infinite dimensional Lie theory”, Progress in Mathematics, Vol. 288, Birkh¨auser, 2010,53–126.[P] I. Penkov, Classically semisimple locally finite Lie superalgebras, Forum Math. (2004), 431–446.[PS] I. Penkov and V. Serganova, Categories of integrable sl ( ∞ ) − , o ( ∞ ) − , sp ( ∞ ) − modules, Contempo-rary Mathematics, to appear.[S] H. Salmasian, Conjugacy of maximal toral subalgebras of direct limits of loop algebras, Contemp.Math., Vol. 490, Amer. Math. Soc., Providence, RI, 2009, 133–150.[SCG] S. Shang, H. Chen and Y. Gao, Central extensions of Steinberg superalgebras of small rank, Comm.Algebra (2007), 4225–4244.[Wei] C. Weibel, An introduction to homological algebra, Cambridge Studies in Advance Mathematics, Vol.38, Cambridge University Press, 1994.[Wel] A. Welte, Central extensions of graded Lie algebras, Ph.D. thesis, University of Ottawa, 2009. E. Neher: University of Ottawa, Ottawa, Ontario, Canada
E-mail address : [email protected] J. Sun: Department of Mathematics, University of California, Berkeley, CA 94720, USA
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