On the universal central extension of superelliptic affine Lie algebras
aa r X i v : . [ m a t h . R A ] D ec On the universal central extension of superelliptic affine Liealgebras
Felipe Albino dos Santos ∗ Dedicated to the memory of Ben Lewis Cox.
Abstract
Let p ( t ) ∈ C [ t ] be a polynomial with distinct roots. We describe in terms ofgenerators and relations the universal central extension for the infinite dimensionalsuperelliptic affine Lie algebras g ⊗ R with finite dimensional simple Lie algebra g ,whose coordinate ring is of the form R = C [ t, t − , u ] where u m = p ( t ). Introduction
Let g be a simple finite-dimensional complex Lie algebra and G = g ⊗ C [ t, t − ] the loopalgebra of g with commutation relations [ x ⊗ f, y ⊗ g ] = [ x, y ] ⊗ f g , where x, y ∈ g and f, g ∈ C [ t, t − ]. We will denote by ˆ G the universal central extension of G , which is theuntwisted affine Kac-Moody Lie algebra of g . In the construction of the loop algebra,we may replace the Laurent polynomial algebra C [ t, t − ] by any other commutativeassociative complex algebra, say R , and consider the universal central extension of g ⊗ R .When R is the ring of meromorphic functions on Riemann surface with a fixed numberof poles, the algebra ˆ G is called a current Krichever-Novikov algebra . These algebrashave been studied extensively (see, for example, the book [Sch14] and the referencestherein). The Krichever-Novikov algebras where introduced by Krichever and Novikovin their study of string theory in Minkowski space [KN87], [KN88].The ring of rational functions on the Riemann sphere regular everywhere except ata finite number of points appears in the study of the tensor module structures for affineLie algebras in Kazhdan and Luszig’s work (see [KL91] and [KL94]). These algebras arecalled N -point algebras and generalize the untwisted affine Kac-Moody Lie algebras.They are examples of Krichever-Novikov algebras for the genus zero. Bremner [Bre94]presented the generators and commutation relations of the universal central extensionof the N -point algebras.Date, Jimbo, Kashiwara and Miwa considered the universal central extension of g ⊗ C [ t, t − , u ] with u = ( t − b )( t − c ) where b ∈ C \ {− c, c } in their study ofLandau-Lifshitz equation [DJKM83]. The algebra above is called the DJKM algebra .This is an example of a Krichever-Novikov algebra with genus different from zero. Thereare interesting and fundamental work has been done by Cox, Futorny and others on ∗ Funding from CNPq process 142053/2017-1 is gratefully acknowledged. ellipticaffine Lie algebras , which are the universal central extension of the Lie algebras g ⊗ R with R = C [ t, t − , u ] and u = k ( t ) ∈ C [ t ] is an elliptic curve. These algebras werestudied by Bremner in [Bre94] and [Bre95], where the explicit description in terms ofgenerators, relations and families of polynomials (ultraspherical and Pollaczek) of thecommutation relations were given. In the case of Lie algebras of the form g ⊗ R where R is the ring of regular functions defined on an algebraic curve with any number ofpoints removed, Bremner computed the dimension of the associated universal centralextension. These results allowed to obtain the free field type realizations of the fourpoint and elliptic affine algebras (see [CJ14], [CF11], [Cox16]).Hyperelliptic affine Lie algebras form a family of Krichever-Novikov Lie algebraswith the hyperelliptic algebra R = C [ t, t − , u ] where u = k ( t ) ∈ C [ t ]. The hyperellipticcurves are the simplest case of superelliptic curves u m = k ( t ), with k ( t ) ∈ C [ t ] and m ≥
2. The superelliptic Lie algebra recently have been considered by Cox, Guo, Lu andZhao in [CGLZ17]. A natural question that arises on the geometric context of algebraiccurves is what of the already developed theory and applications of the hyperellipticcurves can be extended to the superelliptic curves (see [BSZ15] and [MS19]).Given that, by Kassel in [Kas84], the center C of ˆ G is linearly isomorphic to Ω R /dR ,the space of K¨ahler differentials of R modulo exact differentials. Bremner stated andanswered in [Bre94] three questions about the elliptic Lie algebras:(1) describe C , in particular determine its dimension;(2) find a basis for C , and(3) compute the universal cocycle ˆ G × ˆ G → C explicitly.The purpose of this paper is to generalize some of these results answering thesequestions for the superelliptic affine Lie algebras g ⊗ R with R = C [ t, t − , u ], where u m = P di =0 a i t i with a d = 1, at least one of a , a different from 0, and in which 0 hasmultiplicity ≤ Theorem:
1. The differentials t − dt , together with t − u l dt , . . . , t − d u l dt (where we omit t − d u l dt if a = 0), with l ∈ { , , . . . , m − } , give finite basis for Ω R /dR
2. For the resulting superelliptic Lie algebras, we computed the universal cocycleˆ
G × ˆ G → C explicitly.Applying strategies that could be found in [Bre94], we found the first part in Theorem1.12 and the second part in Theorem 2.11.In the last section, we give well known similar results on hyperelliptic Lie algebras.2 Universal Central Extensions.
Let L be a Lie algebra and an abelian Lie algebra K , a central extension ˆ L of L by K is a short exact sequence of Lie algebras0 −→ K i −→ ˆ L π −→ L −→ i ( K ) is central in ˆ L .A central extension ˆ L of L is said to be universal central extension if for everycentral extension ˆ L ′ of L , there exists a unique pair of Lie algebra homomorphism( φ, φ ) : ( K, ˆ L ) ( K ′ , ˆ L ′ ) such that0 K ˆ L L K ′ ˆ L ′ L iφ φ πi ′ π ′ commutes.We now consider rings R of the form C [ t, t − , u ] where u m ∈ C [ t ], with m ≥
2; thus R has a basis consisting of t i , t i u , t i u . . . , t i u m − for i ∈ Z . We will assume that u m = k ( t ) ∈ C [ t ] and that 0 has multiplicity ≤ k ( t ). We write k ( t ) = P di =0 a i t i where a d = 1 and a , a are not both 0. The equation u m = k ( t ) defines a superellipticcurve. If we write R i for C [ t, t − ] u i , then we see that R = R ⊕ R ⊕ · · · ⊕ R m − is a Z /m -graded ring.Let g be a simple finite-dimensional complex Lie algebra. The algebras G = g ⊗ R are examples of superelliptic loop algebras . The Z /m -grading induces the structure of a Z /m -graded Lie algebra on G by setting G i = g ⊗ R i ( i = 0 , , , . . . , m − G for the universal central extension of G , then as vector spaces we haveˆ G = G ⊕ C , where C is the kernel of the surjective homomorphism from ˆ G onto G .That means C is the center of ˆ G . By Kassel’s theorem [Kas84], the kernel C is linearlyisomorphic to Ω R /dR , the space of K¨ahler differentials of R modulo exact differentials.Our goal is determine a basis for Ω R /dR .Let F = R ⊗ R be the left R -module with action f ( g ⊗ h ) = f g ⊗ h for f, g, h ∈ R .Let K be the submodule generated by the elements 1 ⊗ f g − f ⊗ g − g ⊗ f . ThenΩ R = F/K is the module of K¨ahler differentials. We denote the element f ⊗ g + K ofΩ R by f dg . We define a map d : R → Ω R by d ( f ) = df = 1 ⊗ f + K and we denote thecoset of f dg modulo dR by f dg . The commutation relations for ˆ G , the universal centralextension of G , are[ x ⊗ f, y ⊗ g ] = [ xy ] + ( x, y ) f dg, [ x ⊗ f, ω ] = 0 , [ ω, ω ′ ] = 0 (1.2)where x, y ∈ g , f, g ∈ R and ω, ω ′ ∈ Ω R /dR ; here ( x, y ) denotes the Killing form on g .All of these objects have a Z /m -grading induced by that on R .The elements t i u k ⊗ t j u l , with i, j ∈ Z and k, l ∈ { , , . . . , m − } form a basis of R ⊗ R . Lemma 1.3. Ω R is spanned by the differentials t i u k dt and t i u l du with i ∈ Z , k ∈{ , , . . . , m − } , and l ∈ { , , . . . , m − } . roof. We have to show that any basis element of R ⊗ R is congruent modulo K toan element in the span of t i u k ⊗ t and t i u l ⊗ u with i ∈ Z , k ∈ { , , . . . , m − } , and l ∈ { , , . . . , m − } . We easily show by induction that d ( t j u l ) = jt j − u l dt + lt j u l − du. (1.4)Since K is a submodule of R ⊗ R , we can multiply (1.4) by t i u k : t i u k d ( t j u l ) = jt i + j − u k + l dt + lt i + j u k + l − du. Since u m − du = m d ( u m ) ⇒ t i u m − du = m dt i + d − dt + m ( d − a d − t i + d − dt + · · · + m a t i dt. This shows that any element in the basis of R ⊗ R is equal to an element inthe span of t i u k dt , and t i u l du with k ∈ { , , . . . , m − } , and l ∈ { , , . . . , m − } . Lemma 1.5. Ω R is spanned by the differentials t i dt, t i udt, . . . , t i u m − dt , with i ∈ Z ,together with t d − u l du, . . . , tu l du, u l du (where we omit u l du if a = 0 ), with l ∈ { , , . . . , m − } .Proof. We have m ud ( u m ) = u m du . Since u m = P dk =0 a k t k we find that d X k =1 m ka k t k − udt − d X k =0 a k t k du = 0 . (1.6)We multiply equation (1.6) by t i to get d X k =1 m ka k t i + k − udt − d X k =0 a k t i + k du = 0 . (1.7)First assume that a = 0. For i ≥
0, formula (1.7) shows (since a d = 1) that t i + d du is equal to a linear combination of t i + d − du, . . . , t i du and elements of the form t j udt .For i ≤ − a = 0) that t i du is equal to a linear combination of t i +1 du, · · · , t i + d du and elements of the form t j udt . From this we show by inductionthat elements of the form t i du are equal to a linear combination of du, . . . , t d − du andelements of the form t j udt .If a = 0 then for i ≥ t i + d du is equal to a linear combination of t i + d − du, . . . ,t i +1 du and elements of the form t j udt , and for i ≤ −
1, since a = 0, t i +1 du is equal toa linear combination of t i +2 du, . . . , t i + d du and elements of the form t j udt . The rest ofthe argument is similar.We can multiply formula (1.6) by t i u l to get d X k =1 m ka k t i + k − u l dt − d X k =0 a k t i + k u l du = 0 . (1.8)Similarly, from this we show by induction that elements of the form t i u l du are equalto a linear combination of u l du, . . . , t d − u l du (where we omit u l du if a = 0), andelements of the form t j u l dt . Now Lemma 1.3 completes the proof.4 heorem 1.9 ([Bre94], Theorem 2.1) . The dimension of Ω R /dR is g + n − where g is the genus and n is the number of punctures As is well known, the genus of R is g = ( m ( d − − d − gcd ( m, d )) + 1 and thenumber of allowed poles is n = gcd( m, d ) + 1, if a = 0 and n = gcd( m, d ) + m if a = 0(details can be found in [Har77]). Since dim Ω R /dR = 2 g + n −
1, we will have that
Theorem 1.10. dim Ω R /dR = ( m ( d −
1) + 1 if a = 0 ,m ( d −
1) + 1 − ( m − if a = 0 . (1.11) Theorem 1.12.
A finite basis for Ω R /dR is given by t − dt , together with t − u l dt , . . . , t − d u l dt (where we omit t − d u l dt if a = 0 ), with l ∈ { , , . . . , m − } .Proof. The Z /m -grading of Ω R and dR gives Ω R /dR = L m − i =0 (Ω R ) i /d ( R i ) where if k ≥ R ) k /d ( R k ) = span h t j u k dt, t d − u k − du, . . . , tu k − du, u k − du | j ∈ Z i (we omit u k − du if a = 0) and (Ω R ) /d ( R ) := h t j dt | j ∈ Z i .We first consider the space (Ω R ) /d ( R ). We have d ( t i ) = it i − dt for all i ∈ Z . Fromthis we see that t i − dt ≡ dR ) for i = 0. Therefore (Ω R ) /d ( R ) is spanned by t − dt .Next, we consider the space (Ω R ) /d ( R ). This space is spanned by t i udt togetherwith t d − du, . . . , tdu (and du if a = 0). We have d ( t i u ) = it i − udt + t i du , and so t i du ≡ − it i − udt (mod dR ) . (1.13)Thus we only need to consider the elements t i udt . We will show that modulo dR each of these elements is congruent to a linear combination of the finite set listed in thestatement of Theorem 1.12.First suppose that a = 0. We have t i − udt ≡ − (1 /i ) t i du (mod dR ) for i = 0.By formula (1.7), we know that t i du is a linear combination of t i +1 du, . . . , t i + d du and t i udt, . . . , t i + d − dt . Using (1.13), we see that t i du , and hence also t i − udt , is congruentmodulo dR to an element in the span of t i udt, . . . , t i + d − udt . Now using induction,we see that for i ≤ − d −
1, the element t i − udt is congruent modulo dR to a linearcombination of t − d udt, . . . , t − udt .We also have t i + d − udt ≡ − (1 / ( i + d )) t i + d du (mod dR ) for i = − d . By for-mula (1.7) again we know that t i + d du is a linear combination of t i + d − du, . . . , t i du and t i + d − udt, . . . , t i udt . The coefficient of t i + d − udt in this linear combination is ( d/m );hence we can solve for t i + d − udt , showing that it is congruent modulo dR to a linearcombination of the same elements (excluding t i + d − udt ). By (1.13) we see that t i + d − udt is congruent modulo dR to a linear combination of t i + d − udt, . . . , t i − udt . Now setting j = i + d − j ≥ i ≥ − d + 1), the element t j udt is congruent modulo dR to a linear combination of t − udt, . . . , t − d udt .The proof in the case a = 0 (and a = 0) is similar.Then, we consider the spaces (Ω R ) l /d ( R l ) with l ∈ { , , . . . , m − } .Each (Ω R ) l /d ( R l ) is spanned by t i u l dt together with t d − u l − du, . . . , tu l − du (and u l − du a = 0). We have that d ( t i +1 u l ) = ( i + 1) t i u l dt + lt i +1 u l − du , then t i u l dt ≡ − li + 1 t i +1 u l − du (mod dR ) . (1.14)Thus we only need to consider elements t i u l du .Suppose that a = 0. We have t i u l − du ≡ ( − i/l ) t i − u l dt (mod dR ) for i = 0. By for-mula (1.8), we know that t i u l − du is a linear combination of t i +1 u l − du, . . . , t i + d u l − du and t i u l dt, . . . , t i + d − u l dt . Using (1.14), we see that t i +1 u l − du , and hence also t i u l dt ,is congruent modulo dR to an element in the span of t i u l dt, . . . , t i + d − u l dt . Now usinginduction, we see that for i ≤ − d −
1, the element t i − u l dt is congruent modulo dR toa linear combination of t − d u l dt, . . . , t − u l dt .We also have t i + d − u l dt ≡ − ( l/ ( i + d )) t i + d u l − du (mod dR ) for i = − d . By formula(1.8) again we know that t i + d u l − du is a linear combination of t i + d − u l − du, . . . , t i u l − du and t i + d − u l dt, . . . , t i u l dt . The coefficient of t i + d − u l dt in this linear combination is d/m ; hence we can solve for t i + d − u l − dt , showing that it is congruent modulo dR to alinear combination of the same elements (excluding t i + d − u l − dt ). By (1.14) we see that t i + d − u l dt is congruent modulo dR to a linear combination of t i + d − u l dt, . . . , t i − u l dt .Now setting j = i + d − j ≥ i ≥ − d + 1), the element t j u l dt is congruent modulo dR to a linear combination of t − u l dt, . . . , t − d u l dt .The proof in the case a = 0 (and a = 0) is similar.Then (Ω R ) l /d ( R l ) is spanned by t − u l dt , . . . , t − d u l dt (where we omit t − d u l dt if a = 0).The Theorem (1.10) completes the proof. Ω R /dR To make the commutation relations for ˆ G explicit we need to compute f dg for any basiselements f, g ∈ R . Note that f dg is always the linear combination of basis elements forΩ R /dR which gives the congruence class of f dg modulo dR . By Theorem 1 .
12 we knowthat the elements t − dt , together with t − u l dt , . . . , t − d u l dt (where we omit t − d u l dt if a = 0), with l ∈ { , , . . . , m − } give a basis for Ω R /dR . First we give an explicit description of the cocyles contributing to the even part of thesuperelliptic affine Lie algebra.Set ω = t − dt and ω i,j = t i u j dt for j = 0 . (2.1) Proposition 2.2 ([Bre94], Proposition 4.2) . For i, j ∈ Z one has t i d ( t j ) = jδ i + j, ω . Proposition 2.3.
For i, j ∈ Z and l ∈ { , , . . . , m − } we have t i u l d ( t j u l ) = (cid:18) j − i (cid:19) ω i + j − , l . (2.4)6 roof. The congruence follows from the relation t i u l d ( t j u l ) = jt i + j − u l dt + lt i + j u l − du ≡ jt i + j − u l dt − (cid:18) i + j (cid:19) t i + j − u l dt = (cid:18) j − i (cid:19) t i + j − u l dt. Lemma 2.5 ([CF11], Lemma 2.0.2) . If R = C [ t, t − , u ] with u m = k ( t ) ∈ C [ t ] withdegree d , then Ω R /dR , one has (( m + 1) d + mj ) t d + j − udt ≡ − d − X k =0 (( m + 1) k + mj ) a k t k + j − udt (mod dR ) . (2.6)Thus, d X k =0 (( m + 1) k + mj ) a k t k + j − udt ≡ dR ) . (2.7)We define the sequence of polynomials in d + 1 parameters Q k,l ( a d , a d − , . . . , a ) := t k u l dt for k ∈ Z , l ∈ N \ { } and a d , a d − , . . . , a ∈ C by d X k =0 (( m + 1) k + mj ) a k Q k,l ( a d , a d − , . . . , a ) ≡ dR ) . (2.8) Proposition 2.9.
Let Q k := Q k ( a d , a d − , . . . , a ) . For i, j ∈ Z we have t i u l d ( t j ) ≡ (cid:18) − jd + m ( i + j ) (cid:19) d − X k =0 a k Q k + d + i + j − ,l (mod dR ) . (2.10) Proof.
Given the equation (2.7), we have that(( m + 1) d + mj ) a k t d + j − udt ≡ − d − X k =0 (( m + 1) k + mj ) a k t k + j − udt (mod dR ) . If j = d + j −
1, then we have t j udt ≡ − (cid:18) d + m (1 + j ) (cid:19) d − X k =0 ( k + m ( k + j − d + 1)) a k t k + d + j − udt (mod dR ) . t i ud ( t j ) = jt i + j − udt , jt i + j − udt ≡ − (cid:18) jd + m (1 + ( i + j − (cid:19) d − X k =0 ( k + m ( k + ( i + j − − d + 1)) a k · t k + d +( i + j − − udt (mod dR ) , andt i ud ( t j ) ≡ (cid:18) − jd + m ( i + j ) (cid:19) d − X k =0 ( k + m ( k + i + j − d )) a k t k + d + i + j − udt (mod dR ) . We can now give explicit commutation relations for ˆ G . Corollary 2.11.
The superelliptic affine Lie algebra ˆ G has a Z /m -grading in which ˆ G = g ⊗ C [ t, t − ] ⊕ C ω , ˆ G l = g ⊗ C [ t, t − ] u l d M n =1 C ω l,n . The subalgebra ˆ G is an untwisted affine Kac-Moody Lie algebra with commutation re-lations [ x ⊗ t i , y ⊗ t j ] = [ x, y ] ⊗ δ i + j, ( x, y ) jω . The commutation relations are [ x ⊗ t i u l , y ⊗ t j u l ] = [ x, y ] ⊗ ( t i + j u ) + ( x, y ) (cid:18) j − i (cid:19) ω i + j − , l , if l ≤ m − . When l > m − , [ x ⊗ t i u l , y ⊗ t j u l ] = [ x, y ] ⊗ d X k =0 a k t i + j + k u l − m ! + ( x, y ) (cid:18) j − i (cid:19) ω i + j − , l . The last commutation relation is [ x ⊗ t i u n , y ⊗ t j u ] = [ x, y ] ⊗ ( t i + j u n +1 ) + ( x, y ) (cid:18) − jd + m ( i + j ) (cid:19) d − X k =0 a k Q k + d + i + j − ,n . One might want to compare the previous results with well known examples of hyperel-liptic affine Lie algebras. 8 .1 The hyperelliptic case
We will consider rings R of the form C [ t, t − , u ] where u ∈ C [ t, t − ]. Furthermore p ( t ) = P di =0 a i t i ∈ C [ t, t − ] where a d = 1 and a , a are not both 0. The equation u = p ( t ) defines a hyperelliptic curve and we call g ⊗ R an hyperelliptic loop algebra . Theorem 3.1 ([Bre94], Theorem 3.4) . A basis of Ω R /dR is given by t − dt togetherwith t − udt . . . , t − d udt , where we omit t − d udt if a = 0 . This result was generalized by Theorem 1.12.From now on, we will set ω := t − dt ,ω − := t − udt, and ω + := t − udt. (3.2)When considering R = C [ t, t − , u ] where u = ( t − b )( t − c ) we define the Date-Jimbo-Kashiwara-Miwa algebra as g ⊗ R . The DJKM algebra is an example of hyper-elliptic loop algebra. Using Theorem 3.1, it was showed in [CF11] (Theorem 2.0.1) that { t − dt, t − udt, t − udt, t − udt, t − udt } is a basis for Ω R /dR . It could be verified usingTheorem 1.12. In [CF11], Cox and Futorny explicitly described in terms of generatorsand relations the universal central extension of g ⊗ R . Let Σ be a nonsingular compact complex algebraic curve of genus 1. Representing Σas the quotient of complex plane C by the lattice Λ = Z ⊕ Z λ with basis { , λ } where Imλ >
0. From now on we will restrict the discussion to R that are is the ring ofall meromorphic functions on Σ which are holomorphic outside the set { , µ } where µ = (1 + λ ). The ring R is a ring of elliptic functions. Proposition 3.3 ([Bre94], Proposition 4.1) . If b = − m/ (12 m − P ξ ∈ Λ \{ } ξ − ) ,then R ∼ = C [ t, t − , u ] where u = t − bt + t . We call the Lie algebra L ( g ) := g ⊗ R the elliptic loop Lie algebra . The universalcentral extension ˆ g of L ( g ), is called the elliptic affine Lie algebra .Bremner realized the universal central extension of L ( g ) and gave a description ofthe relations satisfied by the basis elements of ˆ g . He gave a description of the relationssatisfied by the basis elements of ˆ g , that could be found using Theorem 3.1 or 1.12.Before show the final result of [Bre95], we recall the 4-parameter Pollaczek polynomials P k ( b ) = P λk ( b ; α, β, γ ) that are defined as a family of polynomials satisfying the recursionformula( k + γ ) P k ( b ) = 2[( k + λ + α + γ − b + β ] P k − ( b ) − ( k + 2 λ + γ − P k − ( b ) . (3.4)Define two sequences of polynomials p k ( b ), q k ( b ) for k ∈ Z by t k − udt = p k ( b ) t k − udt + q k ( b ) t − udt. (3.5)9 emma 3.6 ([Bre94], Lemma 4.4) . The polynomials p k ( b ) and q k ( b ) are Pollaczekpolynomials for the parameter values λ = − / , α = 0 , β = − and γ = 1 / . Theinitial conditions are p ( b ) = 0 , p ( b ) = 1 , q ( b ) = 1 , and q ( b ) = 0 . (3.7)The final result of [Bre94] that gave the commutation relations for the elliptic affineLie algebra is Theorem 3.8 ([Bre94], Theorem 4.6) . The elliptic affine Lie algebra ˆ g has a Z / Z -grading in which ˆ g = g ⊗ C [ t, t − ] ⊕ C ω , ˆ g = g ⊗ C [ t, t − ] u ⊕ C ω − ⊕ C ω + . For x, y ∈ g the commutation relations defining ˆ g are [ x ⊗ t i − u, y ⊗ t j − ] = [ x, y ] ⊗ ( t i + j − − bt i + j + t i + j +1 ) + ( x, y ) − jbω , for i + j = 0 ( j − i ) ω , for | i + j | = 10 , for | i + j | ≥ . [ x ⊗ t i − u, y ⊗ t j u ] = [ x, y ] ⊗ t i + j − u + ( x, y ) j ( p | i + j | ( b ) ω + + q | i + j | ( b ) ω − ) . Let R a be the 4-point ring C [ s, s − , ( s − − , ( s − a ) − ], a ∈ C \ { , } , and S b = C [ t, t − , u ], where u = t − bt + 1 with b a complex number not equal to ± Proposition 3.9 ([Bre95], Proposition 1.1) . If b = ( a + 1) / ( a − with a ∈ C \ { , } ,then R a ∼ = S b and b = ± . Here g still a simple finite-dimensional complex Lie algebra. We call the Lie algebra L ( g ) := g ⊗ R a the . The universal central extension of L ( g ), iscalled the and it will be denoted ˆ g .Bremner realized the universal central extension of L ( g ) and gave a description ofthe realizations satisfied by the basis elements of ˆ g . Theorem 3.10 ([Bre95], Theorem 3.6) . The space Ω R a /dR a has basis { ω , ω − , ω + } . Bremner gave a description of the relations satisfied by the basis elements of ˆ g , thatare x ⊗ t n . x ⊗ t n u , ω , and ω ± . We recall the ultraspherical (Gegenbauer) polynomials P λk ( b ), which are defined to be the coefficient to t k in the Taylor series of P − λ ( b, z ) =(1 − bt + t ) − λ . Setting λ = − , P k ( b ) = P − / k ( b ) and P = P − / k ( b, z ). Define for b = ± Q k ( b ) := − P k + 2( b ) b − b . The final result of [Bre95] is10 orollary 3.12 ([Bre95], Theorem 3.6) . The 4-point affine Lie algebra ˆ g has a Z / Z -grading in which ˆ g = g ⊗ C [ t, t − ] ⊕ C ω , ˆ g = g ⊗ C [ t, t − ] u ⊕ C ω − ⊕ C ω + . For x, y ∈ g the commutation relations defining ˆ g are [ x ⊗ t i − , y ⊗ t j ] = [ x, y ] ⊗ t i + j − + δ i + j ω , [ x ⊗ t i − u, y ⊗ t j − u ] = [ x, y ] ⊗ ( t i + j − − bt i + j + t i + j +1 )+ ( x, y ) ω (cid:18) − jbδ i + j, + 12 ( j − i )( δ i + j, − + δ i + j, ) (cid:19) [ x ⊗ t i − u, y ⊗ t j ] = [ x, y ] ⊗ ( t i + j − u ) + ( x, y ) j (cid:16) Q i + j − ( b )( bω + + ω − ) δ i + j ≥ )+ ω ± δ i + j, ± + Q − i − j − ( b )( ω + + bω ) δ i + j ≤− (cid:17) . When considering R = C [ t, t − , u ] where u = t + 4 t we define the 3-point algebraas g ⊗ R . The 3-point loop algebra is an example of elliptic loop algebra. UsingTheorem 3.1, it was showed in [CJ14] (Proposition 2.2) that { t − dt, t − udt } is a basisfor Ω R /dR . It could be verified using Theorem 1.12. In [CJ14], Cox and Jurisichdescribed the universal central extension of the 3-point current algebra sl (2 , R ) andconstructed realizations of it in terms of sums of partial differential operators. References [Bre94] Murray Bremner. Universal central extensions of elliptic affine Lie algebras.
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Felipe Albino dos Santos,
Departamento de Matem´atica, Universidade de S˜ao Paulo. S˜aoPaulo - SP, Brasil.
E-mail address : [email protected]@gmail.com