The second homology group of current Lie algebras

THE SECOND HOMOL OGY GR OUP OF CURRENT LIE ALGEBRAS P A UL ZUSMANO VICH 0. Introduction It is a w ell kno w n fact that the curren t Lie algebra G ⊗ C [[ t, t − 1 ]] asso ciated to a simple finite-dimensional Lie C -algebra G has a c en tral extens ion leading to the affine non-twisted Kac-Mo o dy algebra G ⊗ C [[ t, t − 1 ]] ⊕ C z with brack et { x ⊗ f , y ⊗ g } = [ x, y ] ⊗ f g + ( x, y ) R es d f dt g z where ( · , · ) is the Killing form on G (cf. [Kac]). In view of the kno wn relatio nship b et w e en cen tral extensions and the second (co)homology group with co efficien ts in the trivial mo dule, one of the main results of this pap er can b e considered as a generalization of this fact for general curren t Lie algebras, i.e., Lie algebras of the form L ⊗ A , where L is a Lie alg ebra a nd A is asso ciativ e comm utativ e algebra, equipp ed with brac k et [ x ⊗ a, y ⊗ b ] = [ x, y ] ⊗ ab. Theorem 0.1. L et L b e an arbitr ary Lie algebr a over a field K of char acteristic p 6 = 2 and A an as s o ciative c ommutative algeb r a with unit o v e r K . Then ther e is an isomorphism of K -ve ctor s p ac es: (0.1) H 2 ( L ⊗ A ) ≃ H 2 ( L ) ⊗ A ⊕ B ( L ) ⊗ H C 1 ( A ) ⊕ ∧ 2 ( L/ [ L, L ]) ⊗ K er ( S 2 ( A ) → A ) ⊕ S 2 ( L/ [ L, L ]) ⊗ T ( A ) wher e the mapping S 2 ( A ) → A induc e d by multiplic ation in A an d T ( A ) = h ab ∧ c + ca ∧ b + bc ∧ a | a, b, c ∈ A i . Here B ( L ) is the space o f coin v arian ts of the L -action on S 2 ( L ), H C 1 ( A ) is the first- order cyclic homology group of A , and ∧ 2 and S 2 denote the sk ew and symmetric pro ducts, resp ectiv ely . Not ice that in the case L = [ L, L ], the third and fourth terms in the rig h t-hand side of (0.1) v anish. Man y particular cases of this theorem w ere pro v ed b y differen t authors previously . An exhaustiv e description of all previous w orks on this theme ma y b e found in [H] and [S]. F or the first time, a cohomology form ula of the t yp e (0.1) has app eared in [S], where Theorem 0 .1 w a s prov ed assuming that L is 1- generated o ver an augmen tation ideal of its en v eloping algebra. A. Haddi [H] obtained a result similar to Theorem 0 .1 in the case where K is a field of c haracteristic zero (ho we ver, it seems that his argumen ts work ov er any field of c haracteristic p 6 = 2 , 3 ) . Our metho d of pro of differs from all previous ones and is based on the Hopf formula expressing H 2 ( L ) in terms of a presen tatio n 0 → I → L ( X ) → L → 0, where L = L ( X ) is the free Lie algebra o v er K freely generated b y the set X : (0.2) H 2 ( L ) ≃ ([ L , L ] ∩ I ) / [ L , I ] (see, fo r example, [K S ]). S. M. F. Ast´ erisque 2 26 (199 4 ). 1 2 P . ZUSMANOVICH The con ten ts of the pap er are as follo ws. § 1 is dev oted to some tec hnical preliminary results. In § 2 we determine the presen tation of a curren t Lie algebra L ⊗ A . In § 3 Theorem 0.1 is prov ed. As it corollary w e get in § 4 a description o f the space B ( L ⊗ A ). In § 5 a ”noncommu tativ e v ersion” of Theorem 0.1 is pro v ed (Th eorem 5.1). Namely , w e de riv e the formula for the second homology group of the Lie algebra ( A ⊗ B ) ( − ) , where A , B are asso ciativ e (noncommutativ e) algebras with unit, a nd (–) in superscript denotes passing to the asso ciated Lie algebra. The tec hnique used here is no longer based on the Hopf form ula, but on more or less direct computatio ns in some factorspaces of cycles. Ho wev er, argumen t s used in pro of, resem ble, to a great exten t , the previous ones. G etting a pa rticular case B = M n ( K ), w e recov er, after a sligh t mo dification, an isomorphism H 2 ( sl n ( A )) ≃ H C 1 ( A ) obtained in [KL]. The follo wing notatio nal con ven tion will b e used: the letters a, b, c, . . . , p o ssibly with sub- and sup erscripts, denote elemen ts o f algebra A , while letters u, v , w , . . . denote elemen ts of the free Lie algebra L ( X ) with the set of g enerators X = { x i } , if the otherwise is not stated. L n ( X ) denotes the n th term in the deriv ed series of L ( X ). The arro ws ֌ and ։ denote injection and surjection, respectiv ely . All o t her undefined notions and nota tion are standard, and ma y b e found, for example, in [F] for Lie algebra (co)homology , and in [LQ] for cyclic homology . In some places w e use diagram c hasing and 3 × 3- Lemma without explicitly men tio ning it. A cknowledgements This w ork w as started during m y graduate studies in the Ins tit ute of Mathematics and Mec hanics of the Kazakh Academ y of Sciences and finished in the Bar-Ilan Unive rsity . My thanks are due t o Ask ar Dzhumadil’daev and Stev en Shnider, m y fo rmer and presen t sup er- visors, for v arious sort of help, mathematical and other, and for interes t in m y w or k, a nd sp ecially t o Stev en Shnider for careful reading of preliminary vers io ns o f man uscript. Also the financial supp ort of Bar-Ila n Unive rsity and Bat -Shev a Ro t shild f oundation is gratefully ac know ledged. F inally , I am grateful to the organizers of the Conference o n K-Theory held in June-July 1 992 in Strasb ourg for inv itation and opp ortunit y to presen t t here the results of this pap er. 1. P reliminaries Lo oking at form ula (0.1), one can distinguish b etw een the first t w o “principal” terms and other tw o “non-principal” ones. In order to simplify calculations, we will obtain a v aria n t of the Hopf formu la leading to the app earance of “principal” terms only , and then the general case will b e derive d. Eac h nonp erfect Lie a lg ebra L , i.e., not coinciding with its commutan t [ L, L ], p ossesses a “trivial” homology classe s o f 2-cycles with co efficien ts in the mo dule K , namely , classes whose represen t ativ es do not lie in L ∧ [ L, L ]. More precisely , consider a natural homomor phism ψ : H 2 ( L ) → H 2 ( L/ [ L, L ]) ≃ ∧ 2 ( L/ [ L, L ]) and denote H ess 2 ( L ) = K er ψ , the homology classes of “essen tial” cycles. Lemma 1.1. One has an exact se quenc e 0 → H ess 2 ( L ) → H 2 ( L ) ψ → ∧ 2 ( L/ [ L, L ]) π → [ L, L ] / [[ L, L ] , L ] → 0 wher e π is in duc e d by multiplic ation in L . Pr o of. This is just an ob vious cons equence of a 5 -term exact sequenc e deriv ed from the Ho c hsc hild- Serre spectral sequence H n ( L/ [ L, L ] , H m ([ L, L ])) ⇒ H n + m ( L ).  F urther, w e need a vers ion of Hopf form ula for H ess 2 ( L ). Lemma 1.2. Given a pr e s e ntation 0 → I → L → L → 0 o f a Lie algebr a L , one has THE SECOND HOMOLO GY GROUP OF CURRENT LIE ALGEBRA S 3 (1.1) H ess 2 ( L ) ≃ L 3 ∩ I L 3 ∩ [ L , I ] . Pr o of. Since L/ [ L, L ] ≃ L / ( L 2 + I ), the Hopf form ula (0.2) b eing applied to the algebra L/ [ L, L ] g iv es H 2 ( L/ [ L, L ]) ≃ L 2 / [ L , L 2 + I ], and K er ψ = K er  L 2 ∩ I [ L , I ] → L 2 [ L , L 2 + I ]  ≃ L 2 ∩ I ∩ [ L , L 2 + I ] [ L , I ] ≃ L 3 ∩ I L 3 ∩ [ L , I ] .  No w consider an action of a Lie algebra L on S 2 ( L ) via [ z , x ∨ y ] = [ z , x ] ∨ y + x ∨ [ z , y ] . Let B ( L ) = S 2 ( L ) / [ L, S 2 ( L )] b e the space of coinv arian ts of this action. The dual B ( L ) ∗ is the space of symmetric bilinear in v aria nt forms on L . Let I , J b e ideals of L . Define B ( I , J ) to b e the space of coin v arian t s o f the action of L on I ∨ J . One has a natural em b edding B ( I , J ) → B ( L ). The natural map L ∨ J → ( L/I ) ∨ (( I + J ) /I ) defines a surjection B ( L, J ) → B ( L/I , ( I + J ) /I ). Lemma 1.3. The short se quenc e (1.2) 0 → B ( L, I ∩ J ) + B ( I , J ) → B ( L, J ) → B ( L/I , ( I + J ) /I ) → 0 is exact. Pr o of. Since K er ( L ∨ J → L/I ∨ ( I + J ) /I ) = L ∨ ( I ∩ J ) + I ∨ J , the factorization through [ L, S 2 ( L )] yields K er ( B ( L, J ) → B ( L/I , ( I + J ) /I )) = ( L ∨ ( I ∩ J ) + I ∨ J + [ L, S 2 ( L )]) / [ L, S 2 ( L )] ≃ B ( L, I ∩ J ) + B ( I , J ) .  R emark . Actually w e need the follo wing tw o cases o f this Lemma: (1) J = [ L, L ]. Since I ∨ [ L, L ] and [ I , L ] ∨ L are congruen t mo dulo [ L, S 2 ( L )] and [ I , L ] ⊆ I ∩ [ L, L ], then B ( I , [ L, L ]) ⊆ B ( L, I ∩ [ L, L ]) and w e get a short exact sequenc e (1.3) 0 → B ( L, I ∩ [ L, L ]) → B ( L, [ L, L ]) → B ( L/I , [ L/I , L/I ]) → 0 . (2) I = [ L, L ] and J = L . Then taking into accoun t tha t for an ab elian Lie algebra M , B ( M ) ≃ S 2 ( M ), the short exact sequenc e (1.2) b ecomes (1.4) 0 → B ( L, [ L, L ]) → B ( L ) → S 2 ( L/ [ L, L ]) → 0 . 2. P resent a tion of L ⊗ A In this section starting from a presen tation of L w e construct a presen tation of L ⊗ A . Let 0 → I → L ( X ) p → L → 0 b e a presen ta t io n of the Lie algebra L . T ensoring b y A , w e get a short exact sequence (2.1) 0 → I ⊗ A → L ( X ) ⊗ A p ⊗ 1 → L ⊗ A → 0 . Let X ( A ) b e a set of sym b ols x ( a ) , x ∈ X , a ∈ A . D efine a homomorphism φ : L ( X ( A )) → L ( X ) ⊗ A by φ : u ( x 1 ( a 1 ) , . . . , x n ( a n )) 7→ u ( x 1 , . . . , x n ) ⊗ a 1 . . . a n . 4 P . ZUSMANOVICH Ob viously this mapping is surjectiv e, a nd taking in to accoun t (2.1), g iv es rise to the fo llo wing exact sequence: (2.2) 0 → φ − 1 ( I ⊗ A ) → L ( X ( A )) ( p ⊗ 1) ◦ φ − → L ⊗ A → 0 whic h give s the presen ta tion of L ⊗ A . In order to determine the structure of φ − 1 ( I ⊗ A ), let us in tro duce one notation. F or eac h homogeneous elemen t u = u ( x 1 , . . . , x n ) of L ( X ), define u ( a ) to b e u ( x 1 ( a ) , x 2 (1) , . . . , x n (1)). No w having an a rbitrary elemen t u ∈ L ( X ), define u ( a ) as u 1 ( a ) + · · · + u k ( a ), where u = u 1 + · · · + u k is decomp osition of u in to the sum of homogeneous comp onen ts. Lemma 2.1. (1) K er φ is line arly g e ner ate d by elemen ts o f the form (2.3) X j u ( x i 1 ( a j 1 ) , . . . , x i n ( a j n )) wher e u ( x i 1 , . . . , x i n ) is homo gene ous element of L ( X ) and P j a j 1 . . . a j n = 0 . (2) φ − 1 ( I ⊗ A ) is line arly gener ate d mo dulo K er φ by elements of the form u ( a ) , w her e u ∈ I . Pr o of. (1) Eviden tly each elemen t of L ( X ( A )) ma y b e ex pressed as a sum o f elemen ts o f the for m u ( a ) and elemen ts of t he form (2.3), the latter lying in K er φ . T o prov e that they exhaust all K er φ , tak e a nonzero elemen t P i P j u i ( a ij ) b elonging to K er φ , where u i ’s are linearly independen t , and o bt a in P i P j u i ⊗ a ij = 0, whic h implies P j a ij = 0 for eac h i . (2) The factorspace φ − 1 ( I ⊗ A ) /K er φ , consisting from cosets u ( a ) + K er φ , maps onto I ⊗ A , whence t he conclusion.  W e a lso need the follow ing tec hnical result. Lemma 2.2. F or any u, v , w ∈ L ( X ) and a, b, c ∈ A , the elements [[ w , u ]( a ) , v ( b )] − [[ w, u ]( b ) , v ( a )] + [[ w , v ]( a ) , u ( b )) − [[ w , v ]( b ) , u ( a )] and [[ u, v ]( ab ) , w ( c )] − [[ u, v )( c ) , w ( ab )] +[[ u, v ]( c a ) , w ( b )] − [[ u, v ]( b ) , w ( ca )] +[[ u, v ]( bc ) , w ( a )] − [[ u, v ]( a ) , w ( bc )] b elon g to [ L ( X ( A )) , K er φ ] . Pr o of. Consider t he first case only , the second one is analogous. W e ha ve mo dulo [ L ( X ( A )) , K er φ ]: [[ w , u ]( a ) , v ( b )] − [[ w , u ]( b ) , v ( a )] + [[ w , v )( a ) , u ( b )] − [[ w , v ]( b ) , u ( a )] ≡ [[ w (1) , u ( a )] , v ( b )] + [[ v ( b ) , w (1)] , u ( a )] + [[ w (1) , v ( a )] , u ( b )] + [[ u ( b ) , w (1)] , v ( a )] ≡ − [[ u ( a ) , v ( b )] , w (1)] + [[ u ( b ) , v ( a )] , w (1)] ≡ 0  THE SECOND HOMOLO GY GROUP OF CURRENT LIE ALGEBRA S 5 3. The second homology of L ⊗ A The a im of this section is to prov e Theorem 0 .1. Consider the following comm utat iv e dia g ram with exact ro ws a nd columns, where φ − 1 stands f or φ − 1 ( I ⊗ A ) (w e will use this no tation in some places further): 0 0   y   y 0 − → L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ] ∩ K e rφ − → L 3 ( X ( A )) ∩ K er φ   y   y 0 − → L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ] − → L 3 ( X ( A )) ∩ φ − 1 − → H ess 2 ( L ⊗ A ) − → 0 φ   y φ   y 0 − → ( L 3 ( X ) ⊗ A ) ∩ [ L ( X ) ⊗ A, I ⊗ A ] − → ( L 3 ( X ) ⊗ A ) ∩ ( I ⊗ A )   y   y 0 0 The middle ro w follo ws from the Lemma 1.2 applied to the presen tation (2.2). Completing this diagram t o the third column, w e get a short exact seque nce (3.1) 0 → L 3 ( X ( A )) ∩ K er φ L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ( I ⊗ A )] ∩ K er φ → H ess 2 ( L ⊗ A ) → L 3 ( X ) ∩ I L 3 ( X ) ∩ [ L ( X ) , I ] ⊗ A → 0 . According to Lemm a 1.2, the righ t term here is nothing but H ess 2 ( L ) ⊗ A . Let us compute the left term. Let F ( Y ) b e a free ske wcomm utative algebra on an alpha b et Y with nonasso ciative pro duct denoted b y [ · , · ]. Define a mapping α : F 2 ( X ( A )) → S 2 ( F ( X )) ⊗ ( A ∧ A ) b y (3.2) α : [ u ( x 1 ( a 1 ) , . . . , x n ( a n )) , v ( x 1 ( b 1 ) , . . . , x m ( b m ))] 7→ ( u ( x 1 , . . . , x n ) ∨ v ( x 1 , . . . , x m )) ⊗ ( a 1 . . . a n ∧ b 1 . . . b m ) . (recall that F 2 ( Y ) is just [ F ( Y ) , F ( Y )]). It is easy to see that this mapping is w ell defined and surjectiv e. Let J ( Y ) b e an ideal of F ( Y ) generated b y eleme n ts of the fo r m [[ u, v ] , w ] + [[ w , u ] , v ] + [[ v , w ] , u ] , u, v , w ∈ F ( Y ) suc h that F ( Y ) /J ( Y ) ≃ L ( Y ) . Lemma 3.1. α ( J ( X ( A ))) = ( J ( X ) ∨ F ( X ) + [ F ( X ) , S 2 ( F ( X ))]) ⊗ ( A ∧ A ) + ( F 2 ( X ) ∨ F ( X )) ⊗ T ( A ) . Pr o of. W riting the generic ele men t in J ( X ( A )), it is easy to see, b y considering graded degree, t hat eve r y elemen t in α ( J ( X ( A ))) can b e written as a sum of an elemen t lying in ( J ( X ) ∨ F ( X )) ⊗ ( A ∧ A ) and an elemen t of the form (3.3) ([ u, v ] ∨ w ) ⊗ ( ab ∧ c ) + ([ w , u ] ∨ v ) ⊗ ( ca ∧ b ) + ([ v , w ] ∨ u ) ⊗ ( bc ∧ a ) for certain u, v , w ∈ F ( X ) and a, b, c ∈ A . Substituting in (3.3) b = c = 1 , w e get an elemen t ([ u, v ] ∨ w + [ w , u ] ∨ v − [ v , w ] ∨ u ) ⊗ (1 ∧ a ) . No w p erm uting the letters u, v in the last expression, one easily get 6 P . ZUSMANOVICH ( F 2 ( X ) ∨ F ( X )) ⊗ (1 ∧ A ) ⊂ α ( J ( A ( X ))) . Substituting in (3.3) c = 1 and taking in to accoun t the last relation, w e get (3.4) ([ w , u ] ∨ v + u ∨ [ w , v ]) ⊗ ( A ∧ A ) ⊂ α ( J ( A ( X ))) . An y elemen t in (3 .3) is cong r uent mo dulo (3.4) to an elemen t of the f o rm ( F 2 ( X ) ∨ F ( X )) ⊗ ( ab ∧ c + ca ∧ b + bc ∧ a ) pro ving the Lemma.  No w factoring the surjection α through J ( X ( A )) and using Lemma 3.1, w e get a mapping α : L 2 ( X ( A )) − → B ( L ( X )) ⊗ H C 1 ( A ) + ( K X ∨ K X ) ⊗ ( A ∧ A ) , ( K X denotes the space of linear terms in F ( X ) suc h that F ( X ) = K X + F 2 ( X )), whic h b eing restricted to L 3 ( X ( A )), giv es rise to the surjection α : L 3 ( X ( A )) − → B ( L ( X ) , L 2 ( X )) ⊗ H C 1 ( A ) , where H C 1 ( A ) = ( A ∧ A ) /T ( A ) is a first or der cyclic homolo gy of A . F urther, the restriction of the mapping φ defined in § 2 to L 3 ( X ( A )) leads to a surjection φ : L 3 ( X ( A )) → L 3 ( X ) ⊗ A . Lemma 3.2. α ( L 3 ( X ( A )) ∩ K er φ ) = α ( L 3 ( X ( A ))) . Pr o of. The Lemma follows immediately from Lemma 2.1 and equalit y α [ u ( a ) , v ( b )] = 1 2 α ([ u ( a ) , v ( b )] − [ u ( b ) , v ( a )]) , where the argumen t in the righ t- hand side lies in K er φ .  Lemma 3.3. α ( L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ( I ⊗ A )]) = B ( L ( X ) , I ∩ L 2 ( X )) ⊗ H C 1 ( A ) . Pr o of. According to Lemma 2.1, α ( L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ( I ⊗ A )]) consists from the linear span of the following elemen ts: u ∨ v ⊗ a ∧ b where either u ∈ L 2 ( X ) , v ∈ I or u ∈ L ( X ) , v ∈ I ∩ L 2 ( X ), and X j u ∨ v ⊗ a ∧ b j where P j b j = 0. The last expres sion ob viously v a nishes. Mo dulo [ L ( X ) , S 2 ( L ( X ))] we ha ve: L 2 ( X ) ∨ I ≡ L ( X ) ∨ [ I , L ( X )] ⊆ L ( X ) ∨ ( I ∩ L 2 ( X )) , whic h implies the assertion of Lemma.  Lemma 3.3 implies that the mapping α , being restricted to L 3 ( X ( A )) ∩ φ − 1 ( I ⊗ A ) and factored through L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ( I ⊗ A )], give s rise to a surjection (3.5) β : L 3 ( X ( A )) ∩ φ − 1 ( I ⊗ A ) L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ( I ⊗ A )] → B ( L ( X ) , L 2 ( X )) B ( L ( X ) , I ∩ L 2 ( X )) ⊗ H C 1 ( A ) . The rig h t-hand side here is b y (1.3) isomorphic to B ( L, [ L, L ]) ⊗ H C 1 ( A ). THE SECOND HOMOLO GY GROUP OF CURRENT LIE ALGEBRA S 7 F urther, according to Lemma 3.2, β can b e restricted to a surjection (3.6) β : L 3 ( X ( A )) ∩ K er φ L 3 ( X ( A )) ∩ [ L ( X ( A )) , φ − 1 ( I ⊗ A )] ∩ K er φ → B ( L, [ L, L ]) ⊗ H C 1 ( A ) . Lemma 3.4. β in ( 3 .6) is inje ctive. Pr o of. Denoting the left- hand side in (3.5) as F r ac , consider the follo wing diagram: K er ([ L, L ] ⊗ A ∧ L ⊗ A → L ⊗ A ) h − → H ess 2 ( L ⊗ A ) j − → F r ac i x   β   y L ∨ [ L, L ] ⊗ A ∧ A n − → B ( L, [ L, L ]) ⊗ H C 1 ( A ) where h is the ob vious factorization, j is the isomorphism following from Lemma 1.2 applied to presen tation (2.2), n = l ⊗ s , where l : L ∨ [ L, L ] → B ( L, [ L, L ]) and s : A ∧ A → H C 1 ( A ) are obvious factorizations, and i is defined as (3.7) i : ( x ∨ y ) ⊗ ( a ∧ b ) 7→ 1 2 ( x ⊗ a ∧ y ⊗ b − x ⊗ b ∧ y ⊗ a ) for x ∈ [ L, L ] , y ∈ L . The fo llo wing calculation ve r ifies the comm utativit y of this diagram: β ◦ j ◦ h ◦ i (( x ∨ y ) ⊗ ( a ∧ b )) = 1 2 β ◦ j ◦ h ( x ⊗ a ∧ y ⊗ b − x ⊗ b ∧ y ⊗ a ) = 1 2 β ◦ j ( x ⊗ a ∧ y ⊗ b − x ⊗ b ∧ y ⊗ a ) = 1 2 β ◦ j ( ( u ( a ) + φ − 1 ) ∧ ( v ( b ) + φ − 1 ) − ( u ( b ) + φ − 1 ) ∧ ( v ( a ) + φ − 1 )) = 1 2 β ( [ u ( a ) , v ( b )] − [ u ( b ) , v ( a )]) = 1 2 ( ( x ∨ y ) ⊗ ( a ∧ b ) − ( x ∨ y ) ⊗ ( b ∧ a )) = x ∨ y ⊗ a ∧ b = n (( x ∨ y ) ⊗ ( a ∧ b )) where the ov erlined elemen t s denote cosets in the corresp onding factorspaces, and x = u + I , y = v + I . It is also clear from the previous calculation a nd Lemmas 2.1 and 3.2 that the image of j ◦ h ◦ i coincides with the left-hand side of (3.6). Th us the ke rnel of the ma pping (3.6) can b e ev aluated as 8 P . ZUSMANOVICH K er β = j ◦ h ◦ i ( K er n ) = j ◦ h ◦ i ( h [ z , x ] ∨ y + [ z , y ] ∨ x i ⊗ h a ∧ b i + h [ x, y ] ∨ z i ⊗ h ab ∧ c + ca ∧ b + bc ∧ a i ) = j ( h [ z , x ] ⊗ a ∧ y ⊗ b − [ z , x ] ⊗ b ∧ y ⊗ a + [ z , y ] ⊗ a ∧ x ⊗ b − [ z , y ] ⊗ b ∧ x ⊗ a i + h [ x, y ] ⊗ ab ∧ z ⊗ c − [ x, y ] ⊗ c ∧ z ⊗ ab + [ x, y ] ⊗ ca ∧ z ⊗ b − [ x, y ] ⊗ b ∧ z ⊗ ca + [ x, y ] ⊗ bc ∧ z ⊗ a − [ x, y ] ⊗ a ∧ z ⊗ bc i ) = h [[ w , u ]( a ) , v ( b )] − [[ w, u ]( b ) , v ( a )] + [[ w , v ]( a ) , u ( b )] − [[ w, v ]( b ) , u ( a )] i + h [[ u, v ]( ab ) , w ( c )] − [[ u, v ]( c ) , w ( ab )] + [[ u, v ]( ca ) , w ( b )] − [[ u, v ]( b ) , w ( ca )] + [[ u, v ]( bc ) , w ( a )] − [[ u, v ]( a ) , w ( bc )] i (here u = x + I , v = y + I , w = z + I ). The latter expres sion v anishes thanks to Lemma 2.2.  Putting to g ether (3.1), (3.6) and Lemma 3.4, w e get Prop osition 3.5. H ess 2 ( L ⊗ A ) ≃ H ess 2 ( L ) ⊗ A ⊕ B ( L, [ L, L ]) ⊗ H C 1 ( A ) . By Lemma 1.1 we ha ve an exact sequence (3.8) 0 → H ess 2 ( L ⊗ A ) → H 2 ( L ⊗ A ) → ∧ 2 ( L/ [ L, L ] ⊗ A ) π A → [ L, L ] / [[ L, L ] , L ] ⊗ A → 0 . Lemma 3.6. K er π A ≃ K er ( ∧ 2 ( L/ [ L, L ]) π → [ L, L ] / [[ L, L ] , L ]) ⊗ A ⊕ ∧ 2 ( L/ [ L, L ]) ⊗ K er ( S 2 ( A ) → A ) ⊕ S 2 ( L/ [ L, L ]) ⊗ ∧ 2 ( A ) . Pr o of. The follo wing comm utative diag ram with exact row s and columns 0   y S 2 ( L/ [ L, L ]) ⊗ ∧ 2 ( A ) = − → S 2 ( L/ [ L, L ]) ⊗ ∧ 2 ( A ) π − → 0 i   y   y 0 − → K er π A − → ∧ 2 ( L/ [ L, L ] ⊗ A ) π A − − → [ L, L ] / [[ L, L ] , L ] ⊗ A − → 0 k   y    0 − → K er ( π ⊗ m ) − → ∧ 2 ( L/ [ L, L ]) ⊗ S 2 ( A ) π ⊗ m − − − → [ L, L ] / [[ L, L ] , L ] ⊗ A − → 0   y 0 where i is defined in (3.7 ), and k : x ⊗ a ∧ y ⊗ b 7→ ( x ∧ y ) ⊗ ( a ∨ b ) m : a ∨ b 7→ ab THE SECOND HOMOLO GY GROUP OF CURRENT LIE ALGEBRA S 9 for x, y ∈ L/ [ L, L ] , a, b ∈ A , implies (3.9) K er π A ≃ K er ( π ⊗ m ) ⊕ S 2 ( L/ [ L, L ]) ⊗ ∧ 2 ( A ) . Considering the comm utative diagr am with exact row s and columns 0   y ∧ 2 ( L/ [ L, L ]) ⊗ K er m = − → ∧ 2 ( L/ [ L, L ]) ⊗ K er m π − → 0   y   y 0 − → K er ( π ⊗ m ) − → ∧ 2 ( L/ [ L, L ]) ⊗ S 2 ( A ) π ⊗ m − − − → [ L, L ] / [[ L, L ] , L ] ⊗ A − → 0 1 ⊗ m   y    0 − → K er ( π ⊗ 1) − → ∧ 2 ( L/ [ L, L ]) ⊗ A π ⊗ 1 − − − → [ L, L ] / [[ L, L ] , L ] ⊗ A − → 0   y 0 w e get (3.10) K er ( π ⊗ m ) ≃ ∧ 2 ( L/ [ L, L ]) ⊗ K er ( S 2 ( A ) → A ) ⊕ K er ( ∧ 2 ( L/ [ L, L ]) π → [ L, L ] / [[ L, L ] , L ]) ⊗ A. Putting (3 .9 ) and (3.10) together pr ov es the Lemma.  Com bining Prop osition 3.5, (3.8) and Lemma 3.6, w e get H 2 ( L ⊗ A ) ≃ H ess 2 ( L ) ⊗ A ⊕ K er ( ∧ 2 ( L/ [ L, L ]) → [ L, L ] / [ L, [ L, L ]]) ⊗ A ⊕ B ( L, [ L, L ]) ⊗ H C 1 ( A ) ⊕ S 2 ( L/ [ L, L ]) ⊗ ∧ 2 ( A ) ⊕ ∧ 2 ( L/ [ L, L ]) ⊗ K er ( S 2 ( A ) → A ) . By Lemma 1.1 t he first tw o terms here giv e H 2 ( L ) ⊗ A . Using a (noncanonical) splitting ∧ 2 ( A ) = H C 1 ( A ) ⊕ T ( A ) and the exact seque nce (1.4), the third and fourth terms giv e B ( L ) ⊗ H C 1 ( A ) ⊕ S 2 ( L/ [ L, L ]) ⊗ T ( A ). Com bining these iden tifications gives Theorem 0.1. R emark . It is in teresting to compare Theorem 0.1 with the tw o -dimensional cas e of the homological op eration H n ( L ⊗ A ) → M i + j = n − 1 H C i ( U ( L )) ⊗ H C j ( A ) defined in [FT] ( U ( L ) is the univ ersal en v eloping alg ebra of L and the ground field assumed to b e of characteristic zero). T aking n = 2, w e obtain a mapping (3.11) H 2 ( L ⊗ A ) → H C 1 ( U ( L )) ⊗ H C 0 ( A ) ⊕ H C 0 ( U ( L )) ⊗ H C 1 ( A ) . Cyclic homology o f univ ersal en v eloping algebras was studied in [FT] and [Kas2]. Using their results, w e ma y o bserv e that if S ( L ) denotes the whole symmetric algebra o v er L , then H C 0 ( U ( L )) = H 0 ( L, S ( L )) = S ( L ) / [ L, S ( L )] and H C 1 ( U ( L )) is a certain factorspace of H 1 ( L, S ( L )) containing H 2 ( L ). This implies that in general (3.11) is neither inj ection, nor surjection. Ho w ev er, if L = [ L, L ], then (3.11) is an injection. 10 P . ZUSMANOVICH 4. Comput a tion of B ( L ⊗ A ) Theorem 0.1 allo ws us to compute B ( L ⊗ A ) in terms of L a nd A (of course, a n alternativ e but long er pro of ma y b e giv en b y means of direct computatio ns). The o r em 4.1 . B ( L ⊗ A ) ≃ B ( L, [ L, L ]) ⊗ A ⊕ S 2 ( L/ [ L, L ] ⊗ A ) . Pr o of. It is more con v enien t to use Prop o sition 3.5 rather then Theorem 0.1 to obtain a form ula for B ( L ⊗ A, [ L, L ] ⊗ A ) and then to deriv e the general case. T ak e a n y comm utative unital alg ebra A ′ with H C 1 ( A ′ ) ≃ K . According to Prop o sition 3.5, (4.1) H ess 2 ( L ⊗ A ⊗ A ′ ) ≃ H ess 2 ( L ⊗ A ) ⊗ A ′ ⊕ B ( L ⊗ A, [ L, L ] ⊗ A ) ≃ H ess 2 ( L ) ⊗ A ⊗ A ′ ⊕ B ( L, [ L, L ]) ⊗ H C 1 ( A ) ⊗ A ′ ⊕ B ( L ⊗ A, [ L, L ] ⊗ A ) . On the other ha nd, (4.2) H ess 2 ( L ⊗ A ⊗ A ′ ) ≃ H ess 2 ( L ) ⊗ A ⊗ A ′ ⊕ B ( L, [ L, L ]) ⊗ H C 1 ( A ⊗ A ′ ) ≃ H ess 2 ( L ) ⊗ A ⊗ A ′ ⊕ B ( L, [ L, L ]) ⊗ H C 1 ( A ) ⊗ A ′ ⊕ B ( L, [ L, L ]) ⊗ A. (the last isomorphism follows from the par tial first-order commutativ e case of the K ¨ unneth form ula for cyclic homology (cf. [Kas1]): H C 1 ( A ⊗ A ′ ) ≃ H C 1 ( A ) ⊗ A ′ + A ⊗ H C 1 ( A ′ )). Comparing (4.1) and (4 .2), and using the naturality condition guaranteeing compatibility , one has B ( L ⊗ A, [ L, L ] ⊗ A ) ≃ B ( L, [ L, L ]) ⊗ A. No w the assertion of Theorem easily follow s f r om the last isomorphism and the short exact sequence (1.4) applied to the a lg ebra L ⊗ A .  5. The second homology of A ⊗ B Recall that giv en an asso ciativ e algebra A , w e ma y consider its asso ciated Lie a lgebra A ( − ) with t he same underlying space A and the brack et [ a, b ] = ab − ba , as we ll as a Jordan algebra A (+) with multiplic a tion a ◦ b = 1 2 ( ab + ba ). Recall that T ( A ) = h ab ∧ c + ca ∧ b + bc ∧ a | a, b, c ∈ A i . F or the sake of con v enience w e will also use the follow ing notation: T ( A, [ A, A ]) = T ( A ) + [ A, A ] ∧ A [ A, A ] ∧ A H C 1 ( A, [ A, A ]) = A ∧ A [ A, A ] ∧ A + T ( A ) ≃ ∧ 2 ( A/ [ A, A ]) T ( A, [ A, A ]) (the second one is an a nalogue of H ess 2 ( L ) fo r cyclic homology). The a im of this section is to prov e the fo llowing The o r em 5.1 . L et A, B b e asso ciative algebr as with unit over a field K of char acteristic p 6 = 2 . L et F ( A, B ) denote the dir e ct sum of the fol lowin g four ve c tor sp ac es: (1) A [ A, A ] / [ A, A ] ⊗ H C 1 ( B ) (2) A/ A [ A, A ] ⊗ H 2 ( B ( − ) ) (3) ( K er ( S 2 ( A ) → A/ [ A, A ])) / [ A, S 2 ( A )] ⊗ H C 1 ( B , [ B , B ]) (4) K er ( S 2 ( A/ [ A, A ]) → A/ A [ A, A ]) ⊗ T ( B , [ B , B ]) wher e arr ow s i n (3) and (4) a r e induc e d by (asso ciative or Jor dan) multiplic ation in A . Then H 2 (( A ⊗ B ) ( − ) ) ≃ F ( A, B ) ⊕ F ( B , A ) . THE SECOND HOMOLO GY GROUP OF CURRENT LIE ALGEBRA S 11 The pro of is divided in to sev eral steps. W e employ the follo wing short exact sequenc e: 0 → ∧ 2 A ⊗ S 2 B i → ∧ 2 ( A ⊗ B ) p → S 2 A ⊗ ∧ 2 B → 0 where the middle t erm is identified with the direct sum o f t wo extreme ones via a 1 ⊗ b 1 ∧ a 2 ⊗ b 2 ↔ a 1 ∧ a 2 ⊗ b 1 ∨ b 2 + a 1 ∨ a 2 ⊗ b 1 ∧ b 2 , and i and p are obvious im b edding and pro j ection resp ectiv ely . In what follows , this will b e used without explicitly men tioning it. The a r g umen ts are quite analogous t o the ones at the b eginning of § 3. Here they applied to H 2 (( A ⊗ B ) ( − ) ) ≃ K er d/I m d ( d is the differential in the standard ho mo lo gy complex of ( A ⊗ B ) ( − ) ). The mapping p gives rise to the following short exact sequenc e: (5.1) 0 → K er p ∩ K er d K er p ∩ I m d → H 2 (( A ⊗ B ) ( − ) ) → p ( K er d ) p ( I m d ) → 0 . The Lie brac ke t on A ⊗ B ma y b e written as a sum [ a 1 ⊗ b 1 , a 2 ⊗ b 2 ] = [ a 1 , a 2 ] ⊗ b 1 ◦ b 2 + a 1 ◦ a 2 ⊗ [ b 1 , b 2 ] . The pro of of the follo wing statemen t is quite analogo us to the pro of of (3 .10). L emm a 5.1 . p ( K er d ) ≃ K er ( A ∨ A → A/ [ A, A ]) ⊗ B ∧ B + A ∨ A ⊗ K er ( B ∧ B → B ) wher e the first arr ow is induc e d by (asso ciative or Jor dan ) multiplic ation in algebr a A and the se c o n d one is the Lie multiplic ation in B ( − ) . L emm a 5.2 . p ( I m d ) is a line a r sp an o f the fol lowing elements: (1) [ A, S 2 ( A )] ⊗ B ∧ B (2) [ A, A ] ∨ A ⊗ T ( B ) (3) ( a 1 ∨ a 2 − 1 ∨ a 1 ◦ a 2 ) ⊗ [ B , B ] ∧ B , a i ∈ A (4) A ∨ A ⊗ ([ b 1 , b 2 ] ∧ b 3 + [ b 3 , b 1 ] ∧ b 2 + [ b 2 , b 3 ] ∧ b 1 ) , b i ∈ B . Pr o of. W e adopt the notation x ≡ 0 denoting the fact that certain elemen t x o f A ∨ A ⊗ B ∧ B lies in p ( I m d ). The generic relation defining the quotient b y p ( I m d ) is [ a 1 , a 2 ] ∨ a 3 ⊗ ( b 1 ◦ b 2 ) ∧ b 3 + ( a 1 ◦ a 2 ) ∨ a 3 ⊗ [ b 1 , b 2 ] ∧ b 3 +[ a 3 , a 1 ] ∨ a 2 ⊗ ( b 3 ◦ b 1 ) ∧ b 2 + ( a 3 ◦ a 1 ) ∨ a 2 ⊗ [ b 3 , b 1 ] ∧ b 2 (5.2) +[ a 2 , a 3 ] ∨ a 1 ⊗ ( b 2 ◦ b 3 ) ∧ b 1 + ( a 2 ◦ a 3 ) ∨ a 1 ⊗ [ b 2 , b 3 ] ∧ b 1 ≡ 0 . Symmetrizing this relation with resp ect to a 1 , a 2 , w e get: (5.3) 2( a 1 ◦ a 2 ) ∨ a 3 ⊗ [ b 1 , b 2 ] ∧ b 3 + ([ a 3 , a 1 ] ∨ a 2 − [ a 2 , a 3 ] ∨ a 1 ) ⊗ (( b 3 ◦ b 1 ) ∧ b 2 − ( b 2 ◦ b 3 ) ∧ b 1 ) + (( a 3 ◦ a 1 ) ∨ a 2 + ( a 2 ◦ a 3 ) ∨ a 1 ) ⊗ ([ b 3 , b 1 ] ∧ b 2 + [ b 2 , b 3 ] ∧ b 1 ) ≡ 0 . Cyclic p ermu tations of a 1 , a 2 , a 3 in the last relatio n yield: (( a 1 ◦ a 2 ) ∨ a 3 + ( a 3 ◦ a 1 ) ∨ a 2 + ( a 2 ◦ a 3 ) ∨ a 1 ) ⊗ ([ b 1 , b 2 ] ∧ b 3 + [ b 3 , b 1 ] ∧ b 2 + [ b 2 , b 3 ] ∧ b 1 ) ≡ 0 . This relatio n, in its turn, eviden t ly implies (5.4) A ∧ A ⊗ ([ b 1 , b 2 ] ∧ b 3 + [ b 3 , b 1 ] ∧ b 2 + [ b 2 , b 3 ] ∧ b 1 ) ≡ 0 . 12 P . ZUSMANOVICH No w rewriting (5 .3) mo dulo (5.4) and substituting a 3 = 1 and b 2 = 1, we get, resp ectiv ely: (5.5) ( a 1 ∨ a 2 − 1 ∨ a 1 ◦ a 2 ) ⊗ [ B , B ] ∧ B ≡ 0 and ([ a 3 , a 1 ] ∨ a 2 − [ a 2 , a 3 ] ∨ a 1 ) ⊗ ( b 1 ∧ b 3 + 1 ∧ b 1 ◦ b 3 ) ≡ 0 . Symmetrizing the last relatio n with respect to b 1 , b 3 , one gets: (5.6) ([ a 3 , a 1 ] ∨ a 2 − [ a 2 , a 3 ] ∨ a 1 ) ⊗ B ∧ B ≡ 0 . P articularly , ta king in (5.6) a 2 = 1, one gets (5.7) 1 ∨ [ A, A ] ⊗ B ∧ B ≡ 0 . No w, (5.2) is equiv alen t mo dulo (5.4)–(5.6 ) to [ a 1 , a 2 ] ∨ a 3 ⊗ ( b 1 b 2 ∧ b 3 + b 3 b 1 ∧ b 2 + b 2 b 3 ∧ b 1 ) + 1 ∨ (( a 1 ◦ a 2 ) ◦ a 3 − ( a 2 ◦ a 3 ) ◦ a 1 ) ⊗ [ b 1 , b 2 ] ∧ b 3 (5.8) + 1 ∨ (( a 3 ◦ a 1 ) ◦ a 2 − ( a 2 ◦ a 3 ) ◦ a 1 ) ⊗ [ b 3 , b 1 ] ∧ b 2 ≡ 0 . T aking in to accoun t the iden tity ( a ◦ b ) ◦ c − ( a ◦ c ) ◦ b = 1 4 [ a, [ b, c ]] (cf. [J], p.37) , and (5.7 ), the relatio n (5.8), in its turn, is equiv alen t to (5.9) [ A, A ] ∨ A ⊗ T ( B ) ≡ 0 . Putting to g ether (5.4)–( 5 .6) and (5.9 ), w e get exactly the statemen t o f the Lemma.  L emm a 5.3 . (1) p ( K er d ) /p ( I m d ) ≃ F ( A, B ) . (2) ( K er p ∩ K er d ) / ( K er p ∩ I m d ) ≃ F ( B , A ) . Pr o of. (1) is deriv ed from Lemmas 5.1 and 5.2 after a nu m b er of r o utine transformatio ns. (2) D efine a pro jection p ′ : ∧ 2 ( A ⊗ B ) → ∧ 2 A ⊗ S 2 B . Due to an obvious fact that p ′ is the iden tity on K er p = A ∧ A ⊗ B ∨ B , w e hav e an isomor phism K er d ∩ K er p K er p ∩ I m d ≃ p ′ ( K er d ∩ K er p ) p ′ ( K er p ∩ I m d ) = p ′ ( K er d ) p ′ ( I m d ) But the righ t- hand term here is computed as in the part (1), up to p erm utation of A and B .  No w Theorem 5.1 follo ws immediately from (5.1) a nd Lemma 5.3. R emark . T aking in Theorem 5.1 B = M n ( K ), we get, after a series of elemen ta ry transfor- mations, an isomorphism H 2 ( g l n ( A )) ≃ H C 1 ( A ) ⊕ ∧ 2 ( A/ [ A, A ]) . Using the Ho c hsc hild-Serre spectral sequence asso ciated with cen tral extension 0 → sl n ( A ) → g l n ( A ) → A/ [ A, A ] → 0 of Lie algebras, we deriv e H 2 ( sl n ( A )) ≃ H C 1 ( A ) THE SECOND HOMOLO GY GROUP OF CURRENT LIE ALGEBRA S 13 whic h is a result of C. K a ssel and J.- L. 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F ourier 32 (198 2 ), 119–142 . [KS] M. A. Knus, U. Stamm bach, Anwendungen der Homolo giethe orie der Lie algebr en auf Zentr alr eihen und auf Pr¨ asentierungen , Comm. Math. Helv. 42 (1967 ), 297–3 06. [LQ] J.-L. Loday , D. Quillen, Cyclic homolo gy and Lie algebr a homolo gy of matric es , Comm. Math. Helv. 59 (1984 ), 565–5 91. [S] L. J. Santharoubane, The se c ond c ohomolo gy gr oup for Kac-Mo o dy Lie algebr as and K¨ ahler differ en - tials , J. Algebra 125 (198 9), 13–2 6. P aul Zusmano vic h Departmen t of Mathematics Bar-Ilan Unive rsity Ramat-Gan 52 900, Israel e-mail: zusman@bimacs.cs. biu.ac.il