Commutative 2-cocycles on Lie algebras
aa r X i v : . [ m a t h . R A ] M a y COMMUTATIVE -COCYCLES ON LIE ALGEBRAS ASKAR DZHUMADIL’DAEV AND PASHA ZUSMANOVICHA
BSTRACT . On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfyingthe usual cocycle equation. We note their relationship with antiderivations and compute them for someclasses of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras. I NTRODUCTION A commutative -cocycle on a Lie algebra L over a field K is a symmetric bilinear form ϕ : L × L → K satisfying the cocycle equation(1) ϕ ([ x , y ] , z ) + ϕ ([ z , x ] , y ) + ϕ ([ y , z ] , x ) = x , y , z ∈ L . Commutative 2-cocycles appear at least in two different contexts. First, they appearin the study of nonassociative algebras satisfying certain skew-symmetric identities. It turns out that allskew-symmetric identities of degree 3 reduce to a number of identities, among which [ a , b ] c + [ b , c ] a + [ c , a ] b = a [ b , c ] + b [ c , a ] + c [ a , b ] = [ a , b ] = ab − ba ).Algebras satisfying both these identities are dubbed two-sided Alia algebras in [D3] and [DB]. Notethat the class of two-sided Alia algebras contains both Lie algebras and Novikov algebras (for the latter,see the end of § A is two-sided Alia if and only if the associated “minus”algebra A ( − ) with multiplication defined by the bracket [ · , · ] , is a Lie algebra (in other words, A isLie-admissible), and multiplication in A can be written as(2) ab = [ a , b ] + ϕ ( a , b ) , † where ϕ is an A ( − ) -valued commutative 2-cocycle on A ( − ) .Note, however, that also any commutative (nonassociative) algebra is two-sided Alia, so the questionof description of simple algebras in this class does not make much sense without imposing additionalconditions. One such natural condition is, in a sense, opposite to the condition of commutativity of A – namely, that the Lie algebra A ( − ) is simple. In this way we arrive to the problem of description ofcommutative 2-cocycles on simple Lie algebras.Second, commutative 2-cocycles appear naturally in the description of the second cohomology ofcurrent Lie algebras ([Zu3, Theorem 1]). All this, as well as a sheer curiosity in what happens withthe usual second Lie algebra cohomology when we replace the condition of the skew-symmetricity ofcochains by its opposite – symmetricity – makes them worth to study.It is worth to note that this situation is similar (and somewhat dual) to the question which goes back toA.A. Albert and was a subject of an intensive study later, namely, determination of Lie-admissible thirdpower-associative algebras A whose “minus” algebra A ( − ) belongs to some distinguished class of Liealgebras, such as finite-dimensional simple Lie algebras, or Kac-Moody algebras. It is easy to see that Date : last minor revision September 9, 2017.J. Algebra (2010), N4, 732–748; arXiv:0907.4780 . † Added May 5, 2017: The correct formula is: ab = [ a , b ] + ϕ ( a , b ) . in terms of decomposition (2), for a given Lie algebra L = A ( − ) the third power-associativity implies thecondition [ ϕ ( x , y ) , z ] + [ ϕ ( z , x ) , y ] + [ ϕ ( y , z ) , x ] = x , y , z ∈ L (see, for example, [Be, p. 39]). The latter condition, together with (1), could be viewedas two parts of the usual 2-cocycle equation on a Lie algebra with coefficients in the adjoint module.Of course, the very definition of commutative 2-cocycles begs for a proper generalization – to definehigher cohomology groups such that commutative 2-cocycles constitute cohomology of low degree.However, all “naive” attempts to construct such higher cohomology seemingly fail. Maybe it could bedeveloped in the framework of operadic cohomology of two-sided Alia algebras.The contents of the paper are as follows. In § § sl ( ) and the modular Zassenhaus algebra. We also computethe space of commutative 2-cocycles on the Zassenhaus algebra, and note how this may be utilized inclassification of simple Novikov algebras. § § §
5, and modular semisimple Lie algebras in § OTATION AND CONVENTIONS
Throughout the paper, all algebras and vector spaces are defined over a ground field K of characteristic = ,
3, except in § L , the space of all commutative 2-cocycles on it will be denoted as Z comm ( L ) .Occasionally, we will consider the M-valued commutative -cocycles , that is, bilinear forms L × L → M ,where M is an L -module (or, rather, just a vector space), satisfying the cocycle equation (1). It isimmediate that if either L or M is finite-dimensional, then the space of all such cocycles is isomorphicto Z comm ( L ) ⊗ M .Recall that a Lie algebra L is called perfect if [ L , L ] = L . Obviously, a symmetric bilinear form on L that vanishes whenever one of the arguments belongs to [ L , L ] , is a commutative 2-cocycle. Such cocy-cles will be called trivial and they exist on any non-perfect Lie algebra. The space of trivial commutative2-cocycles is isomorphic to S ( L / [ L , L ]) ∗ .The rest of our notation and definitions is mostly standard. H n ( L , M ) and Z n ( L , M ) denote the usual Chevalley–Eilenberg n th order cohomology and the space of n th order cocycles, respectively, of a Lie algebra L with coefficients in a module M .When being considered as an L -module, the ground field K is always understood as the one-dimensi-onal trivial module. HC ( A ) denotes the first order cyclic cohomology of an associative commutative algebra A . Recallthat it is nothing but the space of all skew symmetric bilinear forms α : A × A → K such that α ( ab , c ) + α ( ca , b ) + α ( bc , a ) = a , b , c ∈ A . Note an obvious but useful fact: if A contains a unit 1, then α ( , A ) = α ∈ HC ( A ) . Der ( A ) denotes the Lie algebra of derivations of A . OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 3
For an associative algebra A , we may consider associated Lie [ · , · ] and Jordan ◦ products on A : [ a , b ] = ab − ba and a ◦ b = ab + ba for a , b ∈ A . The vector space A with the bracket [ · , · ] forms a Liealgebra which is denoted as A ( − ) .For a vector space V , S ( V ) and ∧ ( V ) denotes the symmetric and skew-symmetric product respec-tively (so, the space of symmetric, respectively skew-symmetric bilinear forms on V is isomorphic to S ( V ) ∗ , respectively to ∧ ( V ) ∗ ).Other nonstandard notions (antiderivations, cyclic forms) are introduced in § N EXACT SEQUENCE CONNECTING COMMUTATIVE COCYCLES AND ANTIDERIVATIONS
A relationship between the second cohomology with coefficients in the trivial module H ( L , K ) , andthe first cohomology with coefficients in the coadjoint module H ( L , L ∗ ) was noted many times inslightly different forms in the literature, and goes back to the classical works of Koszul and Hochschild–Serre. Namely, there is an exact sequence(3) 0 → H ( L , K ) u → H ( L , L ∗ ) v → B ( L ) w → H ( L , K ) . Here B ( L ) denotes the space of symmetric bilinear invariant forms on L . The maps are defined asfollows: for the representative ϕ ∈ Z ( L , K ) of a given cohomology class, we have to take the class of u ( ϕ ) , the latter being given by(4) ( u ( ϕ )( x ))( y ) = ϕ ( x , y ) for any x , y ∈ L , v is sending the class of a given cocycle d ∈ Z ( L , L ∗ ) to the bilinear form v ( d ) : L × L → K defined by the formula v ( d )( x , y ) = d ( x )( y ) + d ( y )( x ) , and w is sending a given symmetric bilinear invariant form ϕ : L × L → K to the class of the cocycle ω ∈ Z ( L , K ) defined by(5) ω ( x , y , z ) = ϕ ([ x , y ] , z ) (see, for example, [D2], where a certain long exact sequence is obtained, of which this one is thebeginning, and references therein for many earlier particular variations; this exact sequence was alsoestablished in [NW, Proposition 7.2] with two additional terms on the right).The following is a “commutative” version of this exact sequence, connecting commutative 2-cocyclesand antiderivations. Definition.
An antiderivation of a Lie algebra L to an L-module M is a linear map d : L → M such thatd ([ x , y ]) = y • d ( x ) − x • d ( y ) for any x , y ∈ L, where • denotes the L-action on M. The set of all such maps will be denoted asADer ( L , M ) . When M = L , the adjoint module, we get the notion of an antiderivation of a Lie algebra (to itself)with the defining condition d ([ x , y ]) = − [ d ( x ) , y ] − [ x , d ( y )] , what was the subject of study in a number of papers, including [F].The third ingredient in our exact sequence, a counterpart of symmetric bilinear invariant forms, isdefined as follows. Definition.
A bilinear form ϕ : L × L → K is said to be cyclic if (6) ϕ ([ x , y ] , z ) = ϕ ([ z , x ] , y ) for any x , y , z ∈ L. The space of all cyclic skew-symmetric bilinear forms on a Lie algebra L will bedenoted as C ( L ) . OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 4
Proposition 1.1.
For any Lie algebra L, there is an exact sequence → Z comm ( L ) u → ADer ( L , L ∗ ) v → C ( L ) where the map u is defined by formula (4) , and v is sending a given antiderivation d ∈ ADer ( L , L ∗ ) tothe bilinear form v ( d ) : L × L → K defined by the formulav ( d )( x , y ) = d ( x )( y ) − d ( y )( x ) . Proof. If ϕ ∈ Z comm ( L ) , then the cocycle equation yields ( u ( ϕ )([ x , y ]))( z ) = ϕ ([ x , y ] , z ) = − ϕ ( x , [ y , z ]) + ϕ ( y , [ x , z ]) = ( y • u ( ϕ )( x ))( z ) − ( x • u ( ϕ )( y ))( z ) for any x , y , z ∈ L , where • denotes the standard L -action on L ∗ , i.e. u ( ϕ ) ∈ ADer ( L , L ∗ ) .If d ∈ ADer ( L , L ∗ ) , then v ( d ) is obviously skew-symmetric, and v ( d )([ x , y ] , z ) = d ([ x , y ])( z ) − d ( z )([ x , y ])= − d ( x )([ y , z ]) + d ( y )([ x , z ]) − d ( z )([ x , y ]) = d ([ z , x ])( y ) − d ( y )([ z , x ])= v ( d )([ z , x ] , y ) for any x , y , z ∈ L , i.e. v ( d ) ∈ C ( L ) .Now let us check exactness. Obviously, u is injective, so the sequence is exact at Z comm ( L ) . Next, Im u consists of all antiderivations d ∈ ADer ( L , L ∗ ) such that the bilinear form ( x , y ) d ( x )( y ) satisfiesthe cocycle equation, what is equivalent to d being antiderivation, and is symmetric, what is equivalentto d ( x )( y ) = d ( y )( x ) for any x , y ∈ L . But the latter condition is equivalent to d belonging to Ker v , hence
Im u = Ker v and the sequence is exact at
ADer ( L , L ∗ ) . (cid:3) Unfortunately, we do not know how to extend this exact sequence further. The map defined by theformula (5) maps C ( L ) to the space of skew-symmetric trilinear forms L × L × L → K , but the resultingimages are neither Chevalley–Eilenberg (or Leibniz) cocycles, nor do they satisfy any other naturalcondition. Obviously, this is related to the difficulty to define higher analogs of commutative 2-cocycles.Another difficulty is related to the fact that C ( L ) turns out to be not a very fascinating invariant of aLie algebra: it vanishes in the most interesting cases. Lemma 1.2 (Referee) . If L is a perfect Lie algebra, then C ( L ) = .Proof. For any ϕ ∈ C ( L ) , and any x , y , z , t ∈ L , we have ϕ ([[ x , y ] , z ] , t ) (Jacobi identity) = − ϕ ([[ z , x ] , y ] , t ) − ϕ ([[ y , z ] , x ] , t ) (cyclicity of ϕ ) = − ϕ ([ y , t ] , [ z , x ]) − ϕ ([ x , t ] , [ y , z ]) (skew-symmetry of ϕ ) = − ϕ ([ x , z ] , [ y , t ]) + ϕ ([ y , z ] , [ x , t ]) (cyclicity of ϕ ) = − ϕ ([[ y , t ] , x ] , z ) + ϕ ([[ x , t ] , y ] , z ) (Jacobi identity) = ϕ ([[ x , y ] , t ] , z ) (cyclicity of ϕ ) = ϕ ([ z , [ x , y ]] , t )= − ϕ ([[ x , y ] , z ] , t ) . Therefore, ϕ ([[ L , L ] , L ] , L ) =
0, and the asserted statement follows. (cid:3)
The exact sequence (3) was utilized many times in the literature to evaluate H ( L , K ) basing on H ( L , L ∗ ) (see, for example, references in [D2]). We will utilize Proposition 1.1 in a similar way. Forthis, we shall need to establish some facts about antiderivations of Lie algebras, what is done in the nextsection. OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 5
2. S
IMPLE L IE ALGEBRAS
In [F], Filippov obtained many results about Lie algebras possessing a nonzero antiderivation. A slightmodification of his reasonings allows to extend some of these results to antiderivations with values in aLie algebra module.Recall the standard identity of degree ∑ σ ∈ S ( − ) σ [[[[ y , x σ ( ) ] , x σ ( ) ] , x σ ( ) ] , x σ ( ) ] = , where the summation is performed over all elements of the symmetric group S . The word at the leftside of (7) will be denoted as s ( x , x , x , x , y ) . For a Lie algebra L , let s ( L ) denote the linear spanof values of this word for any x , x , x , x , y ∈ L . By the general result about verbal ideals (see, forexample, [AS, Chapter 14], Theorem 2.8 and remark after it), s ( L ) is an ideal of L . Lemma 2.1.
If a Lie algebra possesses nonzero commutative -cocycles, then it has a nonzero homo-morphic image satisfying the standard identity of degree .Proof. If a Lie algebra L possesses nonzero commutative 2-cocycles, then by Proposition 1.1, L pos-sesses a nonzero antiderivation to its coadjoint module. By [Zu4, Lemma 4.4] (which is just a slightgeneralization of [F, Theorem 4]), s ( L ) lies in the kernel of this antiderivation, and hence is a properideal in L . The required homomorphic image is L / s ( L ) . (cid:3) Lemma 2.2.
A Lie algebra with a subalgebra of codimension which is not an ideal, possesses nonzerocommutative -cocycles.Proof. Let S be a subalgebra of codimension 1 in a Lie algebra L . Pick x ∈ L \ S and let f : S → K bea linear map such that [ x , y ] ∈ f ( y ) x + S for any y ∈ S . The Jacobi identity implies f ([ S , S ]) =
0. Since S is not an ideal in L , f is nonzero. It is easy to see that any bilinear map ϕ : L × L → K satisfying theconditions ϕ ( S , S ) = , ϕ ( x , y ) = ϕ ( y , x ) = f ( y ) for y ∈ S , is a commutative 2-cocycle (in fact, the cocycle equation in that case is equivalent to the Jacobiidentity; ϕ ( x , x ) can take any value). (cid:3) Theorem 2.3.
A simple Lie algebra possesses nonzero commutative -cocycles if and only if it satisfiesthe standard identity of degree .Proof. The “only if” part follows from Lemma 2.1.The “if” part. Suppose a simple Lie algebra L satisfies the identity of degree 5. By Razmyslov’scharacterization of such algebras ([R, Proposition 46.1]), there exists a field extension F of the centroid C of L such that the Lie F -algebra L ⊗ C F contains a subalgebra of codimension 1, and, by Lemma2.2, possesses nonzero commutative 2-cocycles. As the space of commutative 2-cocycles is obviouslypreserved under the ground field extension, the Lie C -algebra L ⊗ K C possesses nonzero commutative2-cocycles. Each such cocycle, being restricted to L , gives rise to a nonzero bilinear K -map L × L → C ,which is a commutative C -valued 2-cocycle on L . (cid:3) Using other characterizations of simple Lie algebras satisfying the standard identity of degree 5, itis possible to give an alternative proof of the “if” part of Theorem 2.3 in the case where the groundfield has positive characteristic. Namely, according to [R, Theorem 46.2], each simple Lie algebra ofdimension > A ∂ = { a ∂ | a ∈ A } of an associative commutative algebra A with unit,generated as an A -module by a single derivation ∂ ∈ Der ( A ) . In addition, A does not have ∂ -invariantideals. For such algebras, we have Lemma 2.4. A ∗ is embedded into Z comm ( A ∂ ) . OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 6
Proof.
Consider the map F : A → A ∂ defined by a a ∂ for a ∈ A . The kernel of this map is a ∂ -invariantideal of A , and hence this map is injective.For any χ ∈ A ∗ , define the map Φ ( χ ) : A ∂ × A ∂ → K as Φ ( χ )( a ∂ , b ∂ ) = χ ( ab ) for a , b ∈ A . Thismap is well defined due to injectivity of F and is a commutative 2-cocycle: Φ ( χ )([ a ∂ , b ∂ ] , c ∂ ) + Φ ( χ )([ c ∂ , a ∂ ] , b ∂ ) + Φ ( χ )([ b ∂ , c ∂ ] , a ∂ )= Φ ( χ )(( a ∂ ( b ) − b ∂ ( a )) ∂ , c ∂ )+ Φ ( χ )(( c ∂ ( a ) − a ∂ ( c )) ∂ , b ∂ )+ Φ ( χ )(( b ∂ ( c ) − c ∂ ( b )) ∂ , a ∂ )= χ ( ac ∂ ( b )) − χ ( bc ∂ ( a )) + χ ( bc ∂ ( a )) − χ ( ab ∂ ( c )) + χ ( ab ∂ ( c )) − χ ( ac ∂ ( b )) = a , b , c ∈ A . It is obvious that Φ is injective. (cid:3) Corollary 2.5.
A finite-dimensional central simple Lie algebra possesses nonzero commutative -co-cycles if and only if it is isomorphic either to a form of sl ( ) or, in the case where the characteristic ofthe ground field is positive, to the Zassenhaus algebra W ( n ) .Proof. Obviously, we may pass to the algebraic closure of the ground field. Finite-dimensional simpleLie algebras over an algebraically closed field satisfying the standard identity of degree 5, are precisely sl ( ) and the Zassenhaus algebra. This well known fact can be derived in several ways, perhaps theeasiest one is to invoke once again [R, Proposition 46.1] to establish that such Lie algebras have a sub-algebra of codimension 1, and then refer to the known classification of such algebras (see, for example,[E] and references therein). (cid:3) This generalizes [DB, Theorem 1.1], where the same result is proved for Lie algebras of classical typeby performing computations with the corresponding root system. Another proof for such Lie algebrascould be derived by combining results of [L] and [Zu3]. In a sense, root space computations in [DB] areequivalent to the appropriate part of computations in [L].Note that the same reasoning (Theorem 2.3 coupled with Razmyslov’s results) shows that amonginfinite-dimensional Lie algebras of Cartan type, only the Witt algebra possesses nonzero commuta-tive 2-cocycles. The latters are described in [D3, Theorem 6.7]. Another important class of infinite-dimensional Lie algebras – Kac-Moody algebras – is treated in § sl ( ) and the Zassenhaus algebra are characterized among simplefinite-dimensional Lie algebras in various interesting ways: these are algebras having a subalgebra ofcodimension 1, algebras having a solvable maximal subalgebra (see [S, Corollary 9.1.0]), algebras withcertain properties of the lattice of subalgebras (see, for example, [BTW] and references therein), algebrashaving non-trivial δ -derivations ([Zu4, § Z comm ( sl ( )) forms a 5-dimensional hyperplane in the 6-dimensionalspace of symmetric bilinear maps ϕ : sl ( ) × sl ( ) → K , determined by the equation ϕ ( h , h ) = ϕ ( e − , e + ) in the sl ( ) -basis { e − , e + , h } with multiplication table [ h , e ± ] = ± e ± , [ e − , e + ] = h .The famous Zassenhaus algebra W ( n ) can be realized in two different ways. One realization isthe algebra of derivations of the divided powers algebra O ( n ) = { x i | ≤ i ≤ p n − } , where p is thecharacteristic of the ground field, with multiplication given by x i x j = (cid:18) i + jj (cid:19) x i + j , of the form O ( n ) ∂ , where the derivation ∂ ∈ Der ( O ( n )) acts as follows: ∂ ( x i ) = x i − . Anotherrealization is the algebra with the basis { e α | α ∈ G } , where G is an additive subgroup of order p n of theground field K , with multiplication(8) [ e α , e β ] = ( β − α ) e α + β OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 7 for α , β ∈ G . We formulate the result in terms of the first realization, but perform actual computationswith the second one. Proposition 2.6. Z comm ( W ( n )) ≃ O ( n ) ∗ , each cocycle being of the form ( a ∂ , b ∂ ) f ( ab ) where a , b ∈ O ( n ) , for some f ∈ O ( n ) ∗ .Proof. The proof consists of straightforward computations reminiscent both the computation of thesecond cohomology of W ( n ) in [Bl, §
5] and computation of the space of commutative 2-cocycles onthe infinite-dimensional Witt algebra in [D3, Theorem 6.7].Let ϕ ∈ Z comm ( W ( n )) . Writing the cocycle equation in terms of the basis with multiplication (8) for e , e α , e β , where α , β ∈ G such that α = β , we get(9) ϕ ( e α , e β ) = ϕ ( e , e α + β ) . Now, taking this into account, and writing the cocycle equation for e α , e β , e α + β , where α , β ∈ G suchthat α , β =
0, we get ϕ ( e α + β , e α + β ) = ϕ ( e , e α + β ) . Consequently, (9) holds for any α , β ∈ G . Conversely, each symmetric bilinear map satisfying thecondition (9) is easily seen to satisfy the cocycle equation. Each such map can be decomposed into thesum of the maps of the form ( e α , e β ) ( , α + β = γ , otherwisefor each γ ∈ G . Thus we get | G | = p n linearly independent cocycles, so the whole space Z comm ( W ( n )) is p n -dimensional. Now, switching to the first realization as derivation algebra of O ( n ) , applying Lemma2.4 and comparing dimensions, we see that the embedding of Lemma 2.4 is an isomorphism in thisparticular case. (cid:3) Note that the results of this section allow to provide a somewhat alternative way for classification offinite-dimensional simple Novikov algebras. Recall that an algebra is called
Novikov if it satisfies theidentities x ( yz ) − ( xy ) z = y ( xz ) − ( yx ) z and ( xy ) z = ( xz ) y . Novikov algebras are ubiquitous in various branches of mathematics and physics (see [Bu], [O] and [Ze]with a transitive closure of references therein) † .A well known important fact is that each Novikov algebra A is Lie-admissible, i.e., A ( − ) is a Liealgebra. In [Ze], Zelmanov proved that finite-dimensional simple Novikov algebras over a field ofcharacteristic zero are 1-dimensional (i.e., coincide with the ground field), and in [O], Osborn provedthat if A is a finite-dimensional simple Novikov algebra over a field of positive characteristic, then A ( − ) iseither 1-dimensional, or isomorphic to the Zassenhaus algebra. The latter was the key result in obtaininglater a complete classification of simple Novikov algebras.Another important observation – which is a matter of simple calculations (see, for example, [Bu,Lemma 2.3]) – is that Novikov algebras are two-sided Alia. Further, it follows from the proof of [O,Theorem 3.5], that if A is a finite-dimensional simple Novikov algebra, then the Lie algebra A ( − ) is alsosimple. Thus, as noted in the Introduction, the multiplication in a finite-dimensional simple Novikovalgebra A could be written in the form (2), where [ · , · ] defines a simple Lie algebra structure on A , and † Actually, what we define here are left Novikov algebras.
Right Novikov algebras are defined by the opposite identities,interchanging left and right multiplications. Often in the literature, left Novikov and right Novikov are confused, even ifauthors make an explicit attempt not to do so (for example, [Ze] treats right Novikov algebras, while [O] and [Bu] – leftNovikov ones). However, almost every result about left Novikov algebras is easily transformed into one about right Novikovalgebras, and vice versa.
OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 8 ϕ defines a commutative 2-cocycle on that Lie algebra. Note further that ϕ is a nonzero cocycle, asotherwise A is a Lie algebra, what quickly leads to contradiction.Then, instead of appealing to some structural results about graded and filtered Lie algebras, like in[O], we may invoke Corollary 2.5 to see that A ( − ) is isomorphic either to 1-dimensional abelian algebra,or to sl ( ) , or to the Zassenhaus algebra. Straightforward calculations based on [D3, Theorem 6.5]exclude the case of sl ( ) , and in this way we get the main results of [Ze] and [O]. Proposition 2.6 maybe utilized in the subsequent classification.3. C HARACTERISTIC § L , considerthe standard Der ( L ) -action on the space of bilinear forms on L : for ϕ being such a form, and D ∈ Der ( L ) ,(10) ( D • ϕ )( x , y ) = − ϕ ( D ( x ) , y ) − ϕ ( x , D ( y )) , x , y ∈ L . It is well known (and easy to verify) that the space of symmetric bilinear invariant forms on L is closed under this action. It turns out that the same is true for the space of commutative 2-cocycles: Lemma 3.1.
For any Lie algebra L, Z comm ( L ) is closed under the action (10).Proof. This follows from the following equality, valid for any bilinear form ϕ : L × L → K , any D ∈ Der ( L ) , and any x , y , z ∈ L : d ( D • ϕ )( x , y , z ) = − d ϕ ( x , y , D ( z )) − d ϕ ( z , x , D ( y )) − d ϕ ( y , z , D ( x )) . where d ϕ ( x , y , z ) denotes the left-hand side of equality (1). (cid:3) In particular, Z comm ( L ) is closed under the action of L (via inner derivations). This is, essentially, thesame observation which is used in deriving a very useful fact about triviality of the Lie algebra actionon its cohomology. Note, however, that, unlike for cohomology, in the case of commutative 2-cocycleswe do not have coboundaries, so we cannot normalize the cocycle appropriately to derive the trivialityof this action. Moreover, for invariants of this action we have the following obvious dichotomy: Z comm ( L ) L = ( B ( L ) , p = { ϕ : L × L → K | ϕ is symmetric, ϕ ([ L , L ] , L ) = } , p = . Still, we can make use of Lemma 3.1 in the same way as in cohomological considerations: when T is a torus in a Lie algebra L such that L decomposes into the direct sum of the root spaces withrespect to the action of T (what always takes places if L is finite-dimensional and the ground field isalgebraically closed), then Z comm ( L ) decomposes into the direct sum of root spaces with respect to theinduced T -action.Naturally, in order to compute the space of commutative 2-cocycles on any class of Lie algebras ofcharacteristic 3, it would be beneficial to elucidate first what the space of symmetric bilinear invariantforms on these algebras looks like. Conjecture 3.2.
The space of symmetric bilinear invariant forms on any central Lie algebra of classicaltype over a field of arbitrary characteristic is -dimensional. Note that in small characteristics (including characteristic 3) not all Lie algebras obtained via the usualChevalley basis construction, are simple (see, for example, [S, § p = p > OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 9 may vanish (see [GP] and references therein), so this conjecture, perhaps, comes as a bit of surprise. Itis, however, supported by computer calculations † . Conjecture 3.3.
The space of commutative -cocycles on a finite-dimensional Lie algebra of classicaltype different from sl ( ) , over the field of characteristic , coincides with the space of symmetric bilinearinvariant forms (and, hence, by Conjecture 3.2, is -dimensional). Note that for Lie algebras not of classical type, it is no longer true that every commutative 2-cocycleis a symmetric bilinear invariant form: for example, for the Zassenhaus algebra W ( n ) , Proposition 2.6is valid also in characteristic 3 (with, essentially, the same proof). On the other hand, since W ( n ) issimple, the space of symmetric bilinear invariant forms on it is at most 1-dimensional (it is, in fact,1-dimensional in characteristic 3, as shown in [D1, §
2, Corollary] or [SF, § A , B and G . For these algebras, one may verify on acomputer that both the spaces of commutative 2-cocycles (for p =
3) and of symmetric bilinear invariantforms (for any p ), are 1-dimensional. The general case should follow then from considerations of 2-sections in a Cartan decomposition, with the help of Lemma 3.1.However, there are lot of subtleties when dealing with structure constants of classical Lie algebrasin small characteristics, as demonstrated by a noticeable amount of errors in works devoted to suchalgebras. We postpone this laborious task to the future.4. C URRENT L IE ALGEBRAS
In this section we consider the current Lie algebras, i.e. Lie algebras of the form L ⊗ A where L is aLie algebra and A is an associative commutative algebra, with multiplication defined by [ x ⊗ a , y ⊗ b ] = [ x , y ] ⊗ ab for x , y ∈ L , a , b ∈ A . Theorem 4.1.
Let L be a Lie algebra, A an associative commutative algebra, and at least one of L,A is finite-dimensional. Then each commutative -cocycle on L ⊗ A can be represented as a sum ofdecomposable cocycles ϕ ⊗ α , ϕ : L × L → K, α : A × A → K of one of the following types: (i) ϕ ([ x , y ] , z ) + ϕ ([ z , x ] , y ) + ϕ ([ y , z ] , x ) = and α ( ab , c ) = α ( ca , b ) , (ii) ϕ ([ x , y ] , z ) = ϕ ([ z , x ] , y ) and α ( ab , c ) + α ( ca , b ) + α ( bc , a ) = , (iii) ϕ ([ L , L ] , L ) = , (iv) α ( AA , A ) = ,and each of these four types splits into two subtypes: with both ϕ and α symmetric, and with both ϕ and α skew-symmetric.Proof. The proof goes almost verbatim to the proof of Theorem 1 in [Zu3]. (cid:3)
Corollary 4.2.
Suppose all assumptions of Theorem 4.1 hold, and, additionally, A contains a unit. ThenZ comm ( L ⊗ A ) ≃ Z comm ( L ) ⊗ A ∗ ⊕ C ( L ) ⊗ HC ( A ) ⊕ ( S ( L / [ L , L ])) ∗ ⊗ Ker ( S ( A ) → A ) ∗ ⊕ ( ∧ ( L / [ L , L ])) ∗ ⊗ { ab ∧ c + ca ∧ b + bc ∧ a | a , b , c ∈ A } ∗ , where the map S ( A ) → A is induced by multiplication in A. † A simple-minded GAP code for calculation of the spaces of commutative 2-cocycles and of symmetric bilinear invariantforms on a Lie algebra, is available at http://justpasha.org/math/comm2.gap . [Added January 4, 2016: currentlyavailable as an ancillary file accompanying the arXiv version of this paper]. The code works by writing the correspondingconditions in terms of a certain basis of a Lie algebra, and solving the arising linear homogeneous system.
OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 10
Proof.
Consider the cases of Theorem 4.1.Case (i). Substituting c = α ( ab , c ) = α ( ca , b ) , we get α ( a , b ) = β ( ab ) for a certainlinear form β : A → K . Consequently, both ϕ and α are necessarily symmetric, hence ϕ ∈ Z comm ( L ) andthe linear span of cocycles of this type is isomorphic to Z comm ( L ) ⊗ A ∗ .Case (ii). Similarly, substituting b = c = α ( ab , c ) + α ( ca , b ) + α ( bc , a ) = , and assuming that α is symmetric, we get α ( a , ) =
0. Now substituting just c = α vanishes. Consequently, for cocycles of this type both ϕ and α are necessarily skew-symmetric, hence ϕ ∈ C ( L ) and α ∈ HC ( A ) .Case (iv). If A contains a unit, then AA = A , so cocycles of this type vanish.Singling out from the linear span of the remaining cocycles of type (iii) the direct sum complement tothe linear span of cocycles of type (i) and (ii), and rearranging it in an obvious way, we get the desiredisomorphism. (cid:3) Evidently, the last two direct summands at the right-hand side of the isomorphism of Corollary 4.2constitute trivial cocycles.It is possible to extend Theorem 4.1 and Corollary 4.2 to various generalizations of current Lie alge-bras, such as twisted algebras, extended affine Lie algebras, toroidal Lie algebras, Lie algebras gradedby root systems, etc. Some of computations could be quite cumbersome, but all of them seem to beamenable to the technique used in [Zu3].It is possible also to consider a sort of noncommutative version of Corollary 4.2, namely, commuta-tive 2-cocycles on the Lie algebra sl ( n , A ) for an associative (and not necessarily commutative) algebra A with unit. It is possible to show that any homomorphic image of such an algebra is closely related tothe algebra of the form sl ( n , B ) , where B is a homomorphic image of A . If sl ( n , A ) possesses nonzero2-commutative cocycles, then by Lemma 2.1, at least one of these homomorphic images satisfies thestandard identity of degree 5. As sl ( n ) is a subalgebra of sl ( n , B ) , this implies that there are no com-mutative 2-cocycles if n >
2. The algebra sl ( , A ) , on the contrary, possesses nonzero 2-commutativecocycles, but in general there seems no nice expression for them in terms of A . As the final answer turnsout not to be very interesting in either case, we are not going into details.5. K AC -M OODY ALGEBRAS
If we want to apply the results of the preceding section to Kac-Moody algebras, we should deal notwith the current Lie algebras and their twisted analogs, but their extensions by means of central elementsand derivations. To this end, we make the following elementary observations.
Lemma 5.1.
Let L be a Lie algebra and I an ideal of L. Then Z comm ( L / I ) is embedded into Z comm ( L ) .Proof. There is an obvious bijection between Z comm ( L / I ) and the set of cocycles ϕ ∈ Z comm ( L ) such that ϕ ( L , I ) = (cid:3) Of course, the similar embedding exists for ordinary (skew-symmetric) cocycles, but this embeddingis usually not preserved on the level of cohomology. For example, free Lie algebras possess, in a sense,“the most” of 2-cocycles, accumulating all cocycles from their homomorphic images, but in the skew-symmetric case all cocycles are killed off by coboundaries. This has no parallel in the commutative casedue to absence of “commutative 2-coboundaries”.The following is a very particular complement, in a sense, to Lemma 5.1.
Lemma 5.2.
Let L be a Lie algebra and I a perfect ideal of codimension of L. Then Z comm ( L ) isembedded into Z comm ( I ) ⊕ K, the second direct summand being represented by a trivial cocycle.Proof.
Write L = I ⊕ Kx for some x ∈ L \ I . Let ϕ ∈ Z comm ( L ) . The cocycle equation on L is equivalentto the cocycle equation on I , plus the cocycle equation for a , b ∈ I and x , the latter could be written inthe form ϕ ([ a , b ] , x ) = ϕ ([ a , x ] , b ) − ϕ ([ b , x ] , a ) . OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 11 As [ I , I ] = I , this formula provides a well defined extension of ϕ from I × I to I × Kx .There are no further restrictions on the value of ϕ ( x , x ) , so defining a bilinear form ϕ on L by ϕ ( x , x ) = ϕ ( I , I ) = ϕ ( I , x ) =
0, we get a trivial cocycle which corresponds to the second direct summand. (cid:3)
We use the realization of affine Kac-Moody Lie algebras as extensions of non-twisted and twistedcurrent Lie algebras (see [K, Chapters 7 and 8]). Let g = L j ∈ Z n g j be a simple finite-dimensional Z n -graded Lie algebra (it is possible that g = g , i.e., n = g ⊗ K [ t , t − ] : L ( g , n ) = M i ∈ Z g i ( mod n ) ⊗ t i . This algebra has a non-split perfect central extension e L ( g , n ) . Each affine Kac-Moody algebra can berepresented in the form b L ( g , n ) = e L ( g , n ) ⊕ Kt ddt for a suitable g , where multiplication between elements of e L ( g , n ) and t ddt is defined by the action of thelatter as derivation on the algebra of Laurent polynomials K [ t , t − ] .We will consider first the case of non-twisted affine Kac-Moody algebras in a bit more general situ-ation. Let L be a Lie algebra, h· , ·i a nonzero symmetric bilinear invariant form on L , A a commutativeassociative algebra with unit, D a Lie subalgebra of Der ( A ) , and ξ a nonzero D -invariant element of HC ( A ) . Consider a Lie algebra defined as the vector space ( L ⊗ A ) ⊕ Kz ⊕ D with the following multi-plication: [ x ⊗ a , y ⊗ b ] = [ x , y ] ⊗ ab + h x , y i ξ ( a , b ) z [ x ⊗ a , d ] = x ⊗ d ( a ) for x , y ∈ L , a , b ∈ A , d ∈ D , and z belongs to the center. Note that the semidirect sum ( L ⊗ A ) A D is a quotient by the 1-dimensional central ideal Kz , and ( L ⊗ A ) ⊕ Kz is a subalgebra, the latter beingcentral extension of the current Lie algebra L ⊗ A (for generalities about central extensions of currentLie algebras, see [Zu2, Zu3, NW]).Specializing this construction to the case where K is an algebraically closed field of characteristiczero, L = g , a simple finite-dimensional Lie algebra, h· , ·i is the Killing form on g , A = K [ t , t − ] , D = Kt ddt , and ξ ( f , g ) = Res ( g d fdt ) for f , g ∈ K [ t , t , − ] , we get non-twisted affine Kac-Moody algebras. Lemma 5.3.
Let L be perfect, and one of L, A is finite-dimensional. Then † Z comm (( L ⊗ A ) A D ) ≃ Z comm ( L ) ⊗ { χ ∈ A ∗ | χ ( d ( a ) b − ad ( b )) = for any a , b ∈ A , d ∈ D }⊕ C ( L ) ⊗ { β ∈ HC ( A ) | β ( d ( a ) , b ) − β ( a , d ( b )) = for any a , b ∈ A , d ∈ D }⊕ Z comm ( D ) . Proof.
Let Φ ∈ Z comm (( L ⊗ A ) A D ) . A restriction of Φ to ( L ⊗ A ) × ( L ⊗ A ) is a commutative 2-cocycleon L ⊗ A . By Corollary 4.2, there are ϕ ∈ Z comm ( L ) , χ ∈ A ∗ , α ∈ C ( L ) and β ∈ HC ( A ) such that(11) Φ ( x ⊗ a , y ⊗ b ) = ϕ ( x , y ) χ ( ab ) + α ( x , y ) β ( a , b ) for any x , y ∈ L , a , b ∈ A . Writing the cocycle equation for x ⊗ a , y ⊗ b , d , we get(12) Φ ([ x , y ] ⊗ ab , d ) = ϕ ( x , y ) χ ( d ( a ) b − ad ( b )) + α ( x , y )( β ( d ( a ) , b ) − β ( a , d ( b ))) † Added May 24, 2011: By Lemma 1.2, the term containing C ( L ) is redundant here. The proofs of Lemmata 5.3 and 5.4can be simplified accordingly. OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 12 for any x , y ∈ L , a , b ∈ A , and d ∈ D . Substituting here a = b =
1, we get respectively: Φ ([ x , y ] ⊗ b , d ) = − ϕ ( x , y ) χ ( d ( b )) Φ ([ x , y ] ⊗ a , d ) = ϕ ( x , y ) χ ( d ( a )) , what implies Φ ( L ⊗ A , D ) =
0. Hence the right-hand side of (12) vanishes, and symmetrizing it fur-ther with respect to x , y , we get that both summands vanish separately. Thus we see that χ ∈ A ∗ and β ∈ HC ( A ) in formula (11) satisfy additional conditions χ ( d ( a ) b − ad ( b )) = β ( d ( a ) , b ) − β ( a , d ( b )) = a , b ∈ A .The cocycle equation for one element from L ⊗ A and two elements from D is satisfied trivially, andrestriction of Φ to D × D gives rise to an element from Z comm ( D ) . (cid:3) Lemma 5.4.
Let L be perfect, and one of L, A is finite-dimensional. ThenZ comm (( L ⊗ A ) ⊕ Kz A D ) ≃ Z comm (( L ⊗ A ) A D ) . Proof.
By Lemma 5.1, we have an embedding of Z comm (( L ⊗ A ) A D ) into Z comm (( L ⊗ A ) ⊕ Kz A D ) , and this embedding is an isomorphism if and only if(13) Φ (( L ⊗ A ) ⊕ Kz A D , z ) = Φ ∈ Z comm (( L ⊗ A ) ⊕ Kz A D ) .Writing the cocycle equation for x ⊗ a , y ⊗ b , z , we get(14) Φ ([ x , y ] ⊗ ab + h x , y i ξ ( a , b ) z , z ) = x , y ∈ L , a , b ∈ A . Substituting here b =
1, we get(15) Φ ( L ⊗ A , z ) = . Substituting the latter equality back to (14), we get Φ ( z , z ) = (( L ⊗ A ) ⊕ Kz ) × (( L ⊗ A ) ⊕ Kz ) , Φ gives rise to a commutative 2-cocycle on ( L ⊗ A ) ⊕ Kz , and due to (15), to a commutative 2-cocycle on L ⊗ A . Now we proceed as in the proofof Lemma 5.3: by Corollary 4.2, the equality (11) holds for any x , y ∈ L , a , b ∈ A and appropriate ϕ ∈ Z comm ( L ) , χ ∈ A ∗ , α ∈ C ( L ) , and β ∈ HC ( A ) . Then the cocycle equation for x ⊗ a , y ⊗ b , d , gives(16) Φ ([ x , y ] ⊗ ab , d ) + h x , y i ξ ( a , b ) Φ ( z , d )= ϕ ( x , y ) χ ( d ( a ) b − ad ( b )) + α ( x , y )( β ( d ( a ) , b ) − β ( a , d ( b ))) for any x , y ∈ L , a , b ∈ A , d ∈ D , and substitution of units in this equality yields Φ ( L ⊗ A , D ) = x , y , gives h x , y i ξ ( a , b ) Φ ( z , d ) = ϕ ( x , y ) χ ( d ( a ) b − ad ( b )) for any x , y ∈ L , a , b ∈ A , d ∈ D . Either both sides of this equality vanishes, in which case Φ ( z , d ) = d ∈ D , or ϕ ( x , y ) = λ h x , y i for some nonzero λ ∈ K . But the latter is obviously impossible (notethat the condition that the characteristic of the ground field is different from 3 is crucial here: as notedin [D3, § (cid:3) Affine Kac-Moody algebras, being non-perfect, possess trivial nonzero commutative 2-cocycles. Asthe commutant is always of codimension 1, the space of such cocycles is 1-dimensional. There are noother cocycles, as the following result shows.
Theorem 5.5.
Affine Kac-Moody algebras do not possess non-trivial commutative -cocycles.Proof. First consider the case of non-twisted algebras. By Lemmata 5.3 and 5.4, the space of commuta-tive 2-cocycles on a non-twisted affine Kac-Moody algebra ( g ⊗ K [ t , t − ]) ⊕ Kz A Kt ddt
OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 13 is isomorphic to Z comm ( g ) ⊗ { χ ∈ K [ t , t − ] ∗ | χ ( t d fdt g − f t dgdt ) = f , g ∈ K [ t , t − ] }⊕ C ( g ) ⊗ { β ∈ HC ( K [ t , t − ) | β ( t d fdt , g ) − β ( f , t dgdt ) = f , g ∈ K [ t , t − ] }⊕ Z comm ( Kt ddt ) . Let us look at the second tensor factor in the first direct summand. Substituting f = t i , g = t j in thedefining condition χ ( t d fdt g − f t dgdt ) = χ ∈ K [ t , t − ] ∗ , we get ( i − j ) χ ( t i + j ) = i , j ∈ Z such that i = j . Hence χ =
0, and the firstdirect summand vanishes.By Lemma 1.2, C ( g ) =
0, hence the second direct summand vanishes too.Finally, Z comm ( Kt ddt ) , being the space of commutative cocycles on the 1-dimensional Lie algebra, is1-dimensional and constitute trivial cocycles.Now consider the general (twisted) case. Suppose that b L ( g , n ) possesses non-trivial commutative2-cocycles. Then by Lemma 5.2, e L ( g , n ) possesses nonzero commutative 2-cocycles, and by Lemma2.1, e L ( g , n ) has a nonzero homomorphic image satisfying the standard identity of degree 5. Everyhomomorphic image of e L ( g , n ) is either a homomorphic image of L ( g , n ) , or is a central extension ofsuch. In the latter case, factoring by the central element, we will get again a homomorphic image of L ( g , n ) satisfying the standard identity of degree 5. But g is a homomorphic image of L ( g , n ) under theevaluation homomorphism ∑ i ∈ Z x i ⊗ t i ∑ i ∈ Z x i , where x i ∈ g , and all but a finite number of summands vanish. Thus g satisfies the standard identityof degree 5 too, hence g ≃ sl ( ) , and the corresponding Kac-Moody algebra is of type A ( ) , i.e. is anon-twisted algebra of the form ( sl ( ) ⊗ K [ t , t − ]) ⊕ Kz A Kt ddt . But the case of non-twisted algebras was already covered. (cid:3)
Note that there is some ambiguity in definition of affine Kac-Moody algebras, and sometimes theyare defined without employing the derivation extension, i.e. merely as central extensions of non-twistedor twisted current algebras (though some people argue that these are not “real” Kac-Moody algebras,as they are not Lie algebras corresponding to Cartan matrices). Such algebras are perfect, and possessnonzero commutative 2-cocycles only in the case of non-twisted type A ( ) . The proof is absolutelysimilar to those presented above.According to Lemma 5.4 (with D =
0) and Corollary 4.2, the space of commutative 2-cocycles onthe Kac-Moody algebra ( sl ( ) ⊗ K [ t , t − ]) ⊕ Kz of type A ( ) is infinite-dimensional and is isomorphic to Z comm ( sl ( )) ⊗ K [ t , t − ] ∗ , each cocycle being of the form ( x ⊗ f , y ⊗ g ) ϕ ( x , y ) ⊗ χ ( f g )( x ⊗ f , z ) ( z , z ) x , y ∈ sl ( ) , f , g ∈ K [ t , t − ] , for some ϕ ∈ Z comm ( sl ( )) , χ ∈ K [ t , t − ] ∗ . OMMUTATIVE 2-COCYCLES ON LIE ALGEBRAS 14
6. M
ODULAR SEMISIMPLE L IE ALGEBRAS
Essentially the same approach as in the previous section, allows to compute the space of commutative2-cocycles on finite-dimensional semisimple Lie algebras over the field of positive characteristic p .According to Block’s classical theorem (see, for example, [S, Corollary 3.3.6]), the typical examplesof such algebras are Lie algebras of the form ( S ⊗ O n ) A D where S is a simple Lie algebra, O n = K [ x , . . . , x n ] / ( x p , . . . , x pn ) is the reduced polynomial algebra in n variables, and D is a Lie subalgebraof W n = Der ( O n ) , the simple Lie algebra of the general Cartan type. To ensure semisimplicity, it isassumed that O n does not contain proper D -invariant ideals. Theorem 6.1.
Let n ≥ . Then Z comm (( S ⊗ O n ) A D ) ≃ Z comm ( D ) .Proof. According to Lemma 5.3, Z comm (( S ⊗ O n ) A D ) is isomorphic to Z comm ( S ) ⊗ { χ ∈ O ∗ n | χ ( d ( f ) g − f d ( g )) = f , g ∈ O n , d ∈ D }⊕ C ( S ) ⊗ { β ∈ HC ( O n ) | β ( d ( f ) , g ) − β ( f , d ( g )) = f , g ∈ O n , d ∈ D }⊕ Z comm ( D ) . Here again, the second tensor factor in the first direct summand vanishes. Indeed, substitution of g = χ ∈ O ∗ n , gives χ ( d ( f )) = f ∈ O n and d ∈ D . But then χ ( d ( f ) g ) = χ ( d ( f ) g − f d ( g )) + χ ( d ( f ) g + f ( d ( g ))) = χ ( d ( f g )) = f , g ∈ O n , d ∈ D . The space D ( O n ) O n is evidently a D -invariant ideal of O n , hence it coincideswith the whole O n , and χ = C ( S ) = (cid:3) This is similar in spirit to the computation of the second cohomology of some modular semisimpleLie algebras in [Zu1, § CKNOWLEDGEMENTS
Thanks are due to the anonymous referee for pointing few inaccuracies in and improvements to theprevious version of the manuscript, to Dimitry Leites for helpful comments, and to J. Marshall Osbornwho kindly supplied a reprint of his (hard-to-find) paper [O].R
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