Commutative Lie algebras and commutative cohomology in characteristic 2
aa r X i v : . [ m a t h . K T ] J u l COMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY INCHARACTERISTIC VIKTOR LOPATKIN AND PASHA ZUSMANOVICHA bstract . We discuss a version of the Chevalley–Eilenberg cohomology in characteristic 2, where thealternating cochains are replaced by symmetric ones. I ntroduction Define a commutative Lie algebra as a commutative algebra satisfying the Jacobi identity. While incharacteristic , ff erent: this class of algebras lies between ordinaryLie algebras (where commutativity is replaced by a stronger alternating property) and Leibniz algebras(where commutativity is dropped altogether), both inclusions are strict. The class of commutative Liealgebras admits a good cohomology theory: the cohomology is defined via the standard formula forthe di ff erential in the Chevalley–Eilenberg complex, with the alternating cochains being replaced bysymmetric ones.Why bother with such curiosity? We give four arguments, roughly in increasing degree of persua-siveness.1) From the operadic viewpoint, a “natural” class of algebras should be defined by multilinear identities.Moreover, the class of commutative Lie algebras appears naturally in certain algebraic topologicaland categorical contexts.2) The underlying complex based on symmetric cochains, unlike the usual one based on alternatingcochains, does not necessary vanish in degrees larger than the dimension of the algebra. This situ-ation is similar to those occurring in cohomology of Lie superalgebras or Leibniz algebras (in anycharacteristic), opens new possibilities, and poses new interesting questions.3) Commutative cohomology provides a new invariant of ordinary Lie algebras.4) Commutative cohomology of ordinary Lie algebras appears naturally in some problems related toclassification of simple Lie algebras.The present note is elucidation of points 2–4 (concerning point 1, see an interesting recent preprint[Et] for an operadic context, [L] for an algebraic topological context, and [GV] for a categorical con-text). While elementary in nature, this elucidation captures, in our opinion, some important phenomenapeculiar to characteristic 2 which will be important in the ongoing classification of simple Lie algebrasin that characteristic.Before we plunge into our considerations, a few remarks are in order. • Commutative 2-cocycles of Lie algebras in arbitrary characteristic do appear naturally in some cir-cumstances and were considered in [D], [DB], and [DZ], but, unlike in characteristic 2, they seeminglydo not lead to any cohomology theory. • For abelian (i.e., with trivial multiplication) Lie algebras, commutative cohomology may be defined inany characteristic. An instance of such second-degree cohomology appears in [Z1, §
5] in the context
Date : July 6, 2019 17:46 CEST.The research of the first author was supported by the grant of the Government of the Russian Federation for the statesupport of scientific research carried out under the supervision of leading scientists, agreement 14.W03.31.0030 dated15.02.2018. of calculating structure functions on manifolds of loops with values in compact hermitian symmet-ric spaces. It seems to be worthy to study this cohomology and associated structures further. (Amore-than-decade-ago promise from [Z1] to develop a “symmetric analogue of Spencer cohomologyrelated with symmetric analogue of Cartan prolongations and some Jordan algebras” remained, so far,unfulfilled). • The phenomenon of appearance of not necessary alternating 2-cocycles in characteristic 2 was notedalready in [J, § • Another interesting (and more sophisticated) versions of cohomology theory of Lie (super)algeb-ras attempting to fix deficiencies of the ordinary cohomology in characteristic 2 were suggested in[BGLL, § • Everything here can be dualized to get commutative homology . This is left as an exercise to the reader.We are interested primarily in cohomology, due to its application in structure theory, as explained in § efinitions Commutative Lie algebras.
Throughout this note, the ground field K is assumed to be of charac-teristic 2, unless stated otherwise. A commutative Lie algebra is an algebra L over K with multiplication[ · , · ] satisfying the commutative identity [ x , y ] = [ y , x ]and the Jacobi identity [[ x , y ] , z ] + [[ z , x ] , y ] + [[ y , z ] , x ] = x , y , z ∈ L . The usual Lie-algebraic notions of abelian algebra, simple algebra, center, ideal,quotient, derivations, deformations, module (including the notions of a trivial, adjoint, and dual module),are carried over commutative Lie algebras without any modification. When considered as an L -module, K is always understood as a trivial module.1.2. A note about terminology.
As noted in the introduction, in characteristic di ff erent from 2, com-mutative Lie algebras appeared in the literature under di ff erent names, see [Z3] and references therein.Neither of these names (“mock-Lie”, “Jacobi-Jordan”, “Jordan algebras of nilindex 3”, etc.) adequatelyreflects the characteristic 2 situation.Algebras satisfying the anticommutative identity[ x , y ] = − [ y , x ]and the Jacobi identity, appeared in [L], [GV], and references therein under the name “quasi-Lie al-gebras”. Quasi-Lie algebras in characteristic 2 are commutative Lie algebras in our terminology, andordinary Lie algebras in all other characteristics.1.3. Relation to Lie and Leibniz algebras.
As commutative Lie algebras form a subclass of Leibnizalgebras, the relationships between the classes of commutative and ordinary Lie algebras follow thealready established patterns. The Jacobi identity implies that in any commutative Lie algebra L , thesquares [ x , x ], where x ∈ L , linearly span the central ideal of L , denoted by L sq (cf. [LP, § L ann ). More generally, L sq acts trivially on any L -module M . The quotient L / L sq is a Lie algebra, and one may study commutative Lie algebras byconsidering corresponding extensions of Lie algebras, like it is done, for example, in [DA].In particular, in any simple commutative Lie algebra L this ideal vanishes, and hence L is a Liealgebra. Following [DA], one may consider the next possible minimal situation concerning ideals: acommutative Lie algebra will be called almost simple if each its proper ideal coincides with L sq . Forany almost simple commutative Lie algebra L , the quotient L / L sq is simple. Note that, unlike in Leibnizsetting, L sq is central, so, in the case L is not Lie, L sq is necessarily one-dimensional. OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 3
Commutative cohomology.
Let L be a commutative Lie algebra and M an L -module, with amodule action defined by • . A commutative cohomology of L with coe ffi cients in M , denoted byH • comm ( L , M ), is defined as cohomology of the cochain complex0 → S ( L , M ) d → S ( L , M ) d → S ( L , M ) d → . . . where S n ( L , M ) is the space of n -linear symmetric maps f : L × · · · × L | {z } n times → M , i.e. n -linear mapssatisfying f ( x σ (1) , . . . , x σ ( n ) ) = f ( x , . . . , x n )for any permutation σ ∈ S n . The di ff erential is defined asd ϕ ( x , . . . , x n + ) = X ≤ i < j ≤ n + ϕ ([ x i , x j ] , x , . . . , b x i , . . . , b x j , . . . , x n + ) + n + X i = x i • ϕ ( x , . . . , b x i , . . . , x n + ) . Note that this is the usual formula for di ff erential in the Chevalley–Eilenberg complex of a Lie algebrain characteristic 2 (i.e., all the signs being dropped). The cocycles and coboundaries in this complexwill be customary denoted by Z • comm ( L , M ) and B • comm ( L , M ), respectively.1.5. “De quadratum nihilo exaequari” † . The equality d = x ∈ L let i ( x ) be endomorphism of the vector space S • ( L , M ) = L n ≥ S n ( L , K ) whichmaps S n ( L , M ) to S n − ( L , M ) by the formula( i ( x ) f )( x , . . . , x n ) = f ( x , x , . . . , x n ) , and let θ be the natural representation of L in S n ( L , M ). Then, for any x , y ∈ L , the usual Cartan formulashold: θ ( x ) i ( y ) + i ( y ) θ ( x ) = i ([ x , y ]) i ( x ) d + d i ( x ) = θ ( x )(1.1) θ ( x ) d = d θ ( x )from what the desired equality d = = wouldinvolve expressions of the form [ u , u ], where u is some expression involving x , . . . , x n + . Similarly,the alternating, and not merely symmetric, property of cochains ϕ ’s would be required only in the casewhere d would involve expressions of the form ϕ ( u , u , . . . ). Neither of these is the case, and hence thecommutativity of the Lie bracket, and the symmetricity of cochains is enough.1.6. No derived functor?
The similarity with the Chevalley–Eilenberg cohomology, however, doeshave its limits: it is interesting to see where the standard proof that the Chevalley–Eilenberg cohomologyis the derived functor of the functor of taking the module invariants M M L (cf., e.g. [W, § L sq acts trivially on any module, the usual universal enveloping algebra U ( L / L sq ) should servethe purpose: the categories of representations of L and of U ( L / L sq ) are the same. Define the chaincomplex(1.2) . . . δ → U ( L / L sq ) ⊗ _ ( L ) δ → U ( L / L sq ) ⊗ _ ( L ) δ → U ( L / L sq ) ⊗ L δ → U ( L / L sq ) ε → K → † From Henri Cartan laudatio on then occasion of receiving Doctor Honoris Causa from the Oxford University.
OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 4 where W n ( L ) is the n -fold symmetric product of L , and ε is the augmentation map with kernel U + ( L / L sq ).The di ff erential is defined exactly by the same formula as in the Lie-algebraic (alternating) case: δ (cid:16) u ⊗ ( x ∨ · · · ∨ x n ) (cid:17) = X ≤ i < j ≤ n u ⊗ ([ x i , x j ] ∨ x ∨ · · · ∨ b x i ∨ · · · ∨ b x j ∨ · · · ∨ x n ) + n X i = ux i ⊗ ( x ∨ · · · ∨ b x i ∨ · · · ∨ x n ) , where u ∈ U ( L / L sq ), and x , . . . , x n ∈ L .By the same arguments as in the Lie-algebraic case – involving a version of Cartan formulas (1.1)for the complex (1.2) – we have δ =
0. However, the complex (1.2) is not exact, so, unlike in theLie-algebraic case, it is not a free resolution of the trivial module K . It is not exact already in the caseof abelian L (what, in the Lie-algebraic case, constitute the Koszul complex and essentially serves as an E page of the spectral sequence abutting to the homology in the general case): for example, the chain1 ⊗ ( x ∨ x ), for nonzero x ∈ L , belongs to Ker δ , but not to Im δ , since the latter in the second degree liesin U + ( L / L sq ) ⊗ S ( L ).Replacing in the complex (1.2) the symmetric product by the “alternating” one, i.e., by the quotientof the tensor algebra T • ( L ) by the ideal generated by elements of the form x ∨ x , x ∈ L , will not workeither: in characteristic 2, this “alternating” product is isomorphic to the exterior one, V • ( L ), and for thefinite-dimensional L , the so obtained complex is finite, while the symmetric cohomology apriori maynot vanish in an arbitrarily large degree (and it does not vanish indeed in all examples computed below).1.7. Motivation.
We have encountered commutative cohomology when started a project of descriptionof simple finite-dimensional Lie algebras having a Cartan subalgebra of toral rank 1, of which [GZ] is thebeginning. In the process, one need to compute various low-degree cohomology of current Lie algebras,i.e. Lie algebras of the form L ⊗ A where L is a Lie algebra and A is a commutative associative algebra,for certain particular instances of L and A . When one tries to extend the known formulas for suchcohomology in characteristics , , L . In [GZ], where we dealt with the case where L is the 3-dimensional simple algebra, commutative cohomology appear in disguise in Proposition 2.1. The resultsof this note will be used in subsequent classification e ff orts of simple Lie algebras in characteristic 2.2. E lementary observations Cohomology of low degree.
The usual interpretations of low-degree cohomology are triviallycarried over from Lie (and Leibniz) algebras to the commutative Lie case: H comm ( L , M ) = M L , themodule of invariants, H comm ( L , K ) ≃ ( L / [ L , L ]) ∗ , H comm ( L , L ) coincides with outer derivations of L ,H comm ( L , M ) describes equivalent classes of abelian extensions0 → M → · → L → , H comm ( L , L ) describes infinitesimal deformations of a commutative Lie algebra L , whereas obstructionsto prolongability of infinitesimal deformations to global ones live in H comm ( L , L ).In particular, the problem of description of almost simple commutative Lie algebras reduces to deter-mination of 1-dimensional central extensions 0 → Q sq → Q → L →
0, and hence to computation ofH comm ( L , K ) of all simple Lie algebras L .For any Lie algebra L defined over a field of characteristic ,
2, there is a useful exact sequence(2.1) 0 → H ( L , K ) → H ( L , L ∗ ) → B( L ) → H ( L , K )which goes back to classical works of Koszul and Hochschild–Serre (see, for example, [DZ, §
1] andreferences therein). Here H • ( L , M ) is the usual Chevalley–Eilenberg cohomology with coe ffi cients in an L -module M , and B( L ) is the space of symmetric invariant bilinear forms on L , i.e. symmetric bilinearmaps ϕ : L × L → K such that(2.2) ϕ ([ x , y ] , z ) = ϕ ([ z , x ] , y ) OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 5 for any x , y , z ∈ L . In characteristic 2, however, (2.1) is no longer true, but we have instead Proposition 1.
For any commutative Lie algebra L, there is a short exact sequence → H comm ( L , K ) → H comm ( L , L ∗ ) → B alt ( L ) → H comm ( L , K ) . Here B alt ( L ) denotes the space of all alternating bilinear maps satisfying (2.2). Proof.
The proof repeats the standard arguments used in establishing the exact sequence (2.1) or itscommutative analog in characteristic , (cid:3) Relation to Chevalley–Eilenberg and Leibniz cohomology.
The natural inclusion of alternatingmaps to symmetric ones induces, for any Lie algebra L , L -module M , and n ∈ N , a commutative diagramC n ( L , M ) d −−−−−→ C n + ( L , M ) y y S n ( L , M ) d −−−−−→ S n + ( L , M )where C n ( L , M ) is the usual space of alternating cochains, and d is the usual Chevalley-Eilenberg di ff er-ential. This, in its turn, induces the map(2.3) H n ( L , M ) → H ncomm ( L , M ) . Similarly, the natural inclusion of symmetric maps to all multilinear maps induces, for any commu-tative Lie algebra L and an L -module M , a commutative diagramS n ( L , M ) d −−−−−→ S n + ( L , M ) y y Hom K ( L ⊗ n , M ) d −−−−−→ Hom K ( L ⊗ n + , M )Here d in the bottom row denotes the di ff erential in the Leibniz complex. This, in its turn, induces themap(2.4) H ncomm ( L , M ) → HL n ( L , M ) , where HL • ( L , M ) denotes the Leibniz cohomology.Obviously, for n = , L , any 2-coboundary with arbitrarycoe ffi cients(2.5) d ϕ ( x , y ) = ϕ ([ x , y ]) + x • ϕ ( y ) + y • ϕ ( x ) , and any 3-coboundary with trivial coe ffi cientsd ϕ ( x , y , z ) = ϕ ([ x , y ] , z ) + ϕ ([ z , x ] , y ) + ϕ ([ y , z ] , x )is alternating, and hence the map (2.3) is an embedding for n =
2, and for n = M = K . Similarly,for any commutative Lie algebra L , the Leibniz 2-coboundary is given by the same formula (2.5), andhence the map (2.4) is an embedding for n =
2. In general, however, neither of the maps (2.3) and (2.4)is an embedding or a surjection.2.3.
Extension of the base field.
The standard arguments based on the universal coe ffi cient theorem,the same as in the case of ordinary Chevalley–Eilenberg cohomology, imply that the commutative co-homology does not change under field extension: if L is a commutative Lie algebra over a field K , and K ⊂ K ′ is a field extension, thenH ncomm ( L ⊗ K K ′ , M ⊗ K K ′ ) ≃ H ncomm ( L , M ) ⊗ K K ′ . OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 6
3. T he cup product
For a commutative Lie algebra L over a field K , define the bilinear map ⌣ : S • ( L , K ) × S • ( L , K ) → S • ( L , K )by the formula(3.1) ( ϕ ⌣ ψ )( x , . . . , x p + q ) = X IJ ϕ ( x i , . . . , x i p ) · ψ ( x j , . . . , x j q ) , where the sum is taken over all shu ffl es, i.e. partitions of the sequence { , . . . , p + q } into two disjointincreasing subsequences I = { i , . . . , i p } and J = { j , . . . , j q } .It is obvious that the so defined ⌣ turns S • ( L , K ) into a (graded) associative ring. Proposition 2.
The di ff erential d is a derivation of the ring S • ( L , K ) with respect to the product ⌣ .Proof. We need to prove that for any ϕ ∈ S p ( L , K ) and ψ ∈ S q ( L , K ), it holds that(3.2) d( ϕ ⌣ ψ ) = d ϕ ⌣ ψ + ϕ ⌣ d ψ. This is verified by direct computation: we have(d ϕ ⌣ ψ )( x , . . . , x p + q ) = X IJ (d ϕ )( x i , . . . , x i p ) · ψ ( x j , . . . , x j q ) = X IJ X ≤ r < s ≤ p ϕ ([ x i r , x i s ] , x i , . . . , b x i r , . . . , b x i s , . . . , x i p ) · ψ ( x j , . . . , x j q ) , ( ϕ ⌣ d ψ )( x , . . . , x p + q ) = X IJ ϕ ( x i , . . . , x i p ) · (d ψ )( x j , . . . , x j q ) = X IJ X ≤ l < t ≤ q ϕ ( x i , . . . , x i p ) · ψ ([ x j l , x j t ] , x j , . . . , b x j l , . . . , b x j t , . . . , x j q ) , and (cid:16) d( ϕ ⌣ ψ ) (cid:17) ( x , . . . , x p + q ) = X ≤ α<β ≤ p + q ( ϕ ⌣ ψ )([ x α , x β ] , x , . . . , b x α , . . . , b x β , . . . , x p + q ) = X IJ X ≤ r < s ≤ p ϕ ([ x i r , x i s ] , x i , . . . , b x i r , . . . , b x i s , . . . , x i p ) · ψ ( x j , . . . , x j q ) + X IJ X ≤ l < t ≤ q ϕ ( x i , . . . , x i p ) · ψ ([ x j l , x j t ] , x j , . . . , b x j l , . . . , b x j t , . . . , x j q ) , and the equality (3.2) follows. (cid:3) It is obvious that the derivation d preserves the grading of S • ( L , K ).As for any ring with a derivation D , the kernel Ker D is a subring, and the image of D is an ideal inKer D , we get: Corollary.
For any commutative Lie algebra L: (i)
The space Z • comm ( L , K ) of commutative cocycles is a subring of the ring S • ( L , K ) . (ii) The space B • comm ( L , K ) of commutative coboundaries is an ideal of the ring Z • comm ( L , K ) . (iii) The commutative cohomology H • comm ( L , K ) is a graded associative ring with respect to the product ⌣ .
4. E xamples
In this section we compute the commutative cohomology in several interesting cases.4.1.
Abelian algebra. If L is an abelian (commutative) Lie algebra, the di ff erential in the complexS • ( L , K ) vanishes, and H ncomm ( L , K ) = S n ( L , K ) for any n . OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 7 -dimensional algebra.
Obviously, the 1-dimensional commutative Lie algebra is abelian (andhence is a Lie algebra). For any module M over the 1-dimensional algebra K x , [ x , x ] =
0, we haveS n ( K x , M ) = Hom K (cid:0) ( K x ) ⊗ n , M (cid:1) ≃ M . The di ff erential d : S n ( K x , M ) → S n + ( K x , M ) reduces tod ϕ ( x , . . . , x ) = nx • ϕ ( x , . . . , x ), and hence both Lie commutative and Leibniz complexes are reduced tothe complex 0 → M → M x → M → M x → . . . whose cohomology is H ncomm ( K x , M ) = HL n ( K x , M ) ≃ Ker( x | M ) if n is evenCoker( x | M ) if n is odd . -dimensional algebra. Let L be the 2-dimensional nonabelian Lie algebra with the basis { a , b } ,[ a , b ] = a . Choose a basis in S n ( L , K ) consisting of the cochains χ pq , p + q = n , defined by χ pq ( a , . . . , a | {z } r , b , . . . , b | {z } s ) = p = r and q = s . We have: d χ pq ( a , . . . , a | {z } r , b , . . . , b | {z } s ) = rs χ pq ( a , . . . , a | {z } r − , a , b , . . . , b | {z } s − ) , and hence d χ pq = p ( q + χ p , q + . It follows that B ncomm ( L , K ) has a basis consisting of χ pq , where p + q = n , and both p , q are odd; andZ ncomm ( L , K ) has a basis consisting of χ pq , where p + q = n , and either p is even, or q is odd.Therefore, the cocycles χ pq , where p + q = n , and p is even, can be chosen as basic cocycles whoserepresentatives span H ncomm ( L , K ).To determine the cup product in terms of this basis, note that by (3.1), χ pq ⌣ χ rs ( a , . . . , a | {z } p + r , b , . . . , b | {z } q + s ) = p + rp ! q + sq ! , and hence χ pq ⌣ χ rs = p + rp ! q + sq ! χ p + r , q + s . In particular, χ p ⌣ χ s = χ ps , which shows that the basic cocycles of the form χ p and χ s generate the whole H • comm ( L , K ) as a ring.4.4. Heisenberg algebra.
The (2 ℓ + a , b , . . . , b ℓ , c , . . . , c ℓ , andmultiplication [ b i , a ] = [ c i , a ] = , [ b i , c j ] = a if i = j i , j , is called the Heisenberg algebra , and is denoted by H ℓ .To compute commutative cohomology of H ℓ with coe ffi cients in the trivial module, we will use alge-braic discrete Morse theory, briefly recalled in Appendix (which should be consulted for all undefinednotions and notation in this section). A very similar in spirit computation of the usual Chevalley–Eilen-berg homology of the Heisenberg algebra in characteristic 2, was performed earlier in [S1].Any cochain ϕ ∈ S n ( H ℓ , K ) is determined uniquely by its values on the basic elements:(4.1) ϕ ( a , . . . , a | {z } α , b , . . . , b | {z } β , . . . , b ℓ , . . . , b ℓ | {z } β ℓ , c , . . . , c | {z } γ , . . . , c ℓ , . . . , c ℓ | {z } γ ℓ ) , OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 8 where(4.2) α + β + · · · + β ℓ + γ + · · · + γ ℓ = n . Assuming β = ( β , . . . , β ℓ ) and γ = ( γ , . . . , γ ℓ ), the following shorthand notation will be used: ϕ ( α ; β ; γ ) will denote the corresponding value (4.1), and α + β + γ will denote the left-hand side of(4.2). At the same time, β ± β ′ denotes the vector obtained by the usual coordinate-wise addition orsubtraction of vectors in Z ℓ ≥ , i.e. ( β ± β ′ , . . . , β ℓ ± β ′ ℓ ), similarly for γ ’s. The vector of length ℓ having1 at the i th place, and 0 at all other places, will be denoted by i . Further, define I ( β ) = { i ∈ { , . . . , ℓ } | β i is even } I ( β ) = { i ∈ { , . . . , ℓ } | β i is odd } . For any triple ( α ; β ; γ ) such that α + β + γ = n +
1, we have:(4.3) d ϕ ( α ; β ; γ ) = X ≤ i ≤ ℓβ i > ,γ i > β i γ i ϕ ( α + β − i ; γ − i ) . Now choose a basis X n in S n ( H ℓ , K ) consisting of the cochains χ ( α ; β ; γ ) , α + β + γ = n , defined by χ ( α ; β ; γ ) ( ˜ α ; ˜ β ; ˜ γ ) = α = α, ˜ β = β, ˜ γ = γ . The formula (4.3) implies then d χ (0; β ; γ ) =
0, andd χ ( α ; β ; γ ) ( α − β + i ; γ + i ) = ( β i + γ i + α > ≤ i ≤ ℓ . This, in its turn, implies(4.4) d χ ( α ; β ; γ ) = P ℓ i = ( β i + γ i + χ ( α − β + i ; γ + i ) if α >
00 if α = . Now we are in the position to apply algebraic discrete Morse theory to the cochain complex (cid:0) S • ( H ℓ , K ) , d (cid:1) . In the graph Γ (cid:0) S • ( H ℓ , K ) (cid:1) constructed from this complex with the chosen basis S n ≥ X n ,define the set M consisting of all edges of the form χ ( α ; β ; γ ) → χ ( α − β + k ; γ + k ) , where k = max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) (so both β k , γ k are even), and k = max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) > max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) . The set M can be depicted as horizontal arrows in the following graph (where it is assumed that i , j ∈ I ( β ) ∩ I ( γ )): ... ... x x χ ( α − β + i ; γ + i ) −−−−−→ χ ( α − β + k + i ; γ + k + i ) x x χ ( α ; β ; γ ) −−−−−→ χ ( α − β + k ; γ + k ) x x χ ( α + β − j ; γ − j ) −−−−−→ χ ( α ; β + k − j ; γ + k − j ) x x ... ... OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 9
It is clear that after flipping all the horizontal arrows, the new graph Γ M (cid:0) S • ( H ℓ , K ) (cid:1) does not containdirected cycles. Also, no vertex is incident to more than one edge in M . Therefore, M is an acyclicmatching.The set of vertices in V = S n ≥ X n which do not serve as a tail for any arrow in M , is equal to(4.5) { χ (0; β ; γ ) } ∪ { χ ( α ; β ; γ ) | α > , max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) < max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) } , while the set of vertices which do not serve as a head for any arrow in M , is equal to(4.6) { χ ( α ; β ; γ ) | max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) > max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) } . Thus S n ≥ X Mn , being the intersection of the sets (4.5) and (4.6), is equal to the set C ∪ C , where C = { χ (0; β ; γ ) | max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) > max (cid:0) I ( β ) ∩ I ( γ ) (cid:1) } , and C = { χ ( α ; β ; γ ) | I ( β ) ∩ I ( γ ) = I ( β ) ∩ I ( γ ) = ∅ } . By (4.4), all cochains from both C and C are cocycles, and then by Theorem from Appendix A, C ∪ C forms a basis of the cohomology H • comm ( H ℓ , K ). (To be more precise, a basis of the n th degreecohomology H ncomm ( H ℓ , K ) is formed by cocycles from C with β + γ = n , and by cocycles from C with α + β + γ = n ).Let us look now at the ring structure of H • comm ( H ℓ , K ). For any two triples ( α ; β ; γ ) and ( α ′ ; β ′ ; γ ′ ) wehave: χ ( α ; β ; γ ) ⌣ χ ( α ′ ; β ′ ; γ ′ ) = α + α ′ α ! β + β ′ β ! γ + γ ′ γ ! χ ( α + α ′ ; β + β ′ ; γ + γ ′ ) , where (cid:16) β ′ β (cid:17) is a shorthand for the product (cid:16) β ′ β (cid:17) · · · (cid:16) β ′ ℓ β ℓ (cid:17) , similarly for γ ’s. From this formula it is clearthat C ⌣ C ⊆ C , C ⌣ C ⊆ C , and C ⌣ C ⊆ C , and therefore, as a ring, H • comm ( H ℓ , K ) isdecomposed into the semidirect sum of two subrings:H • comm ( H ℓ , K ) ≃ K C A K C , where K C acts on K C .4.5. Zassenhaus algebras.
The algebra W ( n ) is defined as an algebra of special derivations O ( n ) ∂ of the divided powers algebra O ( n ) (see, e.g., [DA], [J], or [GZ] for details). It has the basis { e i = x ( i + ∂ | − ≤ i ≤ n − } with multiplication[ e i , e j ] = (cid:16) i + j + i + (cid:17) e i + j if − ≤ i + j ≤ n −
20 otherwise . In characteristic 2, unlike in bigger characteristics, the algebra W ( n ) is not simple, but its commutant W ′ ( n ) of dimension 2 n −
1, linearly spanned by elements { e i | − ≤ i ≤ n − } , is. The algebras W ′ ( n )are referred as Zassenhaus algebras . The basic elements provide the standard grading W ′ ( n ) = n − M i = − Ke i . In the first nontrivial case n =
2, the algebra W ′ (2) is 3-dimensional, with multiplication table[ e − , e ] = e − , [ e , e ] = e , [ e − , e ] = e , and is an analog of sl (2) in big characteristics.Another realization of the algebra W ( n ) is defined over the field GF (2 n ) as the algebra with the basis { f α | α ∈ GF (2 n ) } and multiplication [ f α , f β ] = ( α + β ) f α + β for α, β ∈ GF (2 n ). Again, in characteristic 2 this algebra is not simple, but its commutant { f α | α ∈ GF (2 n ) ∗ } , isomorphic to W ′ ( n ), is. OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 10
For any k elements α , . . . , α k ∈ GF (2 n ) ∗ such that the sum of any number of these elements isnonzero, the 2 k − f α i + ··· + α i ℓ , 1 ≤ i ≤ · · · ≤ i ℓ ≤ k , span a subalgebra L ( α , . . . , α k ) of W ′ ( n )isomorphic to W ′ ( k ). Proposition 3. H comm ( W ′ ( n ) , K ) has dimension n. The basic cocycles can be chosen as (4.7) e i ∨ e j if i = j = k − , or { i , j } = {− , k + − } otherwise . for k = , . . . , n − .Proof. It is straightforward to check that the maps (4.7) are indeed commutative 2-cocycles (that boilsdown to the fact that if i , j ≥ i + j = k −
2, then [ e i , e j ] = (cid:16) k i + (cid:17) e k − = ff erent weights withrespect to the standard grading of W ′ ( n ). Thus dim H comm ( W ′ ( n ) , K ) ≥ n . To prove that we have here anequality, we will switch to the basis { f α } .We shall prove that the basic cocycles in H comm ( W ′ ( n ) , K ) can be chosen as(4.8) f α ∨ f β λ α if α = β α , β where α, β ∈ GF (2 n ) ∗ , λ α ∈ K , subject to linear relations(4.9) αλ α + βλ β + ( α + β ) λ α + β = α, β ∈ GF (2 n ) ∗ , α , β .We proceed similarly to [DB] where, in order to prove the vanishing of commutative 2-cocycles onsimple classical Lie algebras in characteristic >
2, first the rank 2 case is established, and the generalcase follows easily.So, first consider the cases n = n =
3. In that cases the statement follows from direct com-putations, similar to those performed in [D, Theorem 6.5] and [DB]. These computations can be alsoperformed on computer, using a simple GAP program for computations of the space of commutative2-cocycles on a given Lie algebra (see [DZ, footnote at § n ≥
3, take arbitrary α, β, γ ∈ GF (2 n ) ∗ , α + β + γ ,
0, and restrict an ar-bitrary cocycle ϕ ∈ Z comm ( W ′ ( n ) , K ) to the 7-dimensional subalgebra L ( α, β, γ ) linearly spanned by f α , f β , f γ , f α + β , f α + γ , f β + γ , f α + β + γ . Obviously, this restriction is a commutative 2-cocycle on L ( α, β, γ ) ≃ W ′ (3), and by the just established case n =
3, we have that first, ϕ ( f α , f β ) = ( α + β ) ω α,β,γ ( f α + β )for some linear map ω α,β,γ : L ( α, β, γ ) → K , and second, that the relation (4.9) holds for λ α = ϕ ( f α , f α ).Embedding the pair α, β into another triple α, β, γ ′ , we see that ω α,β,γ does not depend on γ . In the samevein, it does not depend neither on α , nor on β , so ϕ ( f α , f β ) = d ω ([ f α , f β ]) for any α, β ∈ GF (2 n ) ∗ , α , β ,and some linear map ω : W ′ ( n ) → K . Consequently, ϕ can be represented as the sum of d ω and a mapof the form (4.8). The latter maps are obviously commutative 2-cocycles, and we are done.It remains to determine the dimension of H comm ( W ′ ( n ) , K ). The relation (4.9) can be expanded as λ α + ··· + α k = α α + · · · + α k λ α + · · · + α k α + · · · + α k λ α k for any α , . . . , α k ∈ GF (2 n ) ∗ , α + · · · + α k ,
0, what means that dim H comm ( W ′ ( n ) , K ) is equal to thenumber of the generators of the additive group of GF (2 n ). The latter number is equal to dimension of GF (2 n ) as a vector space over GF (2), and hence is equal to n . (cid:3) Note that since dim B ( W ′ ( n ) , K ) = dim W ′ ( n ) = n −
1, we have dim Z comm ( W ′ ( n ) , K ) = n + n − W ′ ( n ) is a coboundary,and hence H ( W ′ ( n ) , K ) = OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 11
Eick’s algebras.
Commutative cohomology may serve as another invariant helping to distinguishalgebras. In [Ei], a computer-generated list of simple Lie algebras over GF (2) of dimension ≤
20 waspresented, and sophisticated (nonlinear) methods were used to establish non-isomorphism of algebrasin the list.For example, computer calculations with GAP show that the degree 2 commutative cohomology withtrivial coe ffi cients of the two new 15-dimensional simple Lie algebras in Eick’s list, number 7 and 8, isof dimension 1 and 2 respectively. All the other “conventional” “linear” invariants of these two algebraswe can think of (dimension of low-degree Chevalley–Eilenberg cohomology with trivial and adjointcoe ffi cients, dimension of the p -envelope and of the sandwich subalgebra, the absence of nondegeneratesymmetric invariant forms) do coincide. 5. F urther questions Finally, we take a liberty to indicate some avenues for further research. Some of the questions listedhere seem to be of a purely technical character, while others seem to be di ffi cult and probably willrequire new nontrivial approaches.5.1. Is it possible to represent the commutative cohomology as a derived functor? (This question seemsto be tricky, as it is hard to imagine what the other candidate for the role of the universal envelopingalgebra in the commutative case could be, see § comm ( sl n ( A ) , K ) and (a version of) cyclic cohomology of A in the spiritof [KL]. A glance at [GZ, Proposition 2.1] may suggest that the version of cyclic cohomology, peculiarto characteristic 2, which should appear here, is those where the (skew)symmetric cochains are replacedby alternating ones.5.4. Establish an analog of the Hopf formula for the second degree commutative homology with trivialcoe ffi cients.5.5. Define the cup product in § L ⊕ L and L (see, for example, [W,Exercise 7.3.8]). For this, of course, we will need (a version of) the K¨unneth formula for commutativecohomology.5.6. The classical Stallings-Swan theorem says that groups of cohomological dimension 1 are free.In characteristics 0 and 2 it is an open question whether Lie algebras of cohomological dimension1 are free. What about commutative Lie algebras? (Note that since we do not have a definition ofcommutative cohomology as a derived functor, the very notion of cohomological dimension in this caseis a bit problematic).5.7. It is well known (and easy to see) that the Euler-Poincar´e characteristic of cohomology of a finite-dimensional Lie algebra, i.e. the alternating sum of dimensions of cohomology in all degrees, vanishes.The very notion of the Euler–Poincar´e characteristic of the commutative cohomology does not makesense, as the sum dim H comm ( L , M ) − dim H comm ( L , M ) + dim H comm ( L , M ) − . . . is, generally, infinite and thus diverges. Can this sum be assigned a reasonable value using the theory ofdivergent series, similarly how it was (partially) done for cohomology of Lie superalgebras in [Z2]? OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 12 § ffi cientson the Zassenhaus algebra W ′ ( n ), is equal to 2 n + n −
1. Find a link with combinatorial interpretation ofthis number as the shortest length of a sequence of 0 and 1 containing all subsequences of length n (see[OEIS, A052944]).5.9. Whether the variety of commutative Lie algebras is Schreier, i.e., whether a subalgebra of a freecommutative Lie algebra is free?Let us note at the end that recently Friedrich Wagemann has constructed a Hochschild–Serre-likespectral sequence for commutative cohomology. The construction more or less repeats the constructionof the Hochschild–Serre spectral sequence for the Chevalley–Eilenberg cohomology.A cknowledgements Thanks are due to Alexei Lebedev, Dimitry Leites, Ivan Shestakov, and Friedrich Wagemann forstimulating discussions and useful remarks. During the early stages of this work, Lopatkin enjoyed thehospitality of Czech Technical University in Prague, with special thanks to Pavel ˇSˇtov´ıˇcek and ˇCestm´ırBurd´ık; Zusmanovich enjoyed the hospitality of University of S˜ao Paulo. GAP [G] was utilized to checksome computations performed in this paper.A ppendix . A lgebraic discrete M orse theory Algebraic discrete Morse theory is an algebraic version of discrete Morse theory developed indepen-dently by Sk¨oldberg, [S2], and by J¨ollenbeck and Welker, [JW]. It allows one to construct, starting froma chain complex, a new homotopically equivalent smaller complex using directed graphs. Here, forthe convenience of the reader, we present a short version of this machinery adapted for cochain, ratherthan chain, complexes (this can be done formally by considering cochain complexes as chain complexeswith negative indices and reverting arrows, but we prefer to write down everything explicitly). Wefollow closely [JW, Chapter 2], with minor simplifications and variations in notation.Let C : C → C → C → . . . be a cochain complex of vector spaces over a field K (which is assumed here to be of arbitrary char-acteristic; in fact, the whole theory is generalized, with slight modifications, to the case of arbitrarycomplexes of free modules over an arbitrary associative ring).Let X n be a basis of the vector space C n . Write the di ff erentials d n : C n → C n + with respect to thesebases: d n ( c ) = X c ′ ∈ X n + [ c : c ′ ] · c ′ , where c ∈ X n , and [ c : c ′ ] are coe ffi cients from K .From this data, we construct a directed weighted graph Γ (C) = ( V , E ). The set of vertices V of Γ (C)is the basis V = S n ≥ X n , and the set E of weighted edges consists of triples { ( c , c ′ , [ c : c ′ ]) | c ∈ X n , c ′ ∈ X n + , [ c : c ′ ] , } . A finite subset M ⊆ E of the set of edges is called an acyclic matching , if it satisfies the followingtwo conditions:(Matching) Each vertex v ∈ V lies in at most one edge e ∈ M .(Acyclicity) The subgraph Γ M (C) = ( V , E M ) of the graph Γ (C) has no directed cycles, where E M = ( E \ M ) ∪ { ( c ′ , c , − c : c ′ ] ) | ( c , c ′ , [ c : c ′ ]) ∈ M } . For an acyclic matching M on the graph Γ (C), we introduce the following notation:(1) Define X Mn = { c ∈ X n | c does not lie in any edge e ∈ M } . OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 13 (2) Write c ′ ≤ c if c ∈ X n , c ′ ∈ X n + , and [ c : c ′ ] , c , c ′ ) is the set of paths from c to c ′ in Γ M (C).(4) The weight w ( p ) of a path p = c → . . . → c r ∈ Path( c , c r ) is defined as w ( c → . . . → c r ) = r − Y k = w ( c k → c k + ) w ( c → c ′ ) = − c : c ′ ] if c ≤ c ′ [ c : c ′ ] if c ′ ≤ c . The following is a cohomological version of [JW, Theorem 2.2]:
Theorem.
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OMMUTATIVE LIE ALGEBRAS AND COMMUTATIVE COHOMOLOGY IN CHARACTERISTIC 2 14 S oftware and online repositories [G] The GAP Group, GAP – Groups, Algorithms, and Programming , Version 4.7.8, 2015; [OEIS] The On-Line Encyclopedia of Integer Sequences; http://oeis.org/ (Viktor Lopatkin) L aboratory of M odern A lgebra and A pplications , S t . P etersburg S tate U niversity and S t . P eters - burg D epartment of S teklov M athematical I nstitute , S t . P etersburg , R ussia E-mail address : [email protected] (Pasha Zusmanovich) U niversity of O strava , O strava , C zech R epublic E-mail address ::