# Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2

Andrey Krutov, Alexei Lebedev, Dimitry Leites, Irina Shchepochkina

aa r X i v : . [ m a t h . R T ] F e b NONDEGENERATE INVARIANT SYMMETRIC BILINEAR FORMS ONSIMPLE LIE SUPERALGEBRAS IN CHARACTERISTIC 2

ANDREY KRUTOV A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E Abstract.

As is well-known, the dimension of the space spanned by the non-degenerate in-variant symmetric bilinear forms (NISes) on any simple ﬁnite-dimensional Lie algebra or Liesuperalgebra is equal to at most 1 if the characteristic of the algebraically closed ground ﬁeldis not 2.We prove that in characteristic 2, the superdimension of the space spanned by NISes canbe equal to 0, or 1, or 0 |

1, or 1 |

1; it is equal to 1 | Introduction

This paper is a sequel to [BKLS], where non-degenerate invariant symmetric bilinear forms(NISes) on known simple Lie algebras and Lie superalgebras (ﬁnite-dimensional and Z -graded ofpolynomial growth) are listed if the characteristic p of the ground ﬁeld K is distinct from 2, andfor occasional examples where p = 2. Here we mainly consider an algebraically closed ﬁeld K for p = 2. For numerous applications of Lie (super)algebras with a NIS, see [DSB, BeBou, BLS].Our main result is a newly discovered fact in characteristic 2 — the description of the possiblesuperdimension of the space of NISes on a given simple ﬁnite-dimensional Lie superalgebra, seeTheorem 2.1. For descriptions of simple ﬁnite-dimensional Lie superalgebras over algebraicallyclosed ﬁeld of characteristic 2, see [BGLLS] and references therein.For completeness, we consider Theorem 2.1 for any characteristic since we do not know ifit was ever published in full generality, although known as a folklore for p = 2. If p = 2,Theorem 2.1 seems to be “just a direct generalization” of a well-known fact about Lie algebras, unless we realize that we might encounter objects inhomogeneous with respect to the parity .For p = 2, the situation is even more delicate; its complete description is a new result, cf. [SF].For results on NISes on simple ﬁnite-dimensional Lie algebras over algebraically closed ﬁeldsof characteristic p = 2, see [BZ, Bl, GP] and [BKLS].1.1. Generalities.

From the super K -functor point of view, see [Mi], the superdimension of a given superspace V is sdim V := dim V ¯0 + ε dim V ¯1 , where ε = 1; usually one writessdim V = dim V ¯0 | dim V ¯1 , so ε = 0 |

1. We write dim V := dim V ¯0 + dim V ¯1 .Let g be a Lie superalgebra over K . A bilinear form B on g is called homogenous (withrespect to parity) if1) g ¯0 ⊥ g ¯1 , in this case it is called even ; Mathematics Subject Classiﬁcation.

Primary 17B50.

Key words and phrases.

Restricted Lie algebra, simple Lie algebra, characteristic 2, queeriﬁcation, Lie su-peralgebra, non-degenerate invariant symmetric bilinear form.

ANDREY KRUTOV A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E g ¯0 and g ¯1 are isotropic subspaces, in this case it is called odd .Recall that a bilinear form B is called invariant if the following condition holds B ([ x, z ] , y ) − B ( x, [ z, y ]) = 0 for all x, y, z ∈ g . A non-degenerate invariant symmetric bilinear form on a Lie (super)algebra will be brieﬂycalled NIS; just invariant and symmetric will be brieﬂy called IS.

1) Speaking about “the space of NISes”, we exercise the usual abuse ofthe language: we are speaking about the space spanned by all NISes, but not all forms in thisspace have to be non-degenerate. For example, the zero form is never non-degenerate. Observethat all forms in the space are, however, IS.In the case where “the space of NISes” is of superdimension 1 |

1, it contains a 1-dimensionalinhomogeneous (with respect to parity) space of degenerate forms. We mean that, up toa nonzero factor, there is one even non-degenerate form and one odd non-degenerate form.2) The ground ﬁeld is algebraically closed of characteristic p (mainly, p = 2 or 0), unlessotherwise speciﬁed.3) We consider only ﬁnite-dimensional (super)algebras.4) In this note, all (super)commutative (super)algebras are supposed to be associative with 1;their morphisms should send 1 to 1, and the morphisms of supercommutative superalgebrasshould preserve parity.1.3. On the contents of this paper.

Section 2 contains the Main Theorem 2.1 — the mostinteresting result of the paper.Section 3 contains vital material: even the deﬁnition of Lie superalgebra for p = 2 (and 3) isnot obvious. We also recall deﬁnition of classical restrictedness of Lie superalgebra for p = 2;for non-classical ones, see [BLLS].Section 4 contains several examples illustrating the Main Theorem 2.1.In Section5, we examine existence of multiple NISes over non-closed ﬁelds and deﬁne a Liesuperalgebra structure on L ⊗ A in some cases where L is a Lie superalgebra and A is anon-supercommutative associative superalgebra.2. Main theorem (Main Theorem).

Let K be an algebraically closed ﬁeld of characteristic p . If p = 2 , any NIS on a simple ﬁnite-dimensional Lie superalgebra is homogeneous withrespect to parity, and the dimension of the space spanned by NISes is ≤ .More precisely, the superdimension of the space spanned by NISes is either (no NIS), or in this case, all NISes are even ) , or ε ( in this case, all NISes are odd ) . If p = 2 , the superdimension of the space spanned by NISes on a simple ﬁnite-dimensionalLie superalgebra is equal to either (no NIS), or , or ε , or | . This superdimension is equal to | if and only if the Lie superalgebra is a queer-iﬁcation (see Subsection 3.5) of a simple restricted Lie algebra with a NIS . (Hererestrictedness is understood in the classical sense, see Subsection 3.3; for classiﬁcation in variouscases of simple Lie (super)algebras with a NIS, and deﬁnitions of other types of restrictedness,indigenous for p = 2, see [BKLS].)2.2. A reformulation of Main Theorem.

The proof of Theorem 2.1 in case p = 2 actuallyproves the following Theorem 2.2.1, which implies Theorem 2.1 for p = 2 in its turn. IS IN CHARACTERISTIC 2 3 (A version of Main Theorem).

Let K be an algebraically closed ﬁeld ofcharacteristic p = 2 .If a simple ﬁnite-dimensional Lie superalgebra g has a NIS, then • the space spanned by all NIS-forms on g is a superspace; • the dimension of the space of even IS-forms on g is ≤ ; • the dimension of the space of odd IS-forms on g is ≤ ; • i.e., the space spanned by all NIS-forms on g is a superspace whose superdimension is equalto either , or ε , or | since the even and odd components of an IS-form are also IS-forms ) . • Any homogeneous IS-form on g is either or non-degenerate.Proof. First, we prove several lemmas which allow us to restrict ourselves to homogenous NISes. (On homogenous invariant symmetric forms for p = 2). Let g be a simple Liesuperalgebra over a ﬁeld of characteristics p = 2 . Then, any homogenous invariant symmetricform on g is either zero or non-degenerate.Proof. Let ω be a homogenous IS-form on g . Then, Ker ω is a subsuperspace of g invariantwith respect to ad g , i.e., it is an ideal. Since g is simple, either Ker ω = 0, in which case ω isnon-degenerate, or Ker ω = g , in which case ω = 0. (cid:3) The same is not true in characteristic 2. The problem is that even though Ker ω is invariantwith respect to ad g , it may be not an ideal, since it may be not closed under squaring. Beforewe formulate a similar statement for p = 2, let us prove two more lemmas, where we assume p = 2. (The lemmas are true for p = 2 as well, but they are trivial in that case.) (A necessary condition for NIS). Let p = 2 . Let g be a simple Lie superalgebrawith a NIS ( − , − ) . Then, [ g , g ] = g . (Peculiarity of the p = 2 case). One might think “since [ g , g ] is an idealin g which is supposed to be simple, we have nothing to prove”. But for Lie super algebras incharacteristic 2 there is a diﬀerence between the commutant [ g , g ] := Span([ x, y ] | x, y ∈ g ),and the ﬁrst derived algebra. Recall that the derived Lie superalgebras of g are deﬁned to be(for i ≥ g (0) := g , g ( i +1) := ( [ g ( i ) , g ( i ) ] for p = 2,[ g ( i ) , g ( i ) ] + Span { g | g ∈ g ( i )¯1 } for p = 2 . Proof of Lemma . . Suppose [ g , g ] = g . Then, the orthogonal complement to [ g , g ] with re-spect to ( − , − ) contains a non-zero element u . Since g is simple, it has zero center, and hencethere exists an x ∈ g such that [ u, x ] = 0. Since the form ( − , − ) is non-degenerate, there existsa y ∈ g such that ([ u, x ] , y ) = 0. But then ( u, [ x, y ]) = ([ u, x ] , y ) = 0, contradicting the factthat u ∈ [ g , g ] ⊥ . (cid:3) (A technical lemma). Let p = 2 . Let g be a simple Lie superalgebra such that [ g , g ] = g , and S ⊆ g its subsuperspace such that [ S, g ] ⊆ S . Then, either S = 0 or S = g .Proof. If S = 0, let S be the completion of S with respect to squaring. Since S is a subsuper-space of g , is closed under squaring, and [ S, g ] = [ S, g ] ⊆ S , it follows that S is an ideal, andhence S = g . But then g = [ g , g ] = [ S, g ] ⊆ S , and therefore S = g . (cid:3) Now we can formulate the statement we will use instead of Lemma 2.3 when p = 2: ANDREY KRUTOV A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E (Analog of Lemma 2.3). Let p = 2 . Let g be a simple Lie superalgebra and letthere be a NIS on g . Then, any homogenous invariant symmetric form on g is either zero ornon-degenerate.Proof. The proof is analogous to the proof of Lemma 2.3, but we use Lemmas 2.4 and 2.5 toshow that Ker ω is either 0 or g . (cid:3) Completion of the proof of Theorem 2.1 . Now we can restrict ourselves to consideringhomogenous non-degenerate invariant symmetric forms, since for any non-homogenous NIS, itseven and odd components are invariant and symmetric, and hence non-degenerate by Lemmas2.3 and 2.6.Fix a basis in g , and let B and B be Gram matrices of non-degenerate homogenousinvariant symmetric forms ω and ω ; consider the 1-parameter family of invariant symmetricforms ω λ , where λ ∈ K , with Gram matrices B λ = B + λB . Consider B λ just as a matrix,not supermatrix, and calculate its determinant; it is a polynomial of λ .Since K is algebraically closed , there exists a λ ∈ K such that det B λ = 0. Then, the form ω λ is degenerate. If the forms ω and ω are of the same parity, then ω λ = 0 by Lemmas 2.3and 2.6. This means that ω = − λ ω , i.e., any two homogeneous NISes of the same parity areproportional to one another. So, if ω and ω are even, then the superdimension of the spaceof even NISes is equal to 1; if ω and ω are odd, then the superdimension of the space of oddNISes is equal to ε .Let now forms ω and ω be of diﬀerent parity. Then, Ker ω λ is a non-trivial ad g -invariantsubspace, but it is not a subsuperspace. Consider two subspaces V := Ker ω λ ∩ g ¯0 ⊕ Ker ω λ ∩ g ¯1 ,W := pr ¯0 (Ker ω λ ) ⊕ pr ¯1 (Ker ω λ ) , where pr ¯0 and pr ¯1 are projections to g ¯0 and g ¯1 , respectively. Since both V and W are ad g -invariant sub super spaces, they are either 0 or g by Lemmas 2.4 and 2.5. Since ω λ is non-zero, V = g , so V = 0; since ω λ is degenerate, W = 0, so W = g . Hence, there exists an oddisomorphism f between linear superspaces g ¯0 and g ¯1 such thatKer ω λ = Span { x + f ( x ) | x ∈ g ¯0 } . Since Ker ω λ is ad g -invariant, we see that for all a, b ∈ g :(2) [ a, b + f ( b )] = [ a, b ] + [ a, f ( b )] = ⇒ [ a, f ( b )] = f ([ a, b ]);[ a + f ( a ) , b ] = [ a, b ] + [ f ( a ) , b ] = ⇒ [ f ( a ) , b ] = f ([ a, b ]);[ f ( a ) , b + f ( b )] = [ f ( a ) , b ] + [ f ( a ) , f ( b )]= [ f ( a ) , f ( b )] + f ([ a, b ]) = ⇒ f ([ a, b ]) = f ([ f ( a ) , f ( b )]) . The bottom line in (2) implies that(3) [ f ( a ) , f ( b )] = [ a, b ] for all a, b ∈ g ¯0 .Up to this moment our reasoning did not depend on p .Now, let p = 2. Notice that the left-hand side of the equality (3) is symmetric while theright-hand side is anti-symmetric. Hence,[ f ( a ) , f ( b )] = [ a, b ] = 0 and [ f ( a ) , b ] = f ([ a, b ]) = 0 for all a, b ∈ g ¯0 , and so g is commutative. This contradicts the simplicity of g , and hence there can not existtwo NISes of diﬀerent parity on a simple Lie superalgebra over an algebraically closed ﬁeld of IS IN CHARACTERISTIC 2 5 characteristic p = 2. There can not exist an inhomogeneous NIS in this situation either, as wasmentioned above. This completes the proof of the theorem if p = 2.If p = 2, no such conclusion follows from equality (3); it only tells us that g is a queeriﬁcation(see Subsection 3.5 and [BLLS]) of g ¯0 and g ¯0 is (classically) restricted. Besides, the restrictionof the even of the two forms ω i to g ¯0 is a NIS.Conversely, let g be a queeriﬁcation of a simple restricted Lie algebra g ¯0 with a NIS ω . Then, g can be represented in the form g = g ¯0 ⊗ A , where A is an associative and commutative, but not supercommutative, superalgebra spanned by an even element 1 (unit) and an odd one a ,subject to the relation a = 1, and the natural bracket (for squaring, see Subsection 5.2):[ x ⊗ ϕ, y ⊗ ψ ] = [ x, y ] ⊗ ϕψ for any x, y ∈ g ¯0 , ϕ, ψ ∈ A .Determine two bilinear forms on g := g ¯0 ⊗ A :(4) ω i ( x ⊗ ϕ, y ⊗ ψ ) = ω ( x, y ) f i ( ϕψ ) for i = 1 , , where f ( α · β · a ) = α and f ( α · β · a ) = β for any α, β ∈ K . It is clear that both theseforms are non-degenerate, invariant, and symmetric; ω is even and ω is odd. (cid:3) On degenerate invariant symmetric bilinear forms. Let p = 2 and let g be a simpleﬁnite-dimensional Lie superalgebra such that [ g , g ] = g , i.e., there are elements which cannot be obtained by bracketing, only by squaring, and let k := codim [ g , g ]; for example, if g = osp (1) I Π (1 | k = 2. The dimension of the space of invariant degenerate symmetricbilinear forms on g is ≥ k ( k + 1) = dim S ( g / [ g , g ]), because this is the dimension of the spaceof the forms whose kernel contains [ g , g ]. Such forms ( − , − ) are invariant because for them,([ x, y ] , z ) = ( x, [ y, z ]) = 0 for any x, y, z ∈ g .3. Basics

The Sign Rule, skew and anti.

The deﬁnitions of

Lie superalgebra are the same forany p = 2 or 3: they are obtained from the deﬁnition of the Lie algebra using the Sign Rule “ifsomething of parity p is moved past something of parity q , the sign ( − pq accrues; formulasdeﬁned on homogeneous elements are extended to any elements via linearity”.In addition to the Sign Rule, note that morphisms of superalgebras are only even ones.Observe that sometimes applying the Sign Rule requires some dexterity (we have to distin-guish between two versions both of which turn in the nonsuper case into one, called either skew-or anti-commutativity, see [Gr]): ba = ( − p ( b ) p ( a ) ab (super commutativity) ba = − ( − p ( b ) p ( a ) ab (super anti-commutativity) ba = ( − ( p ( b )+1)( p ( a )+1) ab (super skew-commutativity) ba = − ( − ( p ( b )+1)( p ( a )+1) ab (super antiskew-commutativity)In the case of characteristic 2, of main interest to us, the above conditions turn into one, so super commutativity and super antiskew-commutativity are deﬁned as ab = ba for all a, b , and a = 0 for p ( a ) = ¯1,whereas super anti-commutativity and super skew-commutativity are deﬁned as ab = ba for all a, b , and a = 0 for p ( a ) = ¯0. Recall that in any super commutative superalgebra A , we have a = 0 for any a ∈ A ¯1 . For example, setting p ( i ) = ¯1 we endow C with a structure of a non-supercommutative superalgebra over R . ANDREY KRUTOV A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E Lie superalgebra, pre-Lie superalgebra, Leibniz superalgebra.

Generalization ofnotions of this Subsection to super rings and modules over them is immediate.For any p , a Lie superalgebra is a superspace g = g ¯0 ⊕ g ¯1 such that the even part g ¯0 is a Liealgebra, the odd part g ¯1 is a g ¯0 -module (made into the two-sided one by anti -symmetry, i.e.,[ y, x ] = − [ x, y ] for any x ∈ g ¯0 and y ∈ g ¯1 ) and on g ¯1 , a squaring x x and the bracket are deﬁned via a linear map s : S ( g ¯1 ) −→ g ¯0 , where S denotes the operator of raising tosymmetric square, as follows, for any x, y ∈ g ¯1 : x := s ( x ⊗ x );(5) [ x, y ] := s ( x ⊗ y + y ⊗ x ) . (6)The linearity of the g ¯0 -valued function s implies that( ax ) = a x for any x ∈ g ¯1 and a ∈ K , and(7) [ · , · ] is a bilinear form on g ¯1 with values in g ¯0 .(8)The Jacobi identity involving odd elements takes the form of the following two conditions:[ x , y ] = [ x, [ x, y ]] for any x ∈ g ¯1 , y ∈ g ¯0 , (9) [ x , x ] = 0 for any x ∈ g ¯1 .(10)In the literature describing Lie superalgebras in characteristics p = 2 ,

3, the Jacobi identityis usually given for three elements, and when all those elements are odd, it takes the form(11) [ x, [ y, z ]] + [ y, [ z, x ]] + [ z, [ x, y ]] = 0 for any x, y, z ∈ g ¯1 .This equation follows from (10) in any characteristic; also, if p = 2 ,

3, then (10) follows from(11), since, by substituting x = y = z , we get 6[ x, x ] = 0. The more general equation (10) isnecessary for the super version of the Poincare-Birkhoﬀ-Witt theorem to hold, and this is whywe use it.If p = 3, and the Jacobi identity is formulated in the usual way, i.e., for three elements, thenthe condition(12) [ x, [ x, x ]] = 0 for any x ∈ g ¯1 should be added to the antisymmetry and Jacobi identity amended by the Sign Rule, separately.If p = 3, the superalgebra satisfying the antisymmetry and the Jacobi identity, but not thecondition (12) is called pre-Lie superalgebra ; for examples, see [BeBou].If p = 2, the antisymmetry for p = 2 should be replaced by an equivalent for p = 2, butotherwise stronger alternating or antisymmetry condition[ x, x ] = 0 for any x ∈ g ¯0 . For any p , the superalgebra satisfying the Jacobi identity, and without any restriction onsymmetry is called a Leibniz superalgebra .Over Z /

2, the condition (10) must (see Example 3.2.1) be replaced with a more general one:(13) [ x , y ] = [ x, [ x, y ]] for any x, y ∈ g ¯1 .For any other ground ﬁeld this condition is equivalent to condition (10). (On the Jacobi identity over Z / Z / | g withthe even part spanned by elements A and B , the odd part spanned by elements X , Y and Z ,and the algebraic structure given as follows: IS IN CHARACTERISTIC 2 7 • the even part is commutative; • the action of g ¯0 on g ¯1 is given by the following multiplication table X Y ZA Z B Z • the squaring on g ¯1 is given by the formula( aX + bY + cZ ) = a A + b B for all a, b, c ∈ Z / . This algebra g satisﬁes (10) since[( aX + bY + cZ ) , aX + bY + cZ ] = [ a A + b B, aX + bY + cZ ] = ( a b + ab ) Z, and since a, b ∈ Z /

2, we have a = a and b = b , so a b + ab = 0.But the algebra g does not satisfy condition (13): [ X , Y ] = Z = [ X, [ X, Y ]] = 0.3.2.2.

Ideals, simplicity, derived algebras, modules.

By an ideal of a Lie superalgebraone always means an homogeneous ideal; for p = 2, the ideal should be closed with respect tosquaring.The Lie superalgebra g is said to be simple if dim g > g has no nontrivial (distinctfrom 0 and g ) ideals.For p = 2, the deﬁnition of the derived algebra of the Lie superalgebra g changes, see eq. (1).An even linear map r : g −→ gl ( V ) is said to be a representation of the Lie superalgebra g and V is a g -module if r ([ x, y ]) = [ r ( x ) , r ( y )] for any x, y ∈ g ; r ( x ) = ( r ( x )) for any x ∈ g ¯1 .Since we want der g to be a Lie superalgebra for any Lie superalgebra g , we have to adda generalization of conditions (9), (10), namely, the condition(14) D ( x ) = [ D ( x ) , x ] for odd elements x ∈ g and any D ∈ der g . Clearly, (14) turns into conditions (9), (10) for D = ad y , where y ∈ g .3.3. The p | p -structure or restricted Lie superalgebra. Let the ground ﬁeld K be ofcharacteristic p >

0, and g a Lie algebra. For every x ∈ g , the operator (ad x ) p is a derivation of g .If this derivation is an inner one, then there is a map (called p -structure ) [ p ] : g −→ g , x x [ p ] such that [ x [ p ] , y ] = (ad x ) p ( y ) for any x, y ∈ g , ( ax ) [ p ] = a p x [ p ] for any a ∈ K , x ∈ g , ( x + y ) [ p ] = x [ p ] + y [ p ] + P ≤ i ≤ p − s i ( x, y ) for any x, y ∈ g , where is i ( x, y ) is the coeﬃcient of λ i − in (ad λx + y ) p − ( x ), then the Lie algebra g is said to be restricted or having a p -structure .For a Lie superalgebra g in characteristic p >

0, let the Lie algebra g ¯0 be restricted and(15) [ x [ p ] , y ] = (ad x ) p ( y ) for any x ∈ g ¯0 , y ∈ g . This gives rise to the map [2 p ] : g ¯1 → g ¯0 , x ( x ) [ p ] , satisfying the condition(16) [ x [2 p ] , y ] = (ad x ) p ( y ) for any x ∈ g ¯1 , y ∈ g . ANDREY KRUTOV A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E The pair of maps [ p ] and [2 p ] is called a p - structure (or, sometimes, a p | p - structure ) on g ,and g is said to be restricted , cf. [WZ]. It suﬃces to determine the p | p -structure on any basisof g ; on simple Lie superalgebras there is at most one p | p -structure. • If condition (15) is not satisﬁed, the p -structure on g ¯0 does not have to generate a p | p -structure on g : even if the actions of (ad x ) p and ad x [ p ] coincide on g ¯0 , they do not have tocoincide on the whole of g . • The restricted universal enveloping (super)algebra U [ p ] ( g ), deﬁned for restricted Lie alge-bras g as the quotient of the universal enveloping U ( g ) modulo the two-sided ideal generated by g ⊗ p − g [ p ] for any g ∈ g , should be deﬁned for the restricted Lie superalgebra g as the quotientof U ( g ) modulo the two-sided ideal i generated by g ⊗ p − g [ p ] for any g ∈ g ¯0 .The seemingly needed further factorization modulo the two-sided ideal generated by theelements g ⊗ p − g [2 p ] for any g ∈ g ¯1 is not needed: these elements are in i automatically, as isnot diﬃcult to show. • If p = 2, there are other , no less natural, versions of restrictedness, see [BLLS]; we willnot consider them in this text.3.4. Linear (matrix) Lie (super)algebras.

The general linear

Lie superalgebra of all su-permatrices of size Size = ( p , . . . , p | Size | ) corresponding to linear operators in the superspace V = V ¯0 ⊕ V ¯1 over the ground ﬁeld K is denoted by gl (Size), where Size = ( p , . . . , p | Size | ) isan ordered collection of parities of the basis vectors of V for which we take only vectors ho-mogeneous with respect to parity and | Size | := dim V ; usually, for the standard (simplest froma certain point of view) format Size st := (¯0 , . . . , ¯0 , ¯1 , . . . , ¯1), the notation gl (Size st ) is abbre-viated to gl (dim V ¯0 | dim V ¯1 ). Any X ∈ gl (Size) can be uniquely expressed as the sum of itseven and odd parts; in the standard format this is the following block expression; on non-zerosummands the parity is deﬁned: X = (cid:18) A BC D (cid:19) = (cid:18) A D (cid:19) + (cid:18) BC (cid:19) , p (cid:18) (cid:18) A D (cid:19) (cid:19) = ¯0 , p (cid:18) (cid:18) BC (cid:19) (cid:19) = ¯1 . The supertrace is the map gl (Size) −→ K , ( X ij ) P ( − p i ( p ( X )+1) X ii .Thus, in the standard format, str (cid:18) A BC D (cid:19) = tr A − tr D . Observe that for Lie superalgebra gl C ( p | q ) over a supercommutative superalgebra C , i.e., for supermatrices with elements in C , wehave(17) str X = tr A − ( − p ( X ) tr D for any X = (cid:18) A BC D (cid:19) ,where p ( X ) = p ( A ij ) = p ( D kl ) = p ( B il ) + ¯1 = p ( C kj ) + ¯1 , so if C ¯1 = 0, the supertrace coincides with the trace on odd supermatrices.Since str [ x, y ] = 0, the subsuperspace of supertraceless matrices constitutes a Lie subsuper-algebra called special linear and denoted sl (Size).There are, however, at least two super versions of gl ( n ), not one; for reasons, see [Lsos, Ch1,Ch.7]. The other version — q ( n ) — is called the queer Lie superalgebra and is deﬁned as theone that preserves — if p = 2 — the complex structure given by an odd operator J , i.e., q ( n )is the centralizer C ( J ) of J : q ( n ) = C ( J ) = { X ∈ gl ( n | n ) | [ X, J ] = 0 } , where J = − id . IS IN CHARACTERISTIC 2 9

It is clear that by a change of basis we can reduce J to the form (shape) J n := (cid:18) n − n (cid:19) inthe standard format, and then q ( n ) takes the form(18) q ( n ) = (cid:26) ( A, B ) := (cid:18)

A BB A (cid:19) , where A, B ∈ gl ( n ) (cid:27) . (Over any algebraically closed ﬁeld K , instead of J we can take any odd operator K such that K = a id n | n , where a ∈ K × ; and the centralizers of K , Lie superalgebras C ( K ), are isomorphicfor distinct K ; if p = 2, it is natural to select K = id.)The supertrace vanishes on q ( n ), but there is deﬁned the odd queertrace qtr : ( A, B ) tr B which vanishes on the ﬁrst derived of q ( n ), so it is a trace. Denote by sq ( n ) the Lie superalgebraof queertraceless matrices; set psq ( n ) := sq ( n ) / K n .If p = 2, there is a tailor-made trace on sq ( n ); it is not a trace on q ( n ). It is the halftrace ( str) given byhtr( A, B ) := tr A, s e sq ( n ) := { X = ( A, B ) ∈ sq ( n ) | tr( A ) = 0 } and ps e sq (2 n ) := s e sq (2 n ) / K n .Clearly, gl and q correspond to the super version of Schur’s lemma over an algebraically closedﬁeld: an irreducible module over a collection S of homogeneous operators can be absolutelyirreducible , i.e., have no proper invariant subspaces, in which case the only operator commutingwith S is a scalar (the gl case), or can have an invariant subspace which is not a sub super space,in which case the superdimension of the module is of the form n | n and an odd operator J interchanges the homogeneous components of the module (the q case).3.5. Queeriﬁcation for p = 2 (from [BLLS] ). If p = 2, then we can queerify any restrictedLie algebra g as follows. We set q ( g ) ¯0 = g and q ( g ) ¯1 = Π( g ); deﬁne the multiplication involvingthe odd elements as follows:(19) [ x, Π( y )] = Π([ x, y ]); (Π( x )) = x [2] for any x, y ∈ g . Clearly, if g is restricted and i ⊂ q ( g ) is an ideal, then i ¯0 and Π( i ¯1 ) are ideals in g . So, if g isrestricted and simple, then q ( g ) is a simple Lie superalgebra. (Note that g has to be simple asa Lie algebra, not just as a restricted Lie algebra, i.e., g is not allowed to have any ideals, notonly restricted ones.) A generalization of the queeriﬁcation is the following procedure producingas many simple Lie superalgebras as there are simple Lie algebras.3.5.1. Generalized queeriﬁcation.

Let the 1 -step restricted closure g < > of the Lie algebra g be the minimal subalgebra of the (classically) restricted closure g containing g and all theelements x [2] , where x ∈ g . To any Lie algebra g the generalized queeriﬁcation assigns the Liesuperalgebra ˜ q ( g ) := g < > ⊕ Π( g )with squaring given by (Π( x )) = x [2] for any x ∈ g . Obviously, for g restricted, the generalizedqueeriﬁcation coincides with the queeriﬁcation: ˜ q ( g ) = q ( g ). As proved in [BLLS], if g is asimple Lie algebra, then ˜ q ( g ) is a simple Lie superalgebra.3.6. Vectorial Lie superalgebras.

For their deﬁnition and NISes on them, see [BKLS].3.7.

Deformations with odd parameters and deforms.

Which of the inﬁnitesimal defor-mations can be extended to a global one is a separate much tougher question, usually solved ad hoc ; for examples over ﬁelds of characteristics 3 and 2, see [BLW] and references therein.Deformations with odd parameters are always integrable. Let us give two graphic examples. A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E Deformations of representations . Consider a representation ρ : g −→ gl ( V ). Thetangent space of the moduli superspace of deformations of ρ is isomorphic to H ( g ; V ⊗ V ∗ ).For example, if g is the 0 | n -dimensional (i.e., purely odd) Lie superalgebra (with the only bracketpossible: identically equal to zero), its only irreducible representations are the 1-dimensionaltrivial one, , and Π( ). Clearly, ⊗ ∗ ≃ Π( ) ⊗ Π( ) ∗ ≃ , and, because the Lie superalgebra g is commutative, the diﬀerential in the cochain complex iszero. Therefore H ( g ; ) = E ( g ∗ ) ≃ Π( g ∗ ) , so there are dim g odd parameters of deformations of the trivial representation.2) Deformations of the brackets . Let C be a ﬁnitely generated supercommutative su-peralgebra, let Spec C be the aﬃne super scheme deﬁned literally as the aﬃne scheme of anycommutative ring, see [L1].A deformation of a Lie superalgebra g over Spec C is a Lie superalgebra G over C such that forsome closed point p I ∈ Spec C corresponding to a maximal ideal in I ⊂ C , we have G ⊗ I K ≃ g as Lie superalgebras over K . Note that since C /I ≃ K as K -algebras, this deﬁnition impliesthat G ≃ g ⊗ C as C -modules. The deformation is trivial if G ≃ g ⊗ C as Lie superalgebras over C , not just as modules, and non-trivial otherwise. (We superized [Ru], where the non-supercase was considered.)Generally, the deforms — the results of a deformation — of a Lie superalgebra g over K areLie superalgebras G ⊗ I ′ K , where p I ′ is any closed point in Spec C .In particular, consider a deformation with an odd parameter τ of a Lie superalgebra g overﬁeld K . This is a Lie superalgebra G over K [ τ ] such that G ⊗ I K ≃ g , where I = h τ i is the onlymaximal ideal of K [ τ ]. This implies that G isomorphic to g ⊗ K [ τ ] as a module over K [ τ ]; if,moreover, G = g ⊗ K [ τ ] as a Lie superalgebra over K [ τ ], i.e.,[ a ⊗ f, b ⊗ g ] = ( − p ( f ) p ( b ) [ a, b ] ⊗ f g for all a, b ∈ g and f, g ∈ K [ τ ] , then the deformation is considered trivial (and non-trivial otherwise). Observe that g ⊗ τ isnot an ideal of G : any ideal should be a free K [ τ ]-module.3.7.1. Comment: if deformations are with odd parameters or even but formal, thereare no “deforms”.

In a sense, the people who ignore odd parameters of deformations havea point: we (rather they) consider classiﬁcation of simple Lie superalgebras (or whatever otherproblem) over the ground ﬁeld K , right? Not quite. Actually, the odd parameters of deforma-tions are no less natural than the odd part of the Lie superalgebra itself. However, to see theseparameters, we have to consider whatever we are deforming not over K , but over K [ τ ].We do the same when τ is even and we consider formal deformations over K [[ τ ]]. If the formalseries in τ converges in a domain D , we can evaluate τ for any τ ∈ D and — if dim g < ∞ — consider copies g τ , where τ ∈ D , of the same dimension as g . If the parameter is formal orodd, such an evaluation is possible only trivially: τ Examples

For examples (even classiﬁcation in several cases) of deforms of known symmetric simplemodular Lie superalgebras, see [BGL2], where the cocycles we consider below are given explic-itly, in terms of a Chevalley basis. Here we consider one of the simplest examples of deformswith an odd parameter and several other examples.

IS IN CHARACTERISTIC 2 11 (No NIS on oo (1) I Π (1 |

2) and oo (1) II (1 | Consider the Lie superalgebra oo (1) I Π (1 | and its deform with the help of the cocycle c − , and oo (1) II (1 | and its deforms with the help ofthe cocycles c or c , see [BGL2, § § . There is no NIS on any of these deformed Liesuperalgebras g := g c i .Proof. The direct computations show that [ g , g ] = g . Hence, no NIS on g due to Lemma 2.4. (cid:3) On a map sending cochains of F( g ) to cochains of g . If p = 2, and g is a Liesuperalgebra, let F ( g ) be the desuperization of g , i.e., F is the functor that forgets squaringand parity. The forgetful functor gives a K -linear map i : g −→ F ( g ), and with its help thechoice of a basis in g induces the choice of a basis in F ( g ). Actually, i is just the identity mapon g as a vector space, and a basis of F ( g ) consists of the same vectors as a basis in g .If p = 2, then E . ( V ) ( S . ( V ), where E . ( V ) is the exterior algebra and S . ( V ) is the symmetricalgebra of the space V . The map i induces an injective map i ∗ : C . ( F ( g )) −→ C . ( g ) betweenspaces of cochains. The map i ∗ does not necessary commute with the diﬀerential: as it wasnoted in [BGL2, § F ( g ) deﬁnes a cocycle of g .Interestingly, the map i ∗ sometimes allows us to express some of cocycles of g , representingcohomology classes, in terms of the images of cocycles of F ( g ) under i ∗ (plus, perhaps, termsdeﬁning a deformation of the squaring), see examples in Lemmas 4.3.1 and 4.3.2.4.3. Two lemmas about NISes on deforms.

The NISes on deforms of wk ( n ; α ), where n = 3 or 4, can be directly translated to the corresponding superizations — deforms of bgl ( n ; α )— because the squaring is not involved at all in the invariance condition for the bilinear form. (On NISes on bgl (4; α )). For Lie superalgebra g = bgl (4; α ) , where α = 0 , ,all deforms depend on even parameters, see [BGL2] .Choose a basis in g as in [BGL2] . These deforms of g preserve a NIS with the same Grammatrix as that of the NIS on g , except for the deform g c of bgl (4; α ) with cocycle c when theGram matrix Γ c is a diﬀerent one. The desuperization of Γ c coincides with the Gram matrix,described in [BKLS, Claim 3.3] , of the corresponding deform of wk (4; α ) with cocycle c .Proof. For a basis in H ( F ( g ); F ( g )) take (the classes of) c ± , c ± , c ± , c ± , c ± , c ± , c ± , c ± , c ± , c ; for their explicit expressions, see [BGL2, § § H ( g ; g ) consists of the followingcocycles ¯ c − = i ∗ ( c − ) , ¯ c − = i ∗ ( c − ) , ¯ c − = i ∗ ( c − ) , ¯ c − = i ∗ ( c − ) + α (1 + α ) h ⊗ (ˆ x ) ∧ , ¯ c − = i ∗ ( c − ) + (1 + α ) h ⊗ (ˆ x ) ∧ , ¯ c − = i ∗ ( c − ) + α (1 + α )( h + h ) ⊗ (ˆ x ) ∧ , ¯ c − = i ∗ ( c − ) + (1 + α )( h + h ) ⊗ (ˆ x ) ∧ , ¯ c − = i ∗ ( c − ) + (1 + α )( h + h ) ⊗ (ˆ x ) ∧ , ¯ c − = i ∗ ( c − ) + (1 + α ) h ⊗ (ˆ x ) ∧ , ¯ c = i ∗ ( c ) . where x i , h i , y i are elements of the Chevalley basis of g , see [BGL1], and ˆ x i , ˆ h i , ˆ y i are theelements of the corresponding dual basis of Π( g ∗ ). (cid:3) (On NISes on bgl (1) (3; α ) / c ). For Lie superalgebra g = bgl (1) (3; α ) / c , where α = 0 , , all deforms depend on even parameters; see [BGL2] . Choose a basis in g as in [BGL2] .These deforms preserve a NIS with the same Gram matrix as that of the NIS on g , exceptfor the deform g c of bgl (1) (3; α ) / c with cocycle c when the Gram matrix Γ c is a diﬀerent one. A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E The desuperization of Γ c coincides with the Gram matrix, described in [BKLS, Claim 3.4] , ofthe corresponding deform of wk (1) (3; α ) / c with cocycle c .Proof. For the basis in H ( F ( g ); F ( g )) take (the classes of) c ± , c ± , c ± , c ± , c ; for theirexplicit expressions; see [BGL2, § H ( g ; g ) consists of the following cocycles, see [BGL2, § c − = i ∗ ( c ) , ¯ c − = i ∗ ( c − ) , ¯ c = i ∗ ( c ) . (cid:3) (NISes on q ( sl (3))). For g = q ( sl (3)) , consider its two deformations: g A ( τ ) givenby the odd cocyle A with parameter τ and g B ( ε ) with parameter ε given by the even cocycle B ;for explicit expressions of these cocycles, see [BGL2, Lemma 8.3] .The space of NISes on g A ( τ ) is of rank | over K [ τ ] .The spaces of NISes on g B ( ε ) is (1 | -dimensional if ε = 1 , and -dimensional if ε = 1 .Proof. Although the proof is analytical, the above is called Claim because the expressions ofNISes are obtained with the aid of

SuperLie .Choose a basis in g given by the Chevalley basis in sl (3), namely, x i , y i , h i , and Π x i , Π y i ,Π h i . By Theorem 2.1, there are two NISes, ω ¯0 and ω ¯1 , on q ( sl (3)) induced by the even NISon sl (3), see [BKLS], given by the formula (4).A) For the odd cocyle A , a NIS ω on g A ( τ ) is deﬁned as follows ω = a ω ¯0 + a ω ¯1 + τ ( a ω ¯0 + a ω ¯1 + a B + a B ) , a , a , a , a ∈ K , where B (resp. B ) is an odd (resp. even) bilinear form for which B (Π h , h ) = B (Π h , h ) = B (Π h , h ) = B (Π h , h ) = 1 ,B ( y , x ) = B ( y , x ) = B (Π h , Π h ) = B (Π y , Π x ) = 1 . and zero on all other pairs of Chevalley basis. Observe that g A ( τ ) is a free module over K [ τ ]. Anarbitrary element t = a + bτ ∈ K [ τ ] is deﬁned by a pair of numbers a, b ∈ K . Set t = a + a τ and t = a + a τ ∈ Λ[ τ ]. We have ω = t ( ω ¯0 + τ B ) + t ( ω ¯1 + τ B )Therefore, the K [ τ ]-module of NISes on g A ( τ ) is of rank 1 | g B ( ε ) is nonlinear with respect to ε . There are two NISes on g B ( ε ) when ε = 1:1) An even NIS ω e for which (and zero on all other pairs of Chevalley basis) ω e ( h , h ) = 1 , ω e ( x , y ) = 1 + ε ,ω e ( x , y ) = 1 + ε , ω e ( x , y ) = 1 + ε,ω e (Π h , Π h ) = 1 + ε, ω e (Π x , Π y ) = 1 + ε ,ω e (Π x , Π y ) = 1 + ε , ω e (Π x , Π y ) = 1 .

2) An odd NIS ω o for which (and zero on all other pairs of Chevalley basis) ω o ( h , Π h ) = 1 , ω o ( h , Π h ) = 1 + ε,ω o ( x , Π y ) = 1 + ε , ω o ( y , Π x ) = 1 + ε,ω o ( x , Π y ) = 1 + ε , ω o ( y , Π x ) = 1 + ε,ω o ( x , Π y ) = 1 , ω o ( y , Π x ) = 1 . It is easy to see that ω e is a deformation of ω ¯0 , and ω o is a deformation of ω ¯1 .There is no NIS on g B ( ε ) for ε = 1: IS IN CHARACTERISTIC 2 13

Thus, the space of NISes on g B ( ε ) is (1 | ε = 0 and 0-dimensional if ε = 1. (cid:3) It now follows that the deform g B (1) is a true deform, namely, it is not isomorphic to g = q ( sl (3)), since it has no NISes. (On k (1; n | The Lie superalgebra k (1; n | and its ( n − -parametric familyof even deforms described in [KL, Theorem 6.2] have no NIS.Proof. The even part of the Lie superalgebra k (1; n |

1) and of its ( n − g , g ] = g , and hence no NIS due to Lemma 2.4. (cid:3) Remarks

Multiple NISes over non-closed ﬁelds.

NISes on the simple ﬁnite-dimensional (asso-ciative or Lie) algebra A over algebraically non-closed ﬁelds are described in [Kap, pp. 30–31,Exercise 15(b)] in terms of the centroid of A . In characteristic 0, Waterhouse considereda particular case of Kaplansky’s exercise: described NISes on the simple ﬁnite-dimensional Liealgebra; they all come from the Killing form over the centroid of the algebra, see [W]. We thinkthat the description below elucidates the above-cited results [Kap, W], but is more graphic.Superization is immediate; proof is an easy exercise.5.1.1. On NISes over algebraically non-closed ﬁelds.

1) Let h be a simple ﬁnite-dimensionalLie (super)algebra over R with a NIS b , and g := h ⊗ R C . Clearly, B ( x + iy, z + it ) := b ( x, z ) − b ( y, t ) + i ( b ( y, z ) + b ( x, t )) for any x, y, z, t ∈ h is a NIS on g . Then, the realiﬁcation of g , i.e., g considered as real, has at least two linearlyindependent NISes the real and the imaginary parts of B .2) More generally, let g be a simple ﬁnite-dimensional Lie (super)algebra with a NIS B overa ﬁeld K which is not algebraically closed. For simplicity, let p = 2.Let P ∈ K [ x ], where x is even, be an irreducible polynomial of degree d >

1; set A := K [ x ] / ( P ).The algebra A has no non-trivial ideals and usually A ⊗ g is a simple Lie (super)algebra. Let ϕ : A −→ K be a linear map, not identically equal to zero. Then, the bilinear form on A ⊗ g given by the formula B ϕ ( a ⊗ g , a ⊗ g ) := ϕ ( a a ) B ( g , g ) for any a , a ∈ A and g , g ∈ g is a NIS. In this way, we get a d -dimensional space of NISes (of the same parity as B ) on A ⊗ g considered as a Lie (super)algebra over K .5.2. A Lie superalgebra structure on L ⊗ A where L is a restricted Lie superalgebraand A is a non-supercommutative associate superalgebra. A more general, but non-super, setting is discussed in [Z]. In this Subsection, A is not necessarily ﬁnite-dimensional.It is well-known that for any Lie algebra L and any commutative algebra A , one can introducea Lie algebra structure on L ⊗ A by setting[ l ⊗ a , l ⊗ a ] := [ l , l ] ⊗ a a for any l i ∈ L and a i ∈ A . Recall that the centroid of the algebra A is the set of all linear transformations of A that commute with allleft and right multiplications. The centroid is an algebra containing the identity linear transformation. If A hasa unit element, the centroid coincides with the elements that commute and associate with everything. But not always. For example, let g and A ⊗ g be simple. Then, A ⊗ ( A ⊗ g ) is not simple because A ⊗ A isnot simple. A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E The commutativity of A is required for this bracket to be antisymmetric.Analogously, if p = 2, L is any Lie superalgebra, and A is any supercommutative superalge-bra, then one can introduce a Lie superalgebra structure on L ⊗ A by setting[ l ⊗ a , l ⊗ a ] := ( − p ( l ) p ( a ) [ l , l ] ⊗ a a . Again, the supercommutativity of A is required for this bracket to be super antisymmetric.If p = 2, then the bracket on the Lie superalgebra is again just antisymmetric, so one couldassume that the above deﬁnition would work even if A is just commutative, not supercommuta-tive (note that if p = 2, then a supercommutative superalgebra is commutative as well). But if p = 2, one has to deﬁne the squaring on ( L ⊗ A ) ¯1 separately from the bracket. For an arbitraryLie superalgebra L , this can be done only if A is a supercommutative superalgebra, and thedeﬁnition is as follows:( l ⊗ a ) = 0 for l ∈ L ¯0 , a ∈ A ¯1 ;( l ⊗ a ) = l ⊗ a for l ∈ L ¯1 , a ∈ A ¯0 ; X ≤ i ≤ n l i ⊗ a i ! = X ≤ i ≤ n ( l i ⊗ a i ) + X ≤ i

IS IN CHARACTERISTIC 2 15

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Seminar on supersymmetries. Vol. : Algebra and Calculus on supermanifolds , MCCME,Moscow, 2011, 410 pp. (in Russian)[Mi] Milnor J., Introduction to algebraic K -theory , Princeton Univ Press, Princeton 1971.[Ru] Rudakov, A. N., Deformations of simple Lie algebras. Mathematics of the USSR-Izvestiya, 5 (1971),no. 5, 1120–1126[SF] Strade H., Farnsteiner R., Modular Lie algebras and their representations . Marcel Dekker, 1988.viii+301pp.[WZ] Wang Ying, Zhang Yongzheng, A new deﬁnition of restricted Lie superalgebras. Chinese Science Bulletin,45 (2000), no. 4, 316–321[W] Waterhouse, W. C., Invariant bilinear forms on semisimple Lie algebras. Algebras Groups Geom., 9(1992), no. 1, 49–52[Z1] Zusmanovich P., The second homology group of current Lie algebras. In: K-Theory (C. Kassel et al.eds.), Ast´erisque, 226 (1994), 435–452; arXiv:0808.0217 [Z2] Zusmanovich P., Invariants of Lie algebras extended over commutative algebras without unit, Journal ofNonlinear Mathematical Physics, 17 (2010), Suppl. 1 (Special issue in memory of F.A. Berezin), 87–102; arXiv:0901.1395 A, ∗ , ALEXEI LEBEDEV B , DIMITRY LEITES C,D , IRINA SHCHEPOCHKINA E [Z] Zusmanovich P., Lie algebras and around: selected questions; Matematicheskii Zhurnal (Almaty), 16(2016), no. 2 (A.S. Dzhumadil’daev Festschrift), 231–245; arXiv:1608.05863 a Institute of Mathematics, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague, CzechRepublic; krutov@math.cas.cz, b Equa Simulation AB, R˚asundav¨agen 100, Solna, Sweden; alex-eylalexeyl@mail.ru, c New York University Abu Dhabi, Division of Science and Mathematics,P.O. Box 129188, United Arab Emirates; dl146@nyu.edu, d Department of mathematics, Stock-holm University, SE-106 91 Stockholm, Sweden; mleites@math.su.se, e Independent Universityof Moscow, Bolshoj Vlasievsky per, dom 11, RU-119 002 Moscow, Russia; irina@mccme.ru, ∗∗