On the algebraic variety of Hom-Lie algebras
aa r X i v : . [ m a t h . R A ] J un ON THE ALGEBRAIC VARIETY OF HOM-LIE ALGEBRAS
ELISABETH REMM, MICHEL GOZE
Abstract.
The set H Lie n of the n -dimensional Hom-Lie algebras over an algebraicallyclosed field of characteristic zero is provided with a structure of algebraic subvariety of theaffine plane K n ( n − / . For n = 3, these two sets coincide, for n = 4 it is an hypersurfacein K . For n ≥
5, we describe the scheme of polynomial equations which define H Lie n .We determine also what are the classes of Hom-Lie algebras which are P -algebras where P is a binary quadratic operads. Key Words:
Hom-Lie algebras. Operads
Introduction
The notion of Hom-Lie algebras was introduced by Hartwig, Larsson and Silvestrov in [3].Their principal motivation concerns deformation of the Witt algebra. This Lie algebra is thecomplexification of the Lie algebra of polynomial vector fields on a circle. A basis for theWitt algebra is given by the vector fields L n = − z n +1 ∂∂z for any n ∈ Z . The Lie bracket is given by[ L m , L n ] = ( m − n ) L m + n . The Witt algebra is also viewed as the Lie algebra of derivations of the ring C [ z, z − ].Recall that a derivation on an algebra with product denoted by ab is a linear operatorsatisfying D ( ab ) = D ( a ) b + aD ( b ). The Lie bracket of two derivations D and D ′ is [ D, D ′ ] = D ◦ D ′ − D ′ ◦ D. We can also define a new class of linear operators generalizing derivations,the Jackson derivate, given by D q ( f )( z ) = f ( qz ) − f ( z ) qz − z . It is clear that D q is a linear operator, but its behavior on the product is quite different asthe classical derivative: D q ( f g ( z )) = g ( z ) D q ( f ( z )) + f ( qz ) D q ( g ( z )) . The authors interpret this relation by putting(1) D q ( f g ) = gD q ( f ) + σ ( f ) D q ( g )where σ is given by σ ( f )( z ) = f ( qz ) for any f ∈ C [ z, z − ]. Starting from (1) and for agiven σ , one defines a new space of ”derivations” on C [ z, z − ] constituted of linear operator Date : 4 Chevat 5777 . D satisfying this relation. With the classical bracket we obtain is a new type of algebra socalled σ -deformations of the Witt algebra. This new approach leads naturally to considererthe space of σ -derivations, that is, linear operators satisfying (1), to provide it with themultiplication associated with a bracket. This new algebra is not a Lie algebra becausethe bracket doesn’t satisfies the Jacobi conditions. The authors shows that this bracketsatisfies a ”generalized Jacobi condition”. They have called this new class of algebras theclass of Hom-Lie algebras. This notion, introduced in [3], since made the object of numerousstudies and was also generalized (see [4]). We denote by SA lg n the set of the n -dimensionalskew-symmetric algebras ( A , µ ) over an algebraically closed field K of characteristic 0 whosemultiplication µ is skew-symmetric and by H Lie n the subset of Hom-Lie algebras. It isclear that SA lg n is an affine variety isomorphic to K n ( n − / . In this work, we show that H Lie n is an affine algebraic subvariety of SA lg n . We study in particular the case n = 3and n = 4, proving that in dimension 3 any skew-symmetric algebra is a Hom-Lie algebraand in dimension 4, H Lie is an algebraic hypersurface in SA lg . We end this work by thedetermination of binary quadratic operads whose associated algebras are Hom-Lie.2. Hom-Lie algebras [3]Before to begin the study of Hom-Lie algebras, we recall some very classical results on thenotions of algebras and fix the used notations.An algebra over a field K of characteristic 0 is a pair ( A , µ ) where A is K -vector space and µ a bilinear map µ : A × A → A which is classically called the multiplication of A . When any confusion is possible, we shalldenote it A in place of ( A , µ ). Two algebras ( A , µ ) and ( A ′ , µ ′ ) are isomorphic (in terms ofalgebras), if there is a linear isomorphism f : A → A ′ satisfying(2) f ( µ ( X, Y )) = µ ′ ( f ( X ) , f ( Y ))for any X, Y ∈ A .We assume now that all considered algebras are of finite dimension. Let { e , · · · , e n } be abasis of the algebra ( A , µ ). The constants C ki,j given by µ ( e i , e j ) = X k C ki,j e k are called the structure constants of µ (or A ) relative to the basis { e , · · · , e n } . The set A lg n of n -dimensional K -algebras can be identified to the vector space K n , identifying themultiplication µ with its structure constants C ki,j related to a given basis of K n . One of themain problem is to give the classification of the algebras of a given dimension. This problemwas solved for the dimensions 2 and 3 (in this last case for the skew-symmetric algebras).This classification corresponds to the determination of all orbits related to the action (2) ofthe linear group GL ( n, K ) on A lg n . Among all the algebras, certain classes play importing roles, as the class of Lie algebras,associative algebras, pre-Lie algebras. The aim of this work concerns a recent class, theHom-Lie algebras.
N THE ALGEBRAIC VARIETY OF HOM-LIE ALGEBRAS 3
Definition 1.
A Hom-Lie algebra is a triple ( A, µ, α ) consisting of a linear space A , a skew-bilinear map µ : V × V → V and a linear space homomorphism f : A → A satisfying theHom-Jacobi identity (cid:9) x,y,z µ ( µ ( x, y ) , f ( z )) = 0 for all x, y, z in A , where (cid:9) x,y,z denotes summation over the cyclic permutations on x, y, z . For example, a Hom-Lie algebra whose endomorphism f is the identity is a Lie algebra.We deduce, since any 2-dimensional skew-symmetric algebra (the multiplication µ is a skew-symmetric bilinear map) is a Lie algebra, that any 2-dimensional algebra is a Hom-Liealgebra.In the following section we are interested by the determination of all Hom-Lie algebras forsmall dimensions. 3. Hom-Lie algebras of dimension K -algebra ( A , µ ) isdefined by its structure constants { α i , β i , γ i } i =1 , , with respect to a given basis { e , e , e } : µ ( e , e ) = α e + β e + γ e ,µ ( e , e ) = α e + β e + γ e ,µ ( e , e ) = α e + β e + γ e . Let f be an element of gl (3 , K ) and consider its matrix in the same basis { e , e , e } a b c a b c a b c . We then define the vector v f = t ( a , a , a , b , b , b , c , c , c ) . Theorem 2.
Any -dimensional algebra is a Hom-Lie algebra.Proof. Consider a basis { e , e , e } of the algebra ( A, µ ) and let ( α i , β i , γ i ), i = 1 , , f ∈ gl (3 , K ) satisfies the Hom-Jacobi condition if and only if its corresponding vector v f = t ( a , a , a , b , b , b , c , c , c )is in the Kernel of the matrix M µ = (cid:0) . . . . . . . . . (cid:1) where we use the notation ij.k in place of µ ( µ ( e i , e i ) , e k ) and23 . − α β − α γ − β β − β γ − γ β − γ γ , . α α − α γ β α − β γ γ α − γ γ , . α α + α β β α + β β γ α + γ β . α β + α γ β β + β γ γ β + γ γ , . − α α + α γ − β α + β γ − γ α + γ γ , . − α α − α β − β α − β β − γ α − γ β . − α β − α γ − β β − β γ − γ β − γ γ , . α α − α γ β α − β γ γ α − γ γ , . α α + α β β α + β β γ α + γ β . ELISABETH REMM, MICHEL GOZE
Since the matrix M µ belongs to M (3 ,
9) and represents a linear morphism t : K → K . From the rank theorem we havedim Ker t = 9 − dim Im t ≥ . Then this kernel is always non trivial and for any algebra µ , there exists a non trivial elementin the kernel. Then this algebra always admits a non trivial Hom-Lie structure. Consequence: On the classification of Hom-Lie algebras of dimension K -algebra. Since any Hom-Liealgebra is skew-symmetric, we deduce the classification of Hom-Lie algebras.We have seen that the kernel of M µ is of dimension greater or equal to 3. To computethis kernel we present in a vectorial form this matrix, using the notation ij.k in place of µ ( µ ( e i , e i ) , e k ). We have M µ = (cid:0) . . . . . . . . . (cid:1) with 23 . − α β − α γ − β β − β γ − γ β − γ γ , . α α − α γ β α − β γ γ α − γ γ , . α α + α β β α + β β γ α + γ β . α β + α γ β β + β γ γ β + γ γ , . − α α + α γ − β α + β γ − γ α + γ γ , . − α α − α β − β α − β β − γ α − γ β . − α β − α γ − β β − β γ − γ β − γ γ , . α α − α γ β α − β γ γ α − γ γ , . α α + α β β α + β β γ α + γ β . Thus, for a given algebra, it is easy to compute all the endomorphisms which satisfy theJacobi-Hom-Lie condition. Let us note also that the identity whose associated vector is v Id = (1 , , , , , , , , M µ if and only if µ satisfies(23)1 + (31)2 + (12)3 = 0that is if it is a Lie algebra. Let us note also, that the dimension of M µ is an invariantassociated with µ , that is, if two Hom-Lie algebras are isomorphic then the Kernels of theassociated matrices are isomorphic. N THE ALGEBRAIC VARIETY OF HOM-LIE ALGEBRAS 5 Dimension A , µ ) a 4-dimensional K -algebra. Let us choose a basis { e , e , e , e } of A and let us consider thecorresponding constants structure of µ : µ ( e , e ) = α e + β e + γ e + δ e µ ( e , e ) = α e + β e + γ e + δ e µ ( e , e ) = α e + β e + γ e + δ e µ ( e , e ) = α e + β e + γ e + δ e µ ( e , e ) = α e + β e + γ e + δ e µ ( e , e ) = α e + β e + γ e + δ e These algebra is a Hom-Lie algebra if there exists a linear endomorphism f satisfying theHom-Lie Jacobi equations. The endomorphism f is represented in the basis { e , e , e , e } by a square matrix of order 4. The corresponding v f belongs to K : v f = ( a , a , a , a , b , b , b , b , c , c , c , c , d , d , d , d ) . These Hom-Lie Jacobi conditions appear here in the form of a linear system on these coef-ficients, and this linear system which contains 16 equations. Then f satisfies the Hom-Lieconditions if and only if the vector t v f is in the kernel of the associated matrix M HL ( µ ) whichis a square matrix of order 16. Its kernel is not trivial if and only ifdet( M HL ( µ )) = 0 . We deduce
Proposition 3.
The set HL of -dimensional K -Hom-Lie algebras is provided with a struc-ture of affine algebraic variety embedded in K . In fact, det( M HL ( µ )) = 0 is a polynomial equation whose variables are the 24 constants α i , β i , γ i , δ i , i = 1 , , , . Let us note that this equation is homogeneous of degree 16.For any ( A , µ ) ∈ HL we can define the vector space ker M HL ( µ ). We thus define a singularvector bundle K ( HL ) whose fiber over ( A , µ ) is ker M HL ( µ ). This fiber corresponds to theset of Hom-Lie structure which can be defined on a given 4-dimensional algebra. Remark.
In dimension 3, HL is the affine variety SA lg which is isomorphic to the affinespace K . We have seen that, in this case, the fibers of K ( HL ) are vector spaces of dimensiongreater or equal to 6.We assume now that K is algebraically closed. In the previous presentation, it remains aproblem concerning the existence or not of algebra ( A , µ ) for which we havedet( M HL ( µ )) = 0 . We meet then difficulties of calculation because a determinant of order 16 is not easy to treat.Then we shall simplify this matricial approach. Since we assume that K is algebraicallyclosed, the endomorphism f can be reduced in an appropriated basis on the form a b b c
00 0 c d d ELISABETH REMM, MICHEL GOZE
As a result, the matrix is reduced to a matrix M ′ HL ( µ ) of type 16 ×
7. This matrix is . . . . . . . . . . . . . . .
40 34 . . . . . . with . − β α − γ α − δ α − β β − γ β − δ β − β γ − γ γ − δ γ − β δ − γ δ − δ δ , . β α + γ α + δ α β β + γ β + δ β β γ + γ γ + δ γ β δ + γ δ + δ δ , . − α α + γ α + δ α − α β + γ β + δ β − α γ + γ γ + δ γ − α δ + γ δ + δ δ . α α − γ α − δ α α β − γ β − δ β α γ − γ γ − δ γ α δ − γ δ − δ δ , . α α + β α − δ α α β + β β − δ β α γ + β γ − δ γ α δ + β δ − δ δ , . − β α − γ α − δ α − β β − γ β − δ β − β γ − γ γ − δ γ − β δ − γ δ − δ δ . β α + γ α + δ α β β + γ β + δ β β γ + γ γ + δ γ β δ + γ δ + δ δ , . − α α + γ α + δ α − α β + γ β + δ β − α γ + γ γ + δ γ − α δ + γ δ + δ δ , . α α + β α + γ α α β + β β + γ β α γ + β γ + γ γ α δ + β δ + γ δ . − β α − γ α − δ α − β β − γ β − δ β − β γ − γ γ − δ γ − β δ − γ δ − δ δ , . − α α − β α + δ α − α β − β β + δ β − α γ − β γ + δ γ − α δ − β δ + δ δ , . α α + β α − δ α α β + β β − δ β α γ + β γ − δ γ α δ + β δ − δ δ , . α α + β α + γ α α β + β β + γ β α γ + β γ + γ γ α δ + β δ + γ δ , . α α − γ α − δ α α β − γ β − δ β α γ − γ γ − δ γ α δ − γ δ − δ δ , . − α α + γ α + δ α − α β + γ β + δ β − α γ + γ γ + δ γ − α δ + γ δ + δ δ , . − α α − β α + δ α − α β − β β + δ β − α γ − β γ + δ γ − α δ − β δ + δ δ , . α α + β α − δ α α β + β β − δ β α γ + β γ − δ γ α δ + β δ − δ δ , . α α + β α + γ α α β + β β + γ β α γ + β γ + γ γ α δ + β δ + γ δ To a generic point ( α i , β i , γ i , δ i ), i = 1 , · · · , , this matrix is of rank 7. Then there exists a4-dimensional algebra such that for any homomorphism f the corresponding matrix writtenin a basis which reduces f in a canonical form is of maximal rank. Such algebra cannot bea Hom-Lie algebra. Theorem 4.
The set HL of the -dimensional K -Hom-Lie algebras is a affine algebraicvariety strictly contained in the affine plane SA lg isomorphic to K .Proof. From the previous calculus, HL is strictly contained in SA lg . Since HL is deter-minate by the polynomial condition det( M µ ) = 0, we deduce that it is an affine algebraicsub variety of the affine plane SA lg .Let us note that in the original basis { e , e , e , e } the expression of M µ is, using thenotations of the previous section: (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40 0 0 0 34 . . . . . . . . . . . . N THE ALGEBRAIC VARIETY OF HOM-LIE ALGEBRAS 7
In a generic point the rank of this matrix is equal to 16. Let us consider, for example, thefollowing algebra: µ ( e , e ) = e + 2 e − e µ ( e , e ) = e + 2 e − e µ ( e , e ) = 2 e − e + e µ ( e , e ) = − e + e + 2 e µ ( e , e ) = e + 2 e − e + 3 e µ ( e , e ) = − e − e + e + 2 e The associated matrix is − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
33 3 4 4 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
40 0 0 0 1 − − − − − − − − − − − − − − − − − − − − and its determinant is not zero. This algebra is not Hom-Lie. Corollary 5.
In the affine plane SA lg , there is a Zariski open set whose elements are -dimensional algebras without Hom-Lie structure. The general case
We assume in tis section that ( A , µ ) is a n -dimensional K -algebra. Let f be a linearendomorphism of A . We fix a basis { e , · · · , e n } of A and we consider the associated vector v f = ( a , , · · · , a n, , a , , · · · , a n, , · · · , a ,n , · · · , a n,n ) . This vector satisfies the Jacobi-Hom-Lie condition if and only if it is in the kernel of thematrix M µ where M µ is the matrix n -columns and n ( n − n − lines . This matrix can bewritten . · · · .n . · · · .n . · · · .n · · · · · · . · · · .n . · · · . · · · . · · · . · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . · · · .n · · · . · · · .n . · · · .n · · · · · · · · · . · · · .n . · · · .n . · · · .n · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · n − n − .n ELISABETH REMM, MICHEL GOZE where ij.k is the matrix n × l is X s C si,j C ls,k the constant C ki,j being the structure constants of µ in the given basis { e , · · · , e n } . Thekernel of this matrix is not trivial if and only if its rank is smaller or equal to n −
1. This isequivalent to say that all the minor of order n are zero. We obtain a system of polynomialequations on the variables C ki,j of degree n . The number of equation corresponds to thenumber of combinations of n -elements amongst n ( n − n − elements. For example, if n = 5,the matrix M µ is of order 25 ×
50 and we have 50!25!25! -polynomial equations of degree 625on the 50 variables C ki,j .But this system is not reduced. For example, if we consider the firstminor constituted with the first 25 lines. If it is equal to zero, we can assume that the firstline is a combination of the other 24 lines. Then if we assume that this line is zero, all theminor containing this line are necessarily zero. This show that this system can be reducedto n ( n − n − equations as soon as n ≥ M µ . Proposition 6.
The set HL n of the n -dimensional K -Hom-Lie algebras is a affine algebraicvariety strictly contained in the affine plane A lg n isomorphic to K N with N = . n ( n − n − Some associated operads ”An operad is an algebraic device, which encodes a type of algebras. Instead of studyingthe properties of a particular algebra, we focus on the universal operations that can beperformed on the elements of any algebra of a given type. The information contained inan operad consists in these operations and all the ways of composing them” (Loday-Valettebook) It is illusory to think that there is an operad which encodes Hom-Lie algebras, becausethe latter include diverse sorts of algebras. However, as the operations occurring in therelation of Jacobi-Hom-Lie that is in the matrix M µ are quadratic, we shall see which kindof quadratic operas can be highlighted from the calculation of the pikernel of the matrix M µ .Let us remind briefly the definition of a quadratic operad.Let K [ P n ] the K -algebrea of the symmetric group P n . An operas P is defined by asequence P ( n ) , n ≥ K such that P ( n ) is also a K [ P n ]-module andby composition maps: ◦ i : P ( n ) ⊗ P ( m ) → P ( n + m ?1)for i = 1 , ..., n satisfying some associative properties called the May’s axioms.Any K [ P n ]-module E generates a free operad denoted F ( E ) and satisfying F ( E )(1) = K , F ( E )(2) = E. In particular, if E = K [ P ], the free module F ( E )( n ) admits a basis whose elements areall the paranthezed products on n variables indexed by par { , , ..., n } . Let E be a K [ P n ]-module and R a K [ P n ]-submodule of F ( E )(3). On denote by R the ideal generated by R that is the intersection of all the ideals I of F ( E ) such that I (1) = 0 , I (2) = 0 and I (3) = R. N THE ALGEBRAIC VARIETY OF HOM-LIE ALGEBRAS 9
We call binary quadratic operas generated by E and the relations R , the operad P ( K , E, R ),also denoted by F ( E ) / R with P ( K , E, R )( n ) = F ( E )( n ) / R ( n ) . Since we shall like determinate some quadratic operads based on Hom-Lie algebras, weassume that the P -module E is generated by one element. and more precisely that E ? = sgn where sgn is the one-dimensional signum representation. This is equivalent to saythey the product is skew-symmetric. The ideal of relation R will be given consdering specialendomorphism f in the kernel of M µ . For example, if we assume that f = Id ∈ ker M µ .This is implies that the coefficients of M µ satisfy in dimension 3 the following relation23 . . . . . . . . . . . . . . . . This conduces to consider that R is generated by the relation ij.k + jk.i + ki.j. These relations are invariant by the action of P , this implies that R = R and the corre-sponding quadratic operad is the Lie operad L ie .To find other quadratic operas in relayion with he Hom-Lie conditions, let us consider F ( E )(3). This is a 12-dimensional vector space generated by the triple { ij.k, i.jk, i = j = k, ≤ i, j, k ≤ } . We consider that the multiplication is skew-symmetric, that is R (2) = { ij + ji } . We deducethat P (3) = F ( E )(3) / R (3) is a 4-dimensional vector space generated by { ij.k, ≤ i < j < k ≤ } . Thus, the coefficients of the matrix ( 3) which belongs to P (3) are the elements of this lastspace. We deduce that a subclass of Hom-Lie algebras is associated with a quadratic operasif and only the equations M µ ( v f ) = 0constitue a linear system in P (3). We have seen above that the vector v Id satisfies thiscondition. We have to determine all the others. From the form of the matrix ( 3), we deducethat v f = ( a , , , , , b , , , , , c , , , , , d )and the equations of the kernel are a . b . c . a . b . d . a . c . d . b . c . d . P -invariant if and only if(1) a = b = c = d = 1 and f = Id . (2) ij.k = 0 . The last case can be forgotten and we find only the L ie operad as binary quadratic operasin relation with the Hom − Lie algebras.Since M µ contains also termes of type ij.i , we are conduced to consider not only quadraticoperas but also ternary operads. Then we consider P (2) = F ( E )(2) / R (2) generated by { ij } , P (3) = F ( E )(3) / R (3) generated by all the elements, that is { ij.k, ≤ i < j < k ≤ } . . Theideal of relations lies in F ( E )(4). To illustrate this construction, let us take v f = (0 , , , , , , , , M µ if and only if23 . . . . References [1] Goze. M., Remm E. A class of nonassociative algebras.
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