Leibniz algebras of Heisenberg type
aa r X i v : . [ m a t h . R A ] N ov LEIBNIZ ALGEBRAS OF HEISENBERG TYPE
A.J. CALDER ´ON, L.M. CAMACHO, B.A. OMIROVA
BSTRACT . We introduce and provide a classification theorem for the class of Heisenberg-Fock Leibniz algebras.This category of algebras is formed by those Leibniz algebras L whose corresponding Lie algebras are Heisenbergalgebras H n and whose H n -modules I , where I denotes the ideal generated by the squares of elements of L , areisomorphic to Fock modules. We also consider the three-dimensional Heisenberg algebra H and study three classesof Leibniz algebras with H as corresponding Lie algebra, by taking certain generalizations of the Fock module.Moreover, we describe the class of Leibniz algebras with H n as corresponding Lie algebra and such that the action I × H n → I gives rise to a minimal faithful representation of H n . The classification of this family of Leibnizalgebras for the case of n = 3 is given. AMS Subject Classifications (2010): 17A32, 17B30, 17B10.Key words:
Heisenberg algebra, Leibniz algebra, Fock representation, minimal faithful representation.1. I
NTRODUCTION
The term
Leibniz algebra was introduced in the study of a non-antisymmetric analogue of Lie algebras byLoday [35], being so the class of Leibniz algebras an extension of the one of Lie algebras. However this kindof algebras was previously studied under the name of D -algebras by D. Bloh [10, 11, 12]. Since the 1993Loday’s work many researchers have been attracted by this category of algebras, being remarkable the greatactivity in this field developed in the last years. This activity has been mainly focussed in the frameworks oflow dimensional algebras, nilpotence and physics applications (see [2, 5, 6, 14, 15, 16, 17, 21, 22, 23, 25, 28,33, 40, 41]). Definition 1. A Leibniz algebra L is a linear space over a base field F endowed with a bilinear product [ · , · ] satisfying the Leibniz identity [[ y, z ] , x ] = [[ y, x ] , z ] + [ y, [ z, x ]] , for all x, y, z ∈ L .In presence of anti-commutativity, Jacobi identity becomes Leibniz identity and therefore Lie algebras areexamples of Leibniz algebras. Throughout this paper F will be algebraically closed and with zero characteristic.Let L be a Leibniz algebra. The ideal I generated by the squares of elements of the algebra L , that is I isgenerated by the set { [ x, x ] : x ∈ L } , plays an important role in the theory since it determines the (possible)non-Lie character of L . From the Leibniz identity, this ideal satisfies [ L, I ] = 0 . The quotient algebra
L/I is a Lie algebra, called the corresponding Lie algebra of L , and the map I × L/I → I , ( i, [ x ]) [ i, x ] , endows I of a structure of L/I -module (see [4, 37]). Observe that we can write(1) L = V ⊕ I where V is a linear complement of I in L and V is isomorphic as linear space to L/I . From here, Leibnizalgebras give us the opportunity of treating in an unifying way a Lie algebra together with a module over it.On the other hand, we recall that Heisenberg (Lie) algebras play an important role in mathematical physicsand geometry, in particular in Quantum Mechanics (see for instance [1, 8, 9, 19, 20, 24, 26, 27, 29, 30, 31,32, 36, 42, 44]). Indeed, the Heisenberg Principle of Uncertainty implies the non-compatibility of positionand momentum observables acting on fermions. This non-compatibility reduces to non-commutativity of thecorresponding operators. If we represent by x the operator associated to position and by ∂∂x the one associated tomomentum (acting for instance on a space V of differentiable functions of a single variable), then [ x, ∂∂x ] = 1 V which is non-zero. Thus we can identify the subalgebra generated by , x and ∂∂x with the three-dimensional Heisenberg algebra whose multiplication table in the basis { , x, ∂∂x } has as unique non-zero product [ x, ∂∂x ] =1 . For any non-negative integer k the Heisenberg algebra of dimension n = 2 k + 1 (denoted further by H n ) ischaracterized by the existence of a basis(2) B = { , x , δδx , . . . , x k , δδx k } in which the multiplicative non-zero relations are [ x i , δδx i ] = − [ δδx i , x i ] = 1 for ≤ i ≤ k .In the present paper we are focusing in introducing and studying several classes of Leibniz algebras whosecorresponding Lie algebras are Heisenberg algebras H n . Recall that there is a unique irreducible representationof the Heisenberg algebra (at least a unique one that can be exponentiated). This is why physicists are able to usethe Heisenberg commutation relations to do calculations, without worry about what they are being representedon. This representation is called the Fock (or Bargmann-Fock) representation (see [3, 7, 34, 38, 39, 43]).Physically this representation corresponds to an harmonic oscillator, with the vector ∈ C [ x ] as the vacuumstate and x the operator that adds one quantum to the vacuum state. This representation is also sometimes knownas the oscillator representation. For a given Heisenberg algebra H n , n = 2 k + 1 , this representation gives riseto the so-called Fock module on H n , the linear space F [ x , ..., x k ] with the action induced by(3) ( p ( x , ..., x k ) , p ( x , ..., x k )( p ( x , ..., x k ) , x i ) x i p ( x , ..., x k )( p ( x , ..., x k ) , δδx i ) δδx i ( p ( x , ..., x k )) for any p ( x , ..., x k ) ∈ F [ x , ..., x k ] and i = 1 , ..., k .Taking now into account the above comments, we introduce in Section 2 the class of Heisenberg-Fock Leibnizalgebras as those Leibniz algebras whose corresponding Lie algebras are H n and whose H n -modules I areisomorphic to Fock modules, and provide a classification theorem. Thus, we have the opportunity of consideringHeisenberg Lie algebras together with their Fock representations in a unifying viewpoint. In this section wealso consider a generalization of this class of algebras by means of a direct sum of Heisenberg algebras ascorresponding Lie algebras, and provide also a classification theorem.In Section 3, we center in the three-dimensional Heisenberg algebra H and study three classes of Leibnizalgebras with H as corresponding Lie algebra by taking certain generalizations of the Fock module. We alsonote that Sections 2 and 3 allow us to introduce several new classes of infinite-dimensional Leibniz algebras.Finally, in Section 4, we deal with the category of Leibniz algebras with H n as corresponding algebra andsuch that the action I × H n → I gives rise to a minimal faithful representation of H n . A description of thiscategory of algebras is given and also a classification theorem when n = 3 .2. C LASSIFICATION OF H EISENBERG -F OCK TYPE L EIBNIZ ALGEBRAS
Classification of
HF L n . Consider a Heisenberg algebra H n , with n = 2 k + 1 , and its Fock module F [ x , ..., x k ] under the action (3). The Heisenberg-Fock Leibniz algebra
HF L n is defined as the Leibniz algebrawith corresponding Lie algebra H n and such that the action I × H n → I makes of I the Fock module. Since F [ x , ..., x k ] is infinite-dimensional we get a family of infinite-dimensional Leibniz algebras. Theorem 1.
The Heisenberg-Fock Leibniz algebra
HF L n admits a basis { , x i , δδx i , x t x t . . . x t k k | t i ∈ N ∪ { } , ≤ i ≤ k } in such a way that the multiplication table on this basis has the form: [ x i , δδx i ] = 1 , ≤ i ≤ k, [ δδx i , x i ] = − , ≤ i ≤ k, EIBNIZ ALGEBRAS OF HEISENBERG TYPE 3 [ x t x t . . . x t k k ,
1] = x t x t . . . x t k k , [ x t x t . . . x t k k , x i ] = x t . . . x t i − i − x t i +1 i x t i +1 i +1 . . . x t k k , ≤ i ≤ k, [ x t x t . . . x t k k , δδx i ] = t i x t . . . x t i − i − x t i − i x t i +1 i +1 . . . x t k k , ≤ i ≤ k, where the omitted products are equal to zero.Proof. Taking into account Equations (1) and (3) we conclude that { , x i , δδx i , x t x t . . . x t k k | t i ∈ N ∪ { } , ≤ i ≤ k } is a basis of HF L n and [ x t x t . . . x t k k ,
1] = x t x t . . . x t k k , [ x t x t . . . x t k k , x i ] = x t . . . x t i − i − x t i +1 i x t i +1 i +1 . . . x t k k , [ x t x t . . . x t k k , δδx i ] = t i x t . . . x t i − i − x t i − i x t i +1 i +1 . . . x t k k , for ≤ i ≤ k. Observe that we can write [ x i ,
1] = p i ( x , x , . . . , x k ) , ≤ i ≤ k, [ δδx i ,
1] = q i ( x , x , . . . , x k ) , ≤ i ≤ k, [1 ,
1] = r ( x , x , . . . , x k ) , where p i , q i , r ∈ F [ x , ..., x k ] .Taking the following change of basis, x i ′ = x i − p i ( x , x , . . . , x k ) , ≤ i ≤ k,δδx i ′ = δδx i − q i ( x , x , . . . , x k ) , ≤ i ≤ k, ′ = 1 − r ( x , x , . . . , x k ) , we derive [ x i ,
1] = 0 , [ δδx i ,
1] = 0 , [1 ,
1] = 0 , ≤ i ≤ k. Now denote [ x i , x j ] = a i,j ( x , x , . . . , x k ) , [ δδx i , δδx j ] = b i,j ( x , x , . . . , x k ) , ≤ i, j ≤ k, [ δδx i , x j ] = c i,j ( x , x , . . . , x k ) , [ x i , δδx j ] = d i,j ( x , x , . . . , x k ) , ≤ i, j ≤ k, i = j, [ x i , δδx i ] = 1 + e i ( x , x , . . . , x k ) , [ δδx i , x i ] = − f i ( x , x , . . . , x k ) , ≤ i ≤ k, [1 , x i ] = h i ( x , x , . . . , x k ) , [1 , δδx i ] = g i ( x , x , . . . , x k ) , ≤ i ≤ k. The Leibniz identity on the following triples imposes further constraints on the products.Leibniz identity Constraint { x i , x j , } ⇒ a i,j ( x , x , . . . , x k ) = 0 , ≤ i, j ≤ k, { δδx i , δδx j , } ⇒ b i,j ( x , x , . . . , x k ) = 0 , ≤ i, j ≤ k, { δδx i , x j , } ⇒ c i,j ( x , x , . . . , x k ) = 0 , ≤ i, j ≤ k, i = j, { x i , δδx j , } ⇒ d i,j ( x , x , . . . , x k ) = 0 , ≤ i, j ≤ k, i = j, { x i , δδx i , } ⇒ e i ( x , x , . . . , x k ) = 0 , ≤ i ≤ k, { δδx i , x i , } ⇒ f i ( x , x , . . . , x k ) = 0 , ≤ i ≤ k, { , x i , } ⇒ h i ( x , x , . . . , x k ) = 0 , ≤ i ≤ k, { , δδx i , } ⇒ g i ( x , x , . . . , x k ) = 0 , ≤ i ≤ k. The proof is complete. (cid:3)
A.J. CALDER ´ON, L.M. CAMACHO, B.A. OMIROV
Classification of generalized Heisenberg-Fock Leibniz algebras.
In this subsection we are interested inclassifying the class of (infinite-dimensional) Leibniz algebras formed by those Leibniz algebras L satisfyingthat their corresponding Lie algebras are finite direct sums of Heisenberg algebras and that the actions on I areinduced by Fock representations.Since(4) L/I ∼ = H k +1 ⊕ H k +1 ⊕ H k +1 ⊕ · · · ⊕ H k s +1 , we easily get(5) B i := { i , x ,i , x ,i , ..., x k i ,i , δδx ,i , δδx ,i , ..., δδx k i ,i } for the standard basis of H k i +1 , i ∈ { , , ..., s } .We put(6) I = F [ x , ..., x n ] , where n = k + k + · · · + k s .The action I × L/I → I given by ( p ( x , . . . , x n ) , i ) p ( x , . . . , x n )( p ( x , . . . , x n ) , x j,i ) p ( x , . . . , x n ) x k + k + ··· + k i − + j ( p ( x , . . . , x n ) , δδx j,i ) δδx k k ··· + ki − j p ( x , . . . , x n ) for any p ( x , . . . , x n ) ∈ F [ x , ..., x n ] and ( i, j ) with i ∈ { , , ..., s } , j ∈ { , ..., k i } , endows I of a structureof L/I -module. Hence, we get a new family of Heisenberg-Fock type Leibniz algebras which generalize theprevious ones considered in § . (case s = 1 ), that we call generalized Heisenberg-Fock Leibniz algebras , byintroducing the algebras L = L/I ⊕ I with L/I and I as in Equations (4) and (6). We will denote them as HF L k +1 , k +1 ,..., k s +1 . Our aim is to classify this class of Leibniz algebras.By taking into account the previous arguments, it is clear that for any i ∈ { , , ..., s } we have [ H k i +1 , H k i +1 ] ⊂ H k i +1 being the multiplication table among the elements in the basis B i as in Theo-rem 1. Therefore, we only need to study the products [ H k i +1 , H k j +1 ] with i, j ∈ { , , ..., s } and i = j . Lemma 1.
Let a ∈ B i and b ∈ B j , i, j ∈ { , , ..., s } with i = j . Then [ a, b ] = 0 .Proof. For i = j we have [ a, b ] = p and [ b, i ] = q for some p, q ∈ F [ x , ..., x n ] . Taking now into accountTheorem 1 we derive [ a, i ] = 0 and so p = [[ a, b ] , i ] = [[ a, i ] , b ] + [ a, [ b, i ]] = 0 . (cid:3) The next theorem is now consequence of Theorem 1 and Lemma 1.
Theorem 2.
The Leibniz algebra
HF L k +1 , k +1 ,..., k s +1 admits a basis (see Equations (5) and (6)) B ˙ ∪B ˙ ∪ · · · ˙ ∪B s ˙ ∪{ x t x t · · · x t n n | t i ∈ N ∪ { } , ≤ i ≤ n } , where n = k + k + · · · + k s , and in such a way that the multiplication table on this basis has the form: [ x j,i , δδx j,i ] = 1 i , [ δδx j,i , x j,i ] = − i , [ x t x t . . . x t n n , i ] = x t x t . . . x t n n , [ x t x t . . . x t n n , x j,i ] = x t . . . x t k ··· + ki − j − k + ··· + k i − + j − x t k ··· + ki − j +1 k + ··· + k i − + j x t k ··· + ki − j +1 k + ··· + k i − + j +1 . . . x t n n , [ x t x t . . . x t k k , δδx j,i ] = t k + ··· + k i − + j x t . . . x t k ··· + ki − j − k + ··· + k i − + j − x t k ··· + ki − j − k + ··· + k i − + j x t k ··· + ki − j +1 k + ··· + k i − + j +1 . . . x t n n , for ≤ i ≤ s, ≤ j ≤ k i and where the omitted products are equal to zero. EIBNIZ ALGEBRAS OF HEISENBERG TYPE 5
3. S
EVERAL DEGENERATIONS OF THE F OCK REPRESENTATION FOR THE - DIMENSIONAL H EISENBERG ALGEBRA
In this section we consider several degenerations of the Fock representation of the Heisenberg algebra H .First, we study when an extension of the Fock action F [ x ] × H → F [ x ] , (see Equation (3)), by allowing arbitrarypolynomials as results of the action of a fixed element in the basis { , x, δδx } of H over the elements of F [ x ] ,makes of F [ x ] an H -module. Second, the new H -modules obtained in this way give rise to new classes ofLeibniz algebras that will be described.For any linear mapping Ω : F [ x ] → F [ x ] , consider the linear space F [ x ] with the action induced by thefollowing applications: ψ : F [ x ] × H → F [ x ]( p ( x ) , Ω( p ( x ))( p ( x ) , x ) xp ( x )( p ( x ) , δδx ) δδx p ( x ) . ψ : F [ x ] × H → F [ x ]( p ( x ) , p ( x )( p ( x ) , x ) Ω( p ( x ))( p ( x ) , δδx ) δδx p ( x ) .ψ : F [ x ] × H → F [ x ]( p ( x ) , p ( x )( p ( x ) , x ) xp ( x )( p ( x ) , δδx ) Ω( p ( x )) for any p ( x ) ∈ F [ x ] . From now on, let us denote by { x i } i ∈ N ∪{ } the canonical basis of F [ x ] . By considering ψ ( p ( x ) , [ x, δδx ]) , itis immediate to get that the first action ψ makes of F [ x ] an H -module if and only if Ω = 1 F [ x ] . As consequencewe have. Proposition 1.
The Leibniz algebras obtained from the first action ψ are the same as those obtained in Theorem1. Consider now the second action ψ : F [ x ] × H → F [ x ] . Proposition 2.
The action ψ makes of F [ x ] an H -module if and only if (7) Ω( x i ) = x i +1 + i X k =0 c k (cid:16) ik (cid:17) x i − k , where { c k } k ∈ N ∪{ } is a fixed sequence in F and (cid:16) ik (cid:17) are binomial coefficients.Proof. Suppose F [ x ] is an H -module through the action ψ . Then we have x i = [ x i ,
1] = [ x i , [ x, δδx ]] = [[ x i , x ] , δδx ] − [[ x i , δδx ] , x ] = [[ x i , x ] , δδx ] − [ ix i − , x ] and so(8) [[ x i , x ] , δδx ] = x i + [ ix i − , x ] . Taking into account Equation (8), we can easily prove by induction (7). Indeed, for i = 0 we get from (8)that [[1 , x ] , δδx ] = 1 , which implies [1 , x ] = x + c = Ω(1) . For i = 1 the same equation allows us to get [[ x, x ] , δδx ] = x + [1 , x ] = 2 x + c and so [ x, x ] = x + c x + c = Ω( x ) . Let the induction hypothesis true for i = j and we will show it for i = j + 1 . Taking into account (8) wehave [[ x j +1 , x ] , δδx ] = x j +1 + [( j + 1) x j , x ] = x j +1 + ( j + 1)( x j +1 + j P k =0 c k (cid:16) jk (cid:17) x j − k ) == ( j + 2) x j +1 + j P k =0 c k ( j + 1) (cid:16) jk (cid:17) x j − k = A.J. CALDER ´ON, L.M. CAMACHO, B.A. OMIROV = ( j + 2) x j +1 + j P k =0 c k ( j + 1) j ! k !( j − k )! x j − k == ( j + 2) x j +1 + j P k =0 c k ( j +1)! k !( j +1 − k )! ( j + 1 − k ) x j − k . From here [ x j +1 , x ] = x j +2 + j X k =0 c k ( j + 1)! k !( j + 1 − k )! x j +1 − k + c j +1 = x j +2 + j +1 X k =0 c k (cid:16) j + 1 k (cid:17) x j +1 − k , that is, Ω( x j +1 ) = x j +2 + j +1 X k =0 c k (cid:16) j + 1 k (cid:17) x j +1 − k . The converse is of immediate verification. (cid:3)
Proposition 3.
Any Leibniz algebra obtained from the second action ψ admits a basis { , x, δδx } ˙ ∪{ x i : i ∈ N ∪ { }} in such a way that the multiplication table on this basis has the form: [ x i ,
1] = x i , [ x i , x ] = Ω( x i ) , [ x i , δδx ] = ix i − , [ x, δδx ] = 1 , [ δδx , x ] = − , where the omitted products are equal to zero and Ω( x i ) satisfies Equation (7).Proof. By Proposition 2 we have the restriction on Ω( x i ) . On the other hand, we know [ x i ,
1] = x i , [ x i , x ] = Ω( x i ) , [ x i , δδx ] = ix i − , [ x,
1] = p ( x ) , [ x, δδx ] = 1 + q ( x ) , [ x, x ] = a ( x ) , [ δδx ,
1] = r ( x ) , [ δδx , δδx ] = b ( x ) , [ δδx , x ] = − s ( x ) , [1 , x ] = c ( x ) , [1 ,
1] = d ( x ) , [1 , δδx ] = e ( x ) . By making the change of basis ′ = 1 + q ( x ) we can suppose that [ x, δδx ] = 1 . Now, from Leibniz identity we obtain the following equations:Leibniz identity Constraint { , , } ⇒ c ( x ) = [ d ( x ) , x ] , { , , δδx } ⇒ e ( x ) = δδx ( d ( x )) , { , x, δδx } ⇒ [ e ( x ) , x ] = δδx ( c ( x )) − d ( x ) , { x, , x } ⇒ a ( x ) = [ p ( x ) , x ] , { x, , δδx } ⇒ d ( x ) = δδx ( p ( x )) , { x, x, δδx } ⇒ p ( x ) + c ( x ) = δδx ( a ( x )) , { δδx , , x } ⇒ s ( x ) = d ( x ) + [ r ( x ) , x ] , { δδx , , δδx } ⇒ b ( x ) = δδx ( a ( x )) , { δδx , x, δδx } ⇒ [ b ( x ) , x ] = − e ( x ) − r ( x ) + δδx ( a ( x )) . By making the next change of basis: ′ = 1 − δδx ( p ( x )) ,x ′ = x − p ( x ) , δδx ′ = δδx − r ( x ) , we obtain the family of the proposition. (cid:3) EIBNIZ ALGEBRAS OF HEISENBERG TYPE 7
Finally we consider the third action ψ : F [ x ] × H → F [ x ] , being then [ x i ,
1] = x i , [ x i , x ] = x i +1 , [ x i , δδx ] = Ω( x i ) , i ∈ N ∪ { } . By arguing in a similar way to Propositions 2 and 3 we can prove the next results.
Proposition 4.
The action ψ makes of F [ x ] an H -module if and only if (9) Ω( x i ) = ix i − + x i c ( x ) . for a fixed c ( x ) ∈ F [ x ] and i ∈ N ∪ { } . Proposition 5.
Any Leibniz algebra obtained from the third action ψ admits a basis { , x, δδx } ˙ ∪{ x i : i ∈ N ∪ { }} in such a way that the multiplication table on this basis has the form: [ x i ,
1] = x i , [ x i , x ] = x i +1 , [ x i , δδx ] = Ω( x i ) , [ x, δδx ] = 1 , [ δδx , x ] = − , where the omitted products are equal to zero and Ω( x i ) satisfies Equation (9).
4. L
EIBNIZ ALGEBRAS OF MINIMAL FAITHFUL REPRESENTATION -H EISENBERG TYPE
General case.
Let H m +1 be a Heisenberg algebra of dimension m + 1 , then it is well-known that itsminimal faithful representations have dimension m + 2 , (see [13]). From now on, for a more comfortablenotation, we will denote by { x , x , . . . , x m , y , y , . . . , y m , z } the standard basis of H m +1 , (see Equation (2)), where the non-zero products are [ y i , x i ] = − [ x i , y i ] = z. By [18], we can take as minimal faithful representation the linear mapping ϕ : H m +1 → End( I ) , where I is an ( m + 2) -dimensional linear space with a fixed basis { e , e , . . . , e m +2 } , determined by ϕ ( x i ) = E ,i +1 ≤ i ≤ m,ϕ ( y i ) = E i +1 ,m +2 ≤ i ≤ m,ϕ ( z ) = E ,m +2 . Here E i,j denotes the elemental matrix with in the ( i, j ) slot and in the remaining places and we have ϕ ([ x, y ])( e ) = ϕ ( y ) (cid:0) ϕ ( x )( e ) (cid:1) − ϕ ( x ) (cid:0) ϕ ( y )( e ) (cid:1) for any x, y ∈ H m +1 and e ∈ I. Observe that H m +1 corresponds to the ( m + 2) × ( m + 2) matrices a a . . . a m +1 c . . . b . . . b ... ... ... ... ... ... . . . b m +1 . . . . This representation makes of I an H m +1 -module under the action(10) φ : I × H m +1 → I ( e i +1 , x i ) e , ≤ i ≤ m, ( e m +2 , y i ) e i +1 , ≤ i ≤ m, ( e m +2 , z ) e , being zero the remaining products among the bases elements in the action. A.J. CALDER ´ON, L.M. CAMACHO, B.A. OMIROV
In this section we are going to study the Leibniz algebras ( L, [ · , · ]) satisfying that L/I ∼ = H m +1 and wherethe H m +1 -module I is isomorphic to the minimal faithful representation ( I, φ ) . From the above, dim L =3 m + 3 and { x , x , . . . , x m , y , y , . . . , y m , z, e , e , . . . , e m +2 } is a basis of L. We also have [ e i +1 , x i ] = e , ≤ i ≤ m, [ e m +2 , y i ] = e i +1 , ≤ i ≤ m, [ e m +2 , z ] = e . Theorem 3.
Let L be a Leibniz algebra such that L/I ∼ = H m +1 ( m = 1 ) and I is the L/I -module with theminimal faithful representation given by Equation (10). Then L admits a basis { x , x , . . . , x m , y , y , . . . , y m , z, e , e , . . . , e m +2 } in such a way that the multiplications table on this basis has the form [ e i +1 , x i ] = e , [ e m +2 , y i ] = e i +1 , [ e m +2 , z ] = e , [ x i , x j ] = m +1 P s =1 α si,j e s , [ x i , y j ] = γ i,j e , i = j, [ x i , y i ] = − z + δ i e + τ e + m P s =2 ν ,s e s +1 , [ y i , y j ] = β i,j e , [ y , x ] = z, [ y i , x j ] = m +1 P s =1 ν si,j e s , i = j, [ y i , x i ] = z + ( ν i +1 i, − τ ) e + ε i +1 i e i +1 + m P s =2 s = i ( ν i +1 i,s − ν ,s ) e s +1 , i = 1 , [ z, x ] = τ e , [ z, x i ] = ν ,i e , i = 1 , for ≤ i, j ≤ m , where any α rp,q , γ p,q , δ p , τ, ν rp,q , β p,q , ε rp ∈ F and where the omitted products are equal tozero.Proof. We consider the following products: [ y i , x i ] = z + m +2 X k =1 ε ki e k , ≤ i ≤ m. Putting z ′ = z + m +2 P k =1 ε k e k we can assume [ y , x ] = z. Thus, we have [ e i +1 , x i ] = e , [ e m +2 , y i ] = e i +1 , [ e m +2 , z ] = e , [ x i , x j ] = m +2 P k =1 α ki,j e k , [ x i , y j ] = m +2 P k =1 γ ki,j e k , i = j [ x i , y i ] = − z + m +2 P k =1 δ ki e k , [ x i , z ] = m +2 P k =1 η ki e k , [ y i , y j ] = m +2 P k =1 β ki,j e k , [ y i , x j ] = m +2 P k =1 ν ki,j e k , i = j, [ y i , z ] = m +2 P k =1 θ ki e k , [ y , x ] = z, [ y i , x i ] = z + m +2 P k =1 ε ki e k , i = 1 , [ z, x i ] = m +2 P k =1 τ ki e k , [ z, y i ] = m +2 P k =1 λ ki e k , [ z, z ] = m +2 P k =1 µ k e k , with ≤ i, j ≤ m. EIBNIZ ALGEBRAS OF HEISENBERG TYPE 9
We compute all Leibniz identities using the software Mathematica and we get the following restrictions:Leibniz identity Constraint { z, z, y k } ⇒ µ m +2 = λ m +2 k = 0 , ≤ k ≤ m, { z, z, x k } ⇒ µ k +1 = τ m +2 k = 0 , ≤ k ≤ m, { z, y j , x k } ⇒ λ k +1 j = 0 , µ = λ = λ j +1 j , ≤ j, k ≤ m, j = k, { z, x j , x k } ⇒ τ k +1 j = τ j +1 k , ≤ j, k ≤ m, { y i , z, y k } ⇒ θ m +2 i = β m +2 i,k = 0 , ≤ i, j, k ≤ m, { y i , z, x k } ⇒ ν m +2 i,k = θ k +1 i , ≤ i, k ≤ m, i = k, ⇒ θ i +1 i − µ = 0 , µ = θ , ≤ i ≤ m, k = i, { y i , y j , x k } ⇒ β k +1 i,j = ν m +2 i,k = 0 , ≤ i, j, k ≤ m, j = k = i, ⇒ θ i = β j +1 i,j , θ si = 0 , ≤ i, j ≤ m, k = j, i = j, ≤ s ≤ m + 1 , ⇒ β i +1 i,j = λ j , − λ j +1 j − ε m +2 i = 0 , ≤ i, j ≤ m, i = 1 , ⇒ λ j +1 j = 0 , i = 1 , j = 1 , ⇒ θ s = 0 , ≤ s ≤ m + 1 , i = j = k = 1 , { y i , x i , y i } ⇒ θ i = β i +1 i,i , ≤ i ≤ m, { y i , x j , x k } ⇒ ν k +1 i,j = ν j +1 i,k , ≤ i, j, k ≤ m, j = i = k, ⇒ τ sk = 0 , τ k + ε k +1 i − ν i +1 i,k = 0 , ≤ s ≤ m + 1 , ≤ i, k ≤ m, j = i = k, ⇒ τ j = ν ,j , ≤ j ≤ m, i = k = 1 , j = 1 , { x i , z, y k } ⇒ η m +2 i = γ m +2 i,k = 0 , ≤ i, k ≤ m, i = k, ⇒ δ m +2 i = 0 , ≤ i ≤ m, i = k, { x i , z, x k } ⇒ η k +1 i = α m +2 i,k , ≤ i, k ≤ m, { x i , y i , y k } ⇒ λ k = 0 , ≤ k ≤ m, { x i , y j , y k } ⇒ γ k +1 i,j = α m +2 i,k = 0 , ≤ i, j, k ≤ m, i = j = k ⇒ − τ k + δ k +1 i = 0 , ≤ i, k ≤ m, j = i = k, ⇒ γ i +1 i,j = 0 , ≤ ij ≤ m, k = j = i, ⇒ η i = − τ i + δ i +1 i , η si = 0 , ≤ s ≤ m, ≤ i ≤ m, j = k = i, { x i , x j , y k } ⇒ γ ki,j = 0 , ≤ i, j, k ≤ m, i = k = j, ⇒ η i = γ j +1 i,j , ≤ i, j ≤ m, k = j = i, { x i , x j , x k } ⇒ α k +1 i,j = α j +1 i,k , ≤ i, j, k ≤ m. From here, [ e i +1 , x i ] = e , ≤ i ≤ m, [ e m +2 , y i ] = e i +1 , ≤ i ≤ m, [ e m +2 , z ] = e , [ x i , x j ] = m +1 P s =1 α si,j e s , ≤ i, j ≤ m, [ y i , y j ] = β i,j e + θ i e j +1 , ≤ i, j ≤ m, [ x i , y j ] = γ i,j e + η i e j +1 , ≤ i, j ≤ m, i = j [ x , y ] = − z + δ e + ( η + τ ) e + m P s =2 ν ,s e s +1 , ≤ i ≤ m, [ x i , y i ] = − z + δ i e + τ e + ( η i + ν ,i ) e i +1 + m P s =2 s = i ν ,s e s +1 , ≤ i ≤ m, [ y , x ] = z, [ y i , x i ] = z + ε i e + ( ν i +1 i, − τ ) e + ε i +1 i e i +1 + m P s =2 s = i ( ν i +1 i,s − ν ,s ) e s +1 , ≤ i ≤ m, [ y i , x j ] = m +1 P s =1 ν si,j e s , ≤ i, j ≤ m, i = j [ x i , z ] = η i e , ≤ i ≤ m, [ y i , z ] = θ i e , ≤ i ≤ m, [ z, x ] = τ e , [ z, x i ] = ν ,i e , ≤ i ≤ m, with the following restrictions α k +1 i,j = α j +1 i,k , ≤ i, j, k ≤ m,ν k +1 i,j = ν j +1 i,k , ≤ i, j, k ≤ m, j = i = k. Only rest to make the next change of basis x ′ i = x i − η i e m +2 , ≤ i ≤ m,y ′ = y − θ e m +2 ,y ′ j = y j − ε j e j +1 − θ j e m +2 , ≤ j ≤ m, and we obtain the family of the theorem (renaming the parameters). (cid:3) Particular case: Classification of Leibniz algebras when m = 1 . In this subsection we classify theLeibniz algebras such that
L/I ∼ = H and I is the L/I -module with the minimal faithful representation givenby Equation (10). Let us fix { x, y, z, e , e , e } as basis of L. All computations have been made by using thesoftware
M athematica.
We have the following products: [ e , x ] = e , [ e , y ] = e , [ e , z ] = e , [ x, x ] = α e + α e + α e , [ x, y ] = − z + δ e + δ e + δ e , [ x, z ] = η e + η e + η e , [ y, y ] = β e + β e + β e , [ y, x ] = z, [ y, z ] = θ e + θ e + θ e , [ z, x ] = τ e + τ e + τ e , [ z, y ] = λ e + λ e + λ e , [ z, z ] = µ e + µ e + µ e . The Leibniz identity on the following triples imposes further constraints on the products.Leibniz identity Constraint { x, x, y } ⇒ − η = τ − δ , α − η = τ , − η = τ , { x, x, z } ⇒ α = η , { x, y, z } ⇒ µ = δ , µ = − η , µ = 0 , { y, y, z } ⇒ β = θ = 0 , { y, x, y } ⇒ − θ = λ − β , − θ = λ , − θ = λ , { y, x, z } ⇒ µ = θ , µ = 0 , { z, x, y } ⇒ µ = λ , µ = τ , { z, x, z } ⇒ µ = τ , { z, y, z } ⇒ λ = 0 , { x, z, x } ⇒ η = α , { x, z, y } ⇒ µ = δ , µ = − η . EIBNIZ ALGEBRAS OF HEISENBERG TYPE 11
Thus, we get the following family of algebras, L ( α , α , α , β , β , δ , δ , η , θ ) : [ e , x ] = e , [ e , y ] = e , [ e , z ] = e , [ x, x ] = α e + α e + α e , [ x, y ] = − z + δ e + δ e , [ x, z ] = η e + α e , [ y, y ] = β e + β e , [ y, x ] = z, [ y, z ] = θ e , [ z, x ] = ( δ − η ) e − α e , [ z, y ] = ( β − θ ) e . Theorem 4.
Let L be a Leibniz algebra such that L/I ∼ = H and I is the L/I -module with the minimal faithfulrepresentation given by Equation (10). Then L is isomorphic to one of the following pairwise non-isomorphicalgebras: L (0 , , , , , , , , λ ) , λ ∈ F , L (0 , , , , , , , , , L (0 , , , , , , , , ,L (0 , , , , , , , , λ ) , λ ∈ F , L (0 , , , , , , , , , L (0 , , , , , , , , ,L (0 , , , , , , , , , L (0 , , , , , , , , , L (0 , , , , , , , , ,L (0 , , , , , , , , , L (0 , , , , , , , , , L (0 , , , , , , , , ,L (0 , , , , , , , , , L (0 , , , , , , , , , L (0 , , , , , , , , λ ) , λ ∈ F ,L (0 , , , , , , , , , L (0 , , , , , , , , , L (0 , , , , , , , , ,L (0 , , , , , , , , , L (0 , , , , , , , , , L (0 , , , , , , , , . Proof.
We can distinguish two cases:
Case 1: e ∈ [ L, L ] . Then α = 0 . Applying the general change of basis generators: x ′ = A x + A y + A z + X k =1 P i e i , y ′ = B x + B y + B z + X k =1 Q i e i , e ′ = C x + C y + C z + X k =1 R i e i we derive the expressions of the new parameters in the new basis: α ′ = α A B − α A B + δ A A B + A B P + A B P A B , α ′ = α A B R ,β ′ = β B A R , β ′ = β B + Q R ,δ ′ = β A B + δ A B + A Q + A Q A B R , δ ′ = δ A + P R ,η ′ = η A + P R , θ ′ = θ B + Q R , and the following restrictions: C = C = C = B = A = 0 ,R = − A R A ,A B R = 0 . We set P = − δ A ⇒ δ ′ = 0 ,Q = − β B ⇒ β ′ = 0 ,Q = − δ B ⇒ δ ′ = 0 ,P = − ( α B − α B ) A B ⇒ α ′ = 0 , then we get [ e , x ] = e , [ e , y ] = e , [ e , z ] = e , [ x, x ] = α ′ e , [ x, y ] = − z, [ x, z ] = η ′ e , [ y, y ] = β ′ e , [ y, x ] = z, [ y, z ] = θ ′ e , [ z, x ] = − η ′ e , [ z, y ] = − θ ′ e , where α ′ = α A B R , β ′ = β B A R , η ′ = ( η − δ ) A R , θ ′ = ( θ − β ) B R . We observe that the nullities of α , β , η , θ are invariant. Thus, we can distinguish the following non-isomorphic cases. An appropriate choice of the parameter values ( A , B and R ) allows us to obtain thefollowing algebras or families of algebras.Case Algebra α = 0 , β = 0 , η = 0 , L (0 , , , , , , , , λ ) , λ ∈ F ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , L (0 , , , , , , , , λ ) , λ ∈ F ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,α = 0 , β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , . Case 2: e / ∈ [ L, L ] . Then α = 0 . Making the following change of basis in L ( α , α , α , β , β , δ , δ , η , θ ) e ′ = α e + α e + α e ,e ′ = α e ,e ′ = α e , we obtain L (0 , , , β , β , δ , δ , η , θ ) : [ e , x ] = e , [ e , y ] = e , [ e , z ] = e , [ x, x ] = e , [ x, y ] = − z + δ e + δ e , [ x, z ] = η e + e , [ y, y ] = β e + β e , [ y, x ] = z, [ y, z ] = θ e , [ z, x ] = ( δ − η ) e − e , [ z, y ] = ( β − θ ) e . Analogously to the previous case, by making the general change of basis of generators x ′ = A x + A y + A z + X k =1 P i e i , y ′ = B x + B y + B z + X k =1 Q i e i , we derive the expressions of the new parameters in the new basis: β ′ = β B A , β ′ = β B + Q A ,δ ′ = β A B + A B + δ A B + A B Q + A B Q A B , δ ′ = − A B + δ A B + B P A B ,η ′ = − A B + η A B + B P A B , θ ′ = θ B + Q A , with the restriction: (cid:26) A = B = 0 ,A B = 0 . By putting P = A ( B − δ B ) B ⇒ δ ′ = 0 ,Q = − β B ⇒ β ′ = 0 ,Q = − B + δ B B ⇒ δ ′ = 0 , EIBNIZ ALGEBRAS OF HEISENBERG TYPE 13 we deduce [ e , x ] = e , [ e , y ] = e , [ e , z ] = e , [ x, x ] = e , [ x, y ] = − z, [ x, z ] = η ′ e + e , [ y, y ] = β ′ e , [ y, x ] = z, [ y, z ] = θ ′ e , [ z, x ] = − η ′ e − e , [ z, y ] = − θ ′ e , where β ′ = β B A , η ′ = η − δ A , θ ′ = ( θ − β ) B A . We observe that the nullities of β , η , θ are invariant. Thus, we can distinguish the following non-isomorphic cases. An appropriate choice of the parameter values ( A and B ) allows us to obtain the followingalgebras or families of algebras.Case Algebra β = 0 , η = 0 , L (0 , , , , , , , , λ ) , λ ∈ F ,β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , ,β = 0 , η = 0 , θ = 0 , L (0 , , , , , , , , . 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ALDER ´ ON . Dpto. Matem´aticas. Universidad de C´adiz. 11510 Puerto Real, C´adiz. (Spain),e-mail: [email protected] L UISA
M. C
AMACHO . Dpto. Matem´atica Aplicada I. Universidad de Sevilla. Avda. Reina Mercedes, s/n.41012 Sevilla. (Spain), e-mail: [email protected] B AKHROM
A. O
MIROV . Institute of Mathematics. National University of Uzbekistan, F. Hodjaev str. 29,100125, Tashkent (Uzbekistan), e-mail:. Institute of Mathematics. National University of Uzbekistan, F. Hodjaev str. 29,100125, Tashkent (Uzbekistan), e-mail: