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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations

Remm E* and Goze M

Associate Professor, Université de Haute Alsace, 18 Rue des Frères Lumière, 68093 Mulhouse Cedex, France

Corresponding Author:
Remm E
Associate Professor, Université de
Haute Alsace 18 Rue des Frères Lumière
68093 Mulhouse Cedex, France
Tel: +33389336500
E-mail: [email protected]

Received date: October 15, 2015; Accepted date: November 10, 2015; Published date: November 17, 2015

Citation: Remm E, Goze M (2015) A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations. J Generalized Lie Theory Appl 9:235. doi:10.4172/1736-4337.1000235

Copyright: © 2015 Remm E, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group Σ3. The first one corresponds to the Lie-admissible algebras and this class has been studied in a previous paper of Remm and Goze. Here we are interested by the second one corresponding to the third power associative algebras.

Keywords

Nonassociative algebras; Alternative algebras; Third power associative algebras; Operads

Introduction

Recently, we have classified for binary algebras, Cf. [1], relations of nonassociativity which are invariant with respect to an action of the symmetric group on three elements Σ3 on the associator. In particular we have investigated two classes of nonassociative algebras, the first one corresponds to algebras whose associator Aμ satisfies

image (1)

and the second

image (2)

where τij denotes the transposition exchanging i and j, c is the 3-cycle (1,2,3).

These relations are in correspondence with the only two irreducible one-dimensional subspaces of image3] with respect to the action of Σ3, where image3] is the group algebra of Σ3. In studies of Remm [1], we have studied the operadic and deformations aspects of the first one: the class of Lie-admissible algebras. We will now investigate the second class and in particular nonassociative algebras satisfying (2) with nonassociative relations in correspondence with the subgroups of Σ3.

Convention: We consider algebras over a field image of characteristic zero.

Gi -p3-associative Algebras

Definition

Let Σ3 be the symmetric group of degree 3 and image a field of characteristic zero. We denote by image3] the corresponding group algebra, that is the set of formal sums image, aσimageendowed with the natural addition and the multiplication induced by multiplication in Σ3, image and linearity. Let {Gi}i = <1,...,6 be the subgroups of Σ3. To fix notations we define

image

where < σ > is the cyclic group subgroup generated by σ. To each subgroup Gi we associate the vector image of image3]:

image

Lemma 1. The one-dimensional subspace image of image3] generated by

image

is an irreducible invariant subspace of image3] with respect to the right action of Σ3 on image3].

Recall that there exists only two one-dimensional invariant subspaces of image3], the second being generated by the vector imagewhere ∈(σ ) is the sign of σ. As we have precised in the introduction, this case has been studied in literature of Remm [1].

Definition 2. A Gi -p3-associative algebra is a image -algebra (A,μ) whose associator

image

satisfies

image

where image is the linear map

image

Let image be the orbit of image with respect to the right action

image

Then putting image we have

image

Proposition 3. Every Gi-p3-associative algebra is third power associative.

Recall that a third power associative algebra is an algebra (A,μ) whose associator satisfies Aμ (x, x, x) = 0. Linearizing this relation, we obtain

image

Since each of the invariant spaces image contains the vector ,image we deduce the proposition.

Remark. An important class of third power associative algebras is the class of power associative algebras, that is, algebras such that any element generates an associative subalgebra.

What are Gi -p3-associative algebras?

(1) If i =1, since image, the class of G1 -p3-associative algebras is the full class of associative algebras.

(2) If i =2, the associator of a G2 - p3-associative algebra A satisfies

Aμ (x1, x2, x3) + Aμ (x2, x1, x3) = 0

and this is equivalent to

Aμ (x, x, y),

for all x, y ∈image.

(3) If i = 3, the associator of a G3 -p3-associative algebraimage satisfies

Aμ (x1, x2, x3) + Aμ (x1, x3, x2) = 0,

that is,

Aμ (x, y, y),

for all x, y ∈image.

Sometimes G2 -p3-associative algebras are called left-alternative algebras, G3 -p3-associative algebras are right-alternative algebras. An alternative algebra is an algebra which satisfies the G2 and G3 -p3- associativity.

(4) If i = 4, we have Aμ (x, y, x) for all x, y ∈image, and the class of G3 -p3- associative algebras is the class of flexible algebras.

(5) If i = 5, the class of G5-p3-associative algebras corresponds to G5-associative algebras [2].

(6) If i = 6, the associator of a G6-p3-associative algebra satisfies

image

If we consider the symmetric product image and the skew-symmetric product image, then the G6-p3- associative identity is equivalent to

image

Definition 4. A ([ , ],image)-admissible-algebra is a image-vector spaceimage provided with two multiplications

(a) a symmetric multiplication image,

(b) a skew-symmetric multiplication [,] satisfying the identity

image

for any x, y ∈image.

Then a G6 - p3-associative algebra can be defined as ([ , ], image)- admissible algebra.

Remark: Poisson algebras. A image-Poisson algebra is a vector space image provided with two multiplications, an assocative commutative one x • y and a Lie bracket [x, y], which satisfy the Leibniz identity

image

In studies of Remm [3], it is shown that these conditions are equivalent to provide image with a nonassociative multiplication x . y satisfying

image

If we denote by image and image then the previous identity is equivalent to

image

where w1 = 3Id and image In fact the class of Poisson algebras is a subclass of a family of nonassociative algebras defined by conditions on the associator. The product satisfies

image

and

image

so it is a subclass of the class of algebras which are flexible and G5-p3-associative [1].

The Operads Gi -p3Ass and their Dual

For each i∈{1,...,6}, the operad for Gi -p3-associative algebras will be denoted by Gi -p3 Ass. The operads image are binary quadratic operads, that is, operads of the form image = Γ(E) / (R), where Γ(E) denotes the free operad generated by a Σ2-module E placed in arity 2 and (R) is the operadic ideal generated by a Σ3-invariant subspace R of Γ(E)(3). Then the dual operad image! is the quadratic operad image ) , where image is the annihilator of R ⊂ Γ(E)(3) in the pairing

image (3)

and (R) is the operadic ideal generated by R. For the general notions of binary quadratic operads [4,5]. Recall that a quadratic operad image is Koszul if the free image-algebra based on aimage-vector space V is Koszul, for any vector space V. This property is conserved by duality and can be studied using generating functions of image and of image! [4,6]. Before studying the Koszulness of the operads Gi -p3 Ass, we will compute the homology of an associative algebra which will be useful to look if Gi -p3imagess are Koszul or not.

Let A2 the two-dimensional associative algebra given in a basis Let A2 the two-dimensional associative algebra given in a basis {e1,e2} by e1e1 = e2, e1e2 = e2e1 = e2e2 = 0. Recall that the Hochschild homology of an associative algebra is given by the complex image where image and the differentials image are given by

image

Concerning the algebra A2, we have

image

for any i, j = 1,2. Similarly we have

image

and 0 in all the other cases. Then dim Im d2 = 2 and dim Ker d1 = 4. Then H1(A2 A2) is isomorphic to A2. We have also

image

and d3 = 0 in all the other cases. Then dim Im d3 = 4 and dim Ker d2 = 6. Thus H2(A2, A2) is non trivial and A2 is not a Koszul algebra.

Now we will study all the operads Gi -p3imagess.

The operad (G1 -p3imagess)

Since G1 -p3imagess=imagess, where imagess denotes the operad for associative algebras, and since the operad imagess is selfdual, we have

image

We also have

image

where image is the maximal current operad of image defined in [7,8].

The operad (G2 -p3imagess)

The operad G2 -p3imagess is the operad for left-alternative algebras. It is the quadratic operad image = Γ(E) / (R), where the Σ3-invariant subspace R of Γ (E)(3) is generated by the vectors

image

The annihilator R of R with respect to the pairing (3) is generated by the vectors

image (4)

We deduce from direct calculations that dim R = 9 and

Proposition 5. The (G2 -p3imagess)! -algebras are associative algebras satisfying

abc = −bac.

Recall that (G2imagess)!-algebras are associative algebras satisfying

abc = bac.

and this operad is classically denoted imageerm.

Theorem 6. The operad (G2 -p3imagess)! is not Koszul [9].

Proof. It is easy to describe (G2 -p3imagess)! (n) for any n. In fact (G2 -p3imagess)! (4) corresponds to associative elements satisfying

image

and (G2 -p3imagess)! (4) = {0}. Let imagebe (G2 -p3imagess). The generating function of image! = (G2 -p3imagess)! is

image

But the generating function of image = (G2 -p3imagess) is

image

and if (G2 -p3imagess) is Koszul, then the generating functions should be related by the functional equation

image

and it is not the case so both (G2 -p3imagess) and (G2 -p3imagess)! are not Koszul.

By definition, a quadratic operad image is Koszul if any free image-algebra on a vector space V is a Koszul algebra. Let us describe the free algebra

image when dim V =1 and 2.

A (G2 -p3imagess)!-algebraimage is an associative algebra satisfying

xyz = −yxz,

for any x, y, z ∈image. This implies xyzt = 0 for any x, y, z ∈image. In particular we have

image

for any x, y ∈image. If dim V =1, image is of dimension 2 andimage given by

image

In fact if V = image{e1} thus in image we have e13 = 0. We deduce that image and image is not Koszul. It is easy to generalize this construction. If dim V = n, then image and if {e1,...,en} is a basis of V then image for i, j = 1,...,n and l, m, p = 1,...,n with m > l, is a basis of imageFor example, if n = 2, the basis of imageis

image

and the multiplication table is

image

For this algebra we have

image

and Ker d1 is of dim 64. The space Im d2 doesn’t contain in particular the vectors (vi, vi) for i = 1, 2 because these vectors vi are not in the derived subalgebra. Since these vectors are in Ker d1 we deduce that the second space of homology is not trivial.

Proposition 7. The current operad of G2 -p3imagess is

image

This is directly deduced of the definition of the current operad [7].

The operad (G2 - p3imagess)

It is defined by the module of relations generated by the vector

image

and R is the linear span of

image

Proposition 8. A (G2 p3imagess)!-algebra is an associative algebra image satisfying

abc = −acb,

for any a, b, c ∈image.

Since (G3 -p3imagess)! is basically isomorphic to (G2 -p3imagess)! we deduce that (G3 -p3imagess) is not Koszul.

The operad (G4 -p3imagess)

Remark that a (G4 -p3imagess)-algebra is generally called flexible algebra. The relation

image

is equivalent to Aμ (x, y, x) = 0 and this denotes the flexibility of (image, μ).

Proposition 9. A (G4 -p3imagess)!-algebra is an associative algebra satisfying

abc = −cba.

This implies that dim (G4 -p3imagess)! (3) = 3. We have also image for any σ ∈ Σ4. This gives

dim (G4 -p3imagess)! (4) = 1. Similarly

image

(the algebra is associative so we put some parenthesis just to explain how we pass from one expression to an other). We deduce (G4-p 3imagess)! (5) = {0} and more generally (G4 -p3imagess)! (a) = {0} for a ≥ 5.

The generating function of (G4 -p3imagess)! is

image

Let image be the free (G4 -p3imagess)!-algebra based on the vector space V. In this algebra we have the relations

image

for any a, b∈V. Assume that dimV =1. If {e1} is a basis of V, then e13= 0 and image We deduce that image is not a Koszul algebra.

Proposition 10. The operad for flexible algebra is not Koszul.

Let us note that, if dim V=2 and {e1,e2} is a basis of V, then image is generated by image and is of dimension 12.

Proposition 11. We have

image

This means thatimage is an associative algebra image satisfying abc = cba, for any a, b, c ∈ image.

The operad (G5-p3imagess)

It coincides with (G5-imagess) and this last has been studied in studies of Remm [2].

The operad (G6-p3imagess)

A (G6-p3imagess)-algebra (image, μ) satisfies the relation

image

The dual operad (G6 - p3imagess)! is generated by the relations

image

We deduce

Proposition 12. A (G6-p3imagess)!-algebra is an associative algebra image which satisfies

image

for any a, b, c ∈ image. In particular

image

Lemma 13. The operad (G6-p3imagess)! satisfies (G6 - p3imagess)!(4) = {0}.

Proof. We have in (G6-p3imagess)!(4) that

image

so x1x2x3x4 = 0. We deduce that the generating function of (G6-p3imagess)! is

image

If this operad is Koszul the generating function of the operad (G6-p3imagess) should be of the form

image

But if we look the free algebra generated by V with dimV =1, it satisfies a3 = 0 and coincides with image Then (G6-p3imagess) is not Koszul.

Proposition 14. We have

image

that is the binary quadratic operad whose corresponding algebras are associative and satisfying

abc = acb = bac.

Cohomology and Deformations

Let (image,μ) be a image-algebra defined by quadratic relations. It is attached to a quadratic linear operad image. By deformations of (image,μ), we mean [10]

• A image* non archimedian extension field of image, with a valuation v such that, if A is the ring of valuation and image the unique ideal of A, then the residual field A /image is isomorphic to image.

• The A / imagevector space image is image-isomorphic to image.

• For any a, b ∈image we have that

image

belongs to the image-moduleimage (isomorphic to imageimage).

The most important example concerns the case where A is image[[t]], the ring of formal series. In this case image image* = image((t)) the field of rational fractions. This case corresponds to the classical Gerstenhaber deformations. Since A is a local ring, all the notions of valued deformations coincides [11].

We know that there exists always a cohomology which parametrizes deformations. If the operad image is Koszul, this cohomology is the "standard”-cohomology called the operadic cohomology. If the operad image is not Koszul, the cohomology which governs deformations is based on the minimal model of image and the operadic cohomology and deformations cohomology differ [12].

In this section we are interested by the case of left-alternative algebras, that is, by the operad (G2-p3imagess) and also by the classical alternative algebras.

Deformations and cohomology of left-alternative algebras

A image-left-alternative algebra (image,μ) is a image-(G2-p3imagess)-algebra. Then satisfies

image

A valued deformation can be viewed as a image[[t]] -algebra (A ⊗ image[[t]], μt) whose product μt is given by

image

The operadic cohomology: It is the standard cohomology image of the (G2-p3imagess) -algebra (image,μ). It is associated to the cochains complex

image

where image = (G2-p3imagess) and

image

Since (G2-p3imagess)!(4) = 0, we deduce that

image

because the cochains complex is a short sequence

image

The coboundary operator are given by

image

The deformations cohomology: The minimal model of (G2-p3imagess) is a homology isomorphism

image

of dg-operads such that the image of ∂ consists of decomposable elements of the free operad Γ(E). Since (G2-p3imagess)(1) = image, this minimal model exists and it is unique. The deformations cohomology image of image is the cohomology of the complex

image

where

image

The Euler characteristics of E(q) can be read off from the inverse of the generating function of the operad (G2-p3imagess)

image

which is

image

We obtain in particular

image

Each one of the modules E(p) is a graded module (E*( p)) and

image

We deduce

• E(2) is generated by two degree 0 bilinear operation image

• E(3) is generated by three degree 1 trilinear operation image

• E(4) = 0.

Considering the action of Σn on E(n) we deduce that E(2) is generated by a binary operation of degree 0 whose differential satisfies

∂(μ2) = 0,

E(3) is generated by a trilinear operation of degree one such that

image

image

Since E(4) = 0 we deduce

Proposition 15. The cohomology image which governs deformations of right-alternative algebras is associated to the complex

image

with

image

In particular any 4-cochains consists of 5-linear maps.

Alternative algebras

Recall that an alternative algebra is given by the relation

image

Theorem 16. An algebra (image,μ) is alternative if and only if the associator satisfies

image

with image

Proof. The associator satisfies image with image and image The invariant subspace of image3] generated by v1 and v2 is of dimension 5 and contains the vector image From literature of Remm [1], the space is generated by the orbit of the vector v.

Proposition 17. Let imagelt be the operad for alternative algebras. Its dual is the operad for associative algebras satisfying

image

Remark. The current operad image is the operad for associative algebras satisfying abc = bac = cba = acb = bca , that is, 3-commutative algebras so

image

In literature of Dzhumadil’daev and Zusmanovich [9], one gives the generating functions of image =imagelt and image! =imagelt!

image

and conclude to the non-Koszulness of imagelt.

The operadic cohomology is the cohomology associated to the complex

image

Since imagelt! (p) = 0 for p ≥ 6 we deduce the short sequenceimage

image

But if we compute the formal inverse of the function −gAlt (−x ) we obtain

image

Because of the minus sign it can not be the generating function of the operad image! =imagelt!. So this implies also that both operad are not Koszul. But it gives also some information on the deformation cohomology. In fact if Γ(E) is the free operad associated to the minimal model, then

image

Since image, the graded space E(6) is not concentred in degree even. Then the 6-cochains of the deformation cohomology are 6-linear maps of odd degree.

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