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**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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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.

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

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 Σ_{3} on the associator. In particular we have investigated two classes of nonassociative algebras, the first one corresponds to algebras whose associator A_{μ} satisfies

(1)

and the second

(2)

where τ_{ij} 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 [Σ_{3}] with respect to the action of Σ_{3}, where [Σ_{3}] is the group algebra of Σ_{3}. 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 Σ_{3}.

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

**Definition**

Let Σ_{3} be the symmetric group of degree 3 and a field of characteristic zero. We denote by [Σ_{3}] the corresponding group algebra, that is the set of formal sums , a_{σ} ∈ endowed with the natural addition and the multiplication induced by multiplication in Σ_{3}, and linearity. Let {G_{i}}_{i = <1,...,6} be the subgroups of Σ_{3}. To fix notations we define

where < σ > is the cyclic group subgroup generated by σ. To each subgroup G_{i} we associate the vector of [Σ_{3}]:

**Lemma 1.*** The one-dimensional subspace* *of *[Σ_{3}] *generated by*

is an irreducible invariant subspace of [Σ_{3}] with respect to the right action of Σ_{3} on [Σ_{3}].

Recall that there exists only two one-dimensional **invariant subspaces **of [Σ_{3}], the second being generated by the vector where ∈(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 G _{i} -p^{3}-associative algebra is a -algebra (A,μ) whose associator*

*satisfies*

*where is the linear map*

Let be the orbit of with respect to the right action

Then putting we have

**Proposition 3.** *Every G _{i}-p^{3}-associative algebra is third power associative.*

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

Since each of the invariant spaces contains the vector , 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 G _{i} -p^{3}-associative algebras?**

(1) If i =1, since , the class of G_{1} -p^{3}-associative algebras is the full class of associative algebras.

(2) If i =2, the associator of a G_{2} - p^{3}-associative algebra *A* satisfies

A_{μ} (x_{1}, x_{2}, x_{3}) + A_{μ} (x_{2}, x_{1}, x_{3}) = 0

and this is equivalent to

A_{μ} (x, x, y),

for all x, y ∈.

(3) If i = 3, the associator of a G_{3} -p^{3}-associative algebra satisfies

A_{μ} (x_{1}, x_{2}, x_{3}) + A_{μ} (x_{1}, x_{3}, x_{2}) = 0,

that is,

A_{μ} (x, y, y),

for all x, y ∈.

Sometimes G_{2} -p^{3}-associative algebras are called left-alternative algebras, G_{3} -p^{3}-associative algebras are right-alternative algebras. An **alternative algebra **is an algebra which satisfies the G_{2} and G_{3} -p^{3}- associativity.

(4) If i = 4, we have A_{μ} (x, y, x) for all x, y ∈, and the class of G_{3} -p^{3}- associative algebras is the class of flexible algebras.

(5) If i = 5, the class of G_{5}-p^{3}-associative algebras corresponds to G_{5}-associative algebras [2].

(6) If i = 6, the associator of a G_{6}-p^{3}-associative algebra satisfies

If we consider the symmetric product and the skew-symmetric product , then the G_{6}-p^{3}- associative identity is equivalent to

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

(a) a symmetric multiplication ,

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

for any x, y ∈.

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

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

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

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

where w_{1} = 3Id and 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

and

so it is a subclass of the class of algebras which are flexible and G_{5}-p^{3}-associative [1].

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

(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 is Koszul if the free -algebra based on a-vector space V is Koszul, for any vector space V. This property is conserved by duality and can be studied using generating functions of and of ^{!} [4,6]. Before studying the Koszulness of the operads G_{i} -p^{3} Ass, we will compute the **homology **of an associative algebra which will be useful to look if G_{i} -p^{3}ss are Koszul or not.

Let A_{2} the two-dimensional associative algebra given in a basis Let A_{2} the two-dimensional associative algebra given in a basis {e_{1},e_{2}} by e_{1}e_{1} = e_{2}, e_{1}e_{2} = e_{2}e_{1} = e_{2}e_{2} = 0. Recall that the Hochschild homology of an associative algebra is given by the complex where and the differentials are given by

Concerning the algebra A_{2}, we have

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

and 0 in all the other cases. Then dim Im d_{2} = 2 and dim Ker d_{1} = 4. Then H_{1}(A_{2} A_{2}) is isomorphic to A_{2}. We have also

and d_{3} = 0 in all the other cases. Then dim Im d_{3} = 4 and dim Ker d_{2} = 6. Thus H_{2}(A_{2}, A_{2}) is non trivial and A_{2} is not a **Koszul algebra**.

Now we will study all the operads G_{i} -p^{3}ss.

**The operad (G _{1} -p^{3}ss)**

Since G_{1} -p^{3}ss=ss, where ss denotes the operad for associative algebras, and since the operad ss is selfdual, we have

We also have

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

**The operad (G _{2} -p^{3}ss)**

The operad G_{2} -p^{3}ss is the operad for left-**alternative algebras**. It is the quadratic operad = Γ(E) / (R), where the Σ_{3}-invariant subspace R of Γ (E)(3) is generated by the vectors

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

(4)

We deduce from direct calculations that dim R^{⊥} = 9 and

**Proposition 5. ***The (G _{2} -p^{3}ss)^{!} -algebras are associative algebras satisfying*

*abc = −bac.*

Recall that (G2ss)^{!}-algebras are associative algebras satisfying

*abc = bac.*

*and this operad is classically denoted erm.*

**Theorem 6.** *The operad (G _{2} -p^{3}ss)^{!} is not Koszul [9].*

*Proof. *It is easy to describe (G_{2} -p^{3}ss)^{!} (n) for any n. In fact (G_{2} -p^{3}ss)^{!} (4) corresponds to associative elements satisfying

and (G_{2} -p^{3}ss)^{!} (4) = {0}. Let be (G_{2} -p^{3}ss). The generating function of ^{!} = (G_{2} -p^{3}ss)^{!} is

But the generating function of = (G_{2} -p^{3}ss) is

and if (G_{2} -p^{3}ss) is Koszul, then the generating functions should be related by the functional equation

and it is not the case so both (G_{2} -p^{3}ss) and (G_{2} -p^{3}ss)^{!} are not Koszul.

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

when dim V =1 and 2.

A (G_{2} -p^{3}ss)^{!}-algebra is an associative algebra satisfying

xyz = −yxz,

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

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

In fact if V = {e_{1}} thus in we have e_{1}^{3} = 0. We deduce that and is not Koszul. It is easy to generalize this construction. If dim V = n, then and if {e_{1},...,e_{n}} is a basis of V then for i, j = 1,...,n and l, m, p = 1,...,n with m > l, is a basis of For example, if n = 2, the basis of is

and the multiplication table is

For this algebra we have

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

**Proposition 7.** The current operad of G_{2} -p^{3}ss is

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

**The operad (G _{2} - p_{3}ss)**

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

and R is the linear span of

**Proposition 8**. A (G_{2} p_{3}ss)^{!}-algebra is an associative algebra satisfying

*abc = −acb,*

*for any a, b, c ∈.*

Since (G_{3} -p^{3}ss)^{!} is basically isomorphic to (G_{2} -p^{3}ss)^{!} we deduce that (G_{3} -p^{3}ss) is not Koszul.

**The operad (G _{4} -p^{3}ss)**

Remark that a (G_{4} -p^{3}ss)-algebra is generally called flexible algebra. The relation

is equivalent to A_{μ} (x, y, x) = 0 and this denotes the flexibility of (, μ).

**Proposition 9. **A (G_{4} -p^{3}ss)^{!}-algebra is an associative algebra satisfying

*abc = −cba.*

This implies that dim (G_{4} -p^{3}ss)^{!} (3) = 3. We have also for any σ ∈ Σ_{4}. This gives

dim (G_{4} -p^{3}ss)^{!} (4) = 1. Similarly

(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 3ss)^{!} (5) = {0} and more generally (G_{4} -p^{3}ss)^{!} (a) = {0} for a ≥ 5.

The generating function of (G_{4} -p^{3}ss)^{!} is

Let be the free (G_{4} -p^{3}ss)^{!}-algebra based on the vector space V. In this algebra we have the relations

for any a, b∈V. Assume that dimV =1. If {e_{1}} is a basis of V, then e_{1}^{3}= 0 and We deduce that is not a Koszul algebra.

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

Let us note that, if dim V=2 and {e_{1},e_{2}} is a basis of V, then is generated by and is of dimension 12.

**Proposition 11.** *We have*

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

**The operad (G _{5}-p^{3}ss)**

It coincides with (G_{5}-ss) and this last has been studied in studies of Remm [2].

**The operad (G _{6}-p^{3}ss)**

A (G_{6}-p^{3}ss)-algebra (, μ) satisfies the relation

The dual operad (G_{6} - p_{3}ss)^{!} is generated by the relations

We deduce

**Proposition 12. **A (G_{6}-p^{3}ss)^{!}-*algebra is an associative algebra which satisfies*

for any a, b, c ∈ . In particular

**Lemma 13. **The operad (G_{6}-p^{3}ss)^{!} satisfies (G6 - p3ss)^{!}(4) = {0}.

*Proof. *We have in (G_{6}-p^{3}ss)^{!}(4) that

so x_{1}x_{2}x_{3}x_{4} = 0. We deduce that the generating function of (G_{6}-p^{3}ss)^{!} is

If this operad is Koszul the generating function of the operad (G_{6}-p^{3}ss) should be of the form

But if we look the free algebra generated by V with dimV =1, it satisfies a_{3} = 0 and coincides with Then (G_{6}-p^{3}ss) is not Koszul.

**Proposition 14. ***We have*

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

*abc = acb = bac.*

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

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

• The A / vector space is -isomorphic to .

• For any a, b ∈ we have that

belongs to the -module (isomorphic to ⊗ ).

The most important example concerns the case where A is [[t]], the ring of formal series. In this case * = ((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 is Koszul, this cohomology is the "standard”-cohomology called the operadic cohomology. If the operad is not Koszul, the cohomology which governs deformations is based on the minimal model of 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 (G_{2}-p^{3}ss) and also by the classical alternative algebras.

**Deformations and cohomology of left-alternative algebras**

A -left-alternative algebra (,μ) is a -(G_{2}-p^{3}ss)-algebra. Then satisfies

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

**The operadic cohomology:** It is the standard cohomology of the (G_{2}-p^{3}ss) -algebra (,μ). It is associated to the cochains complex

where = (G_{2}-p^{3}ss) and

Since (G_{2}-p^{3}ss)^{!}(4) = 0, we deduce that

because the cochains complex is a short sequence

The coboundary operator are given by

**The deformations cohomology: **The minimal model of (G_{2}-p^{3}ss) is a homology isomorphism

of dg-operads such that the image of ∂ consists of decomposable elements of the free operad Γ(E). Since (G_{2}-p^{3}ss)(1) = , this minimal model exists and it is unique. The deformations cohomology of is the cohomology of the complex

where

The Euler characteristics of E(q) can be read off from the inverse of the generating function of the operad (G_{2}-p^{3}ss)

which is

We obtain in particular

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

We deduce

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

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

• E(4) = 0.

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

∂(μ_{2}) = 0,

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

Since E(4) = 0 we deduce

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

*with*

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

**Alternative algebras**

Recall that an alternative algebra is given by the relation

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

*with *

*Proof.* The associator satisfies with and The invariant subspace of [Σ_{3}] generated by v_{1} and v_{2} is of dimension 5 and contains the vector From literature of Remm [1], the space is generated by the orbit of the vector v.

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

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

In literature of Dzhumadil’daev and Zusmanovich [9], one gives the generating functions of =lt and ^{!} =lt^{!}

and conclude to the non-Koszulness of lt.

The operadic cohomology is the cohomology associated to the complex

Since lt^{!} (p) = 0 for p ≥ 6 we deduce the short sequence

But if we compute the formal inverse of the function −g_{Alt} (−x ) we obtain

Because of the minus sign it can not be the generating function of the operad ^{!} =lt^{!}. 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

Since , 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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