A class of nonassociative algebras including flexible and alternative algebras, operads and deformations

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 on three elements. The first one corresponds to the Lie-admissible algebras that we studied in a previous paper. Here we are interested by the second one corresponding to the third power associative algebras.


Introduction
In [5] we have classified, for binary algebras, 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 and the second 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 K[Σ 3 ] with respect to the action of Σ 3 , where K[Σ 3 ] is the group algebra of Σ 3 . In [5], we have study, the operadic and deformations aspects of the first one which is 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 K of characteristic zero.

.1 Definition
Let {G i } i=1,··· ,6 be the subgroups of Σ 3 . To fix notations we define                where < σ > the cyclic group subgroup generated by σ. To each subgroup G i we associate the vector v G i of K[Σ 3 ] : Definition 2 A G i -p 3 -associative algebra is a K-algebra (A, µ) whose associator 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 F G i contains the vector v Σ 3 , we deduce the proposition.
Remark 4 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.
, ⋆)-admissible-algebra is a K-vector space A provided with two multiplication: (a) a symmetric multiplication ⋆, for any x, y ∈ A.

The operads G i -p Ass and their dual
For each i ∈ {1, · · · , 6}, the operad for G i -p 3 -associative algebras will be denoted by G i -p 3 Ass. The operads {G i -p 3 Ass} i=1,··· ,6 are binary quadratic operads, that is, operads of the form P = 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 P ! is the quadratic operad P ! : and (R ⊥ ) is the operadic ideal generated by R ⊥ . For the general notions of binary quadratic operads see [3,9]. Recall that a quadratic operad P is Koszul if the free P-algebra based on a K-vector space V is Koszul, for any vector space V . This property is conserved by duality and can be studied using generating functions of P and of P ! (see [3] or [11]) Before studying the Kozsulness 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 Ass are Kozsul or not.
Concerning the algebra A 2 , we have and 0 in all the other cases. Then dim Imd 2 = 2 and dim Kerd 1 = 4. Then and d 3 = 0 in all the other cases. Then dim Imd 3 = 4 and dim Kerd 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 Ass.

The operad (G 1 -p 3 Ass)
Since G 1 -p 3 Ass = Ass, where Ass denotes the operad for associative algebras, and since the operad Ass is selfdual, we have We also have where P is the maximal current operad of P defined in [6].

The operad (G 2 -p 3 Ass)
The operad G 2 -p 3 Ass is the operad for left-alternative algebras. It is the is generated by the vectors The annihilator R ⊥ of R with respect to the pairing (3) is generated by the vectors We deduce from direct calculations that dim R ⊥ = 9 and Recall that (G 2 Ass) ! -algebras are associative algebras satisfying abc = bac. and this operad is classically denoted Perm.
Proof. It is easy to describe (G 2 -p 3 Ass) ! (n) for any n. In fact (G 2 -p 3 Ass) ! (4) correspond to associative elements satisfying But the generating function of P = (G 2 -p 3 Ass) is and if (G 2 -p 3 Ass) 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 Ass) and (G 2 -p 3 Ass) ! are not Koszul.

Remark 8
Since the redaction of this work, Dzumadildaev and Zusmanovich have shown and published this result. For this reason we refer this theorem to these authors.
By definition, a quadratic operad P is Koszul if any free P-algebra on a vector space V is a Koszul algebra. Let us describe the free algebra for any x, y, z ∈ A. This implies xyzt = 0 for any x, y, z ∈ A. In particular we have for any x, y ∈ A. If dim V = 1, F (G 2 -p 3 Ass) ! (V ) is of dimension 2 and given by e 1 e 1 = e 2 , e 1 e 2 = e 2 e 1 = e 2 e 2 = 0.
and if {e 1 , · · · , e n } is a basis of V then {e i , e 2 i , e i e j , e l e m e p } , for i, j = 1, · · · , n and l, m, p = 1, · · · , n with m > l, is a basis of F (G 2 -p 3 Ass) ! (V ). For example, if n = 2, the basis of F (G 2 -p 3 Ass) ! (V ) is , 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 9
The current operad of G 2 -p 3 Ass is This is directly deduced of the definition of the current operad [6].

The operad (G 3 -p 3 Ass)
It is defined by the module of relations generated by the vector and R ⊥ is the linear span of Proposition 10 A (G 3 -p 3 Ass) ! -algebra is an associative algebra A satisfying abc = −acb, for any a, b, c ∈ A.

The operad (G 4 -p 3 Ass)
Remark that a (G 4 -p 3 Ass)-algebra is generally called flexible algebra. The relation is equivalent to A µ (x, y, x) = 0 and this denotes the flexibility of (A, µ).
Proposition 12 The operad for flexible algebra is not Koszul.

Proposition 13 We have
This means that a G 4 -p 3 Ass is an associative algebra A satisfying abc = cba, for any a, b, c ∈ A.

The operad (G 5 -p 3 Ass)
It coincides with (G 5 -Ass) and this last has been studied in [4].

The operad (G 6 -p 3 Ass)
A (G 6 -p 3 Ass)-algebra (A, µ) satisfies the relation The dual operad (G 6 -p 3 Ass) ! is generated by the relations
Proof. We have in (G 6 -p 3 Ass) ! (4) that We deduce that the generating function of (G 6 -p 3 Ass) ! is If this operad is Koszul the generating function of the operad (G 6 -p 3 Ass) should be of the form f (x) = x + x 2 + 11 6 x 3 + 25 6 x 4 + 127 12 But if we look the free algebra generated by V with dim V = 1, it satisfies a 3 = 0 and coincides with F (G 2 -p 3 Ass) ! (V ). Then (G 6 -p 3 Ass) is not Koszul.

Proposition 16
We have that is the binary quadratic operad whose corresponding algebras are associative and satisfying abc = acb = bac.

Cohomology and Deformations
Let (A, µ) be a K-algebra defined by quadratic relations. It is attached to a quadratic linear operad P. By deformations of (A, µ), we mean ( [7]) • a K * non archimedian extension field of K, with a valuation v such that, if A is the ring of valuation and M the unique ideal of A, then the residual field A/M is isomorphic to K.
• The A/M vector space A is K-isomorphic to A.
• For any a, b ∈ A we have that belongs to the M-module A (isomorphic to A ⊗ M) .

The most important example concerns the case where
, the ring of formal series. In this case M = i≥1 a i t i , a i ∈ K , K * = K((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 ( [2]).
We know ( [8]) that there exists always a cohomology which parametrizes deformations. If the operad P is Koszul, this cohomology is the "standard"cohomology called the operadic cohomology. If the operad P is not Koszul, the cohomology which governs deformations is based on the minimal model of P and the operadic cohomology and deformations cohomology differ.
In this section we are interested by the case of left-alternative algebras, that is, by the operad (G 2 -p 3 Ass) and also by the classical alternative algebras.

Deformations and cohomology of left-alternative algebras
, µ t ) whose product µ t is given by

a) The operadic cohomology
It is the standard cohomology H * (G 2 -p 3 Ass) (A, A) st of the (G 2 -p 3 Ass)-algebra (A, µ). It is associated to the cochains complex where P = (G 2 -p 3 Ass) and Since (G 2 -p 3 Ass) ! (4) = 0, we deduce that because the cochains complex is a short sequence The coboundary operator are given by The Euler characteristics of E(q) can be read off from the inverse of the generating function of the operad G 2 -p 3 Ass which is We obtain in particular χ(E(4)) = 0.
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 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 (we have (µ 2 • 2 µ 2 ) · τ 12 (a, b, c) = b(ac)) Since E(4) = 0 we deduce Proposition 17 The cohomology H * (A, A) def o which governs deformations or right-alternative algebras is associated to the complex In particular any 4-cochains consists of 5-linear maps.

Alternative algebras
Recall that an alternative algebra is given by the relation Theorem 18 An algebra (A, µ) is alternative if and only if the associator satisfies with v 1 = Id + τ 12 and v 2 = Id + τ 23 . The invariant subspace of K[Σ 3 ] generated by v 1 and v 2 is of dimension 5 and contains the vector σ∈Σ 3 σ. From [5], the space is generated by the orbit of the vector v.
Proposition 19 Let Alt be the operad for alternative algebras. Its dual is the operad for associative algebras satisfying abc − bac − cba − acb + bca + cab = 0.
Since χ(E(6)) = i (−1) i dim E i (6), the graded space E(6) is not concentred in degree even. Then the 6-cochains of the deformation cohomology are 6linear maps of odd degree.