Cohomology and Formal Deformations of Alternative Algebras

The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of alternative algebras. A short review of basics on alternative algebras and their connections to some other algebraic structures is also provided. MSC 2010: 17D05, 17D15, 16S80, 16E40


Introduction
Deformation theory arose mainly from geometry and physics. In the latter field, the non-commutative associative multiplication of operators in quantum mechanics is thought of as a formal associative deformation of the pointwise multiplication of the algebra of symbols of these operators. In the sixties, Murray Gerstenhaber introduced algebraic formal deformations for associative algebras in a series of papers (see [11,12,13,14,15]). He used formal series and showed that the theory is intimately connected to the cohomology of the algebra. The same approach was extended to several algebraic structures (see [2,3,5,6,25]). Other approaches to study deformations exist; see [8,9,10,17,18,19,20,24,28]; also see [26] for a review.
In this paper, we introduce a cohomology and a formal deformation theory of alternative algebras. If A is left alternative algebra, then the algebra defined on the same vector space A with "opposite" multiplication x • y := yx is a right alternative algebra and vice-versa. Hence, all the statements for left alternative algebras have their corresponding statements for right alternative algebras. Thus, we will only consider the left alternative algebra case in this paper. We also review the connections of alternative algebras to other algebraic structures. In Section 2, we review the basic definitions and properties related to alternative algebras. In Section 3, we discuss in particular all the links between alternative algebras and some other algebraic structures such as Moufang loops, Malcev algebras and Jordan algebras. In Section 4, we introduce a cohomology theory of left alternative algebras. We compute the second cohomology group of the 2 by 2 matrix algebra. It is known that, as an associative algebra, its second cohomology group is trivial, but we show that this is not the case as left alternative algebra. Finally, in Section 5, we develop a formal deformation theory for left alternative algebras and show that the cohomology theory introduced in Section 4 fits.

Lemma 3.
A pair (A, μ) is a left alternative K-algebra (resp. right alternative K-algebra) if and only if the identity holds.

Remark 4.
When considering multiplication as a linear map μ : A ⊗ A → A, the condition (2.3) (resp. (2.4)) may be written where id stands for the identity map and σ 1 and σ 2 stand for transpositions generating the permutation group S 3 which are extended to trilinear maps defined by In terms of associators, the identities (2.3) (resp. (2.4)) are equivalent to as + as • σ 1 = 0 resp. as + as • σ 2 = 0 . (2.7) Remark 5. The notions of subalgebra, ideal and quotient algebra are defined in the usual way. For general theory about alternative algebras see [29]. The alternative algebras generalize associative algebras. Recently, in [7], it was shown that their operad is not Koszul. The dual operad of right alternative (resp. left alternative) algebras is defined by associativity and the identity The dual operad of alternative algebras is defined by the associativity and the identity μ μ(x, y), z + μ μ(y, x), z + μ μ(z, x), y + μ μ(x, z), y + μ μ(y, z), x + μ μ(z, y), x = 0.

Structure theorems and examples
We have these following fundamental properties: • Artin's theorem. In an alternative algebra, the subalgebra generated by any two elements is associative. Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written without parenthesis unambiguously in an alternative algebra.
• Generalization of Artin's theorem. Whenever three elements x, y, z in an alternative algebra associate (i.e. as(x, y, z) = 0), the subalgebra generated by those elements is associative. • Corollary of Artin's theorem. Alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative. The converse need not hold: the sedenions are power-associative but not alternative.
Example 6 (4-dimensional alternative algebras). According to A. T. Gainov (see e.g. [16, p. 144]), there are exactly two alternative but not associative algebras of dimension 4 over any field. With respect to a basis {e 0 , e 1 , e 2 , e 3 }, one algebra is given by the following multiplication (the unspecified products are zeros): The other algebra is given by  The octonions algebra is a typical example of alternative algebras. As stated early, the subalgebra generated by any two elements is associative. In fact, the subalgebra generated by any two elements of the octonions is isomorphic to the algebra of reals R, the algebra of complex numbers C or the algebra of quaternions H, all of which are associative. See [4] for the role of the octonions in algebra, geometry and topology, and see also [1] where octions are viewed as quasialgebra.

Connections to other algebraic structures
We begin by recalling some basics of Moufang loops, Moufang algebras and Malcev algebras. Definition 8. Let (M, * ) be a set with a binary operation. It is called a Moufang loop if it is a quasigroup with an identity e such that the binary operation satisfies one of the following equivalent identities: The typical examples include groups and the set of nonzero octonions which gives a nonassociative Moufang loop.
As in the case of Lie group, there exists a notion of analytic Moufang loop (see e.g. [27,30,31]). An analytic Moufang loop M is a real analytic manifold with the multiplication and the inverse, g → g −1 , being analytic mappings. The tangent space TeM is equipped with a skew-symmetric bracket [ , ] : TeM × TeM → TeM satisfying Malcev's identity, that is, for all x, y, z ∈ TeM , and where J corresponds to Jacobi's identity, that is,

Definition 9.
A Malcev K-algebra is a vector space over K and a skew-symmetric bracket satisfying the identity (3.4).
The Malcev algebras are also called Moufang-Lie algebras. We have the following fundamental Kerdman theorem [23].
Theorem 10. For every real Malcev algebra there exists an analytic Moufang loop whose tangent algebra is the given Malcev algebra.
The connection to alternative algebras is given by the following proposition.
Proposition 11. The alternative algebras are Malcev-admissible algebras, that is, their commutators define a Malcev algebra.

Remark 12.
Let A be an alternative algebra with a unit. The set U (A) of all invertible elements of A forms a Moufang loop with respect to the multiplication. Conversely, not any Moufang loop can be imbedded into a loop of type U (A) for a suitable unital alternative algebra A. A counter-example was given in [34]. In [32], the author characterizes the Moufang loops which are imbeddable into a loop of type U (A).
The Moufang algebras which are the corresponding algebras of a Moufang loop are defined as follows.

Definition 13.
A left Moufang algebra (A, μ) is one which is left alternative and satisfying the Moufang identity, that is, The Moufang identities (3.1), (3.2), (3.3) are expressed in terms of associator by It turns out that in a characteristic different from 2, all left alternative algebras are left Moufang algebras. Also, a left Moufang algebra is alternative if and only if it is flexible, that is, as(x, y, x) = 0 for all x, y ∈ A.

Cohomology of left alternative algebras
In this section, we introduce a cohomology theory for left alternative algebras fitting with deformation theory and compute the second cohomology group of 2 × 2-matrix algebra viewed as an alternative algebra.
Let A be a left alternative K-algebra defined by a multiplication μ. A left alternative p-cochain is a linear map from A ⊗p to A. We denote by C p (A, A) the group of all p-cochains.

First differential map
Let id denotes the identity map on A. For f ∈ C 1 (A, A), we define the first differential δ 1 f ∈ C 2 (A, A) by We remark that the first differential of a left alternative algebra is similar to the first differential map of Hochschild cohomology of an associative algebra (1-cocycles are derivations).

Second differential map
Let ϕ ∈ C 2 (A, A), we define the second differential δ 2 φ ∈ C 3 (A, A) by

Remark 14.
The left alternative algebra 2-differential defined in (4.1) may be written using the Hochschild differential δ 2 H as Proposition 15. The composite δ 2 • δ 1 is zero.
Proof. Let x, y, z ∈ A and f ∈ C 1 (A, A). Then In order to simplify the notation, the multiplication is denoted by concatenation of terms and the tensor product is removed. Then, we have After simplifying the terms which cancel in pairs, we group the remaining ones into brackets, so each bracket cancels using the left alternative algebra axiom (see (2.3)). The non-specified terms of these generators are zeros. These generators were obtained independently using the softwares Maple and Mathematica.

Remark 17.
It was implied from [35] that the second cohomology group of A = M 2 (K) is non-trivial. But the exact structure of this group was not known. We completely determine the structure by giving the dimension and generators.

Third differential map and beyond
Let ψ ∈ C 3 (A, A), we define the third differential δ 3 ψ ∈ C 4 (A, A) as that is, for all ψ ∈ C 3 (A, A) and x 1 , . . . , x 4 ∈ A we have Remark 18. The third differential δ 3 ψ ∈ C 4 (A, A) of a left alternative algebra A may be written using the third Hochschild cohomology differential δ 3 H as

Journal of Generalized Lie Theory and Applications
Proposition 19. The composite δ 3 • δ 2 is zero.
Proof. Let x 1 , . . . , x 4 ∈ A and f ∈ C 2 (A, A). Then, by substituting ψ with δ 2 f in the previous formula and rearranging the terms we get Let ψ ∈ C 4 (A, A) and δ 4 H be the fourth Hochschild cohomology differential. We define the fourth differential δ 4 ψ ∈ C 5 (A, A) for a left alternative algebra A as where σ is the extended map, which we still denote by σ, for a permutation σ ∈ S 5 on A ⊗5 defined by By direct calculation, we can prove the following proposition.

Proposition 20. The composite δ 4 • δ 3 is zero.
One may complete the complex by considering δ p = 0 for p > 4. It is shown in [7] that the operad of alternative algebras is not Koszul, thus we think that there exist nontrivial pth differential maps for p > 4 by constructing a minimal model.

Formal deformations of left alternative algebras
In this section, we develop a deformation theory for alternative algebras and show that the cohomology introduced in the previous section fits with formal deformations of left alternative algebras. Let

Definition 21. Let (A, μ 0 ) be a left alternative algebra. A formal left alternative deformation of A is given by the K[[t]]-bilinear map μ t : A[[t]] ⊗ A[[t]] → A[[t]] of the form
]-bilinear), such that for x, y, z ∈ A, the following formal left alternativity condition holds: (5.1)

Deformation equation and obstructions
The first problem is to give conditions about μ i such that the deformation μ t is alternative. Expanding the left-hand side of (5.1) and collecting the coefficients of t k yield i+j=k; i,j≥0 Theory and Applications   7 This infinite system, called the deformation equation, gives the necessary and sufficient conditions for the left alternativity of μ t . It may be written as

Journal of Generalized Lie
The first equation (k = 0) is the left alternativity condition for μ 0 . The second shows that μ 1 must be a 2-cocycle for the alternative algebra cohomology defined above (μ 1 ∈ Z 2 (A, A)). More generally, suppose that μp is the first non-zero coefficient after μ 0 in the deformation μ t . This μp is called the infinitesimal of μ t . Proof. In (5.2) make the following substitutions: k = p and μ 1 = · · · = μ p−1 = 0. Definition 23. The 2-cocycle μp is said integrable if it is the first non-zero term, after μ 0 , of a left alternative deformation.
The integrability of μp implies an infinite sequence of relations which may be interpreted as the vanishing of the obstruction to the integration of μp.
For an arbitrary k, with k > 1, the kth equation of the system (5.2) may be written as Suppose that the truncated deformation μ t = μ 0 + tμ 1 + t 2 μ 2 + · · · + t m−1 μ m−1 satisfies the deformation equation. The truncated deformation is extended to a deformation of order m, that is, satisfying the deformation equation if The right-hand side of this equation is called the obstruction to finding μm extending the deformation. We define a square operation on 2-cochains by Then the obstruction may be written as A straightforward computation gives the following.

Remark 25.
(1) The cohomology class of the element i+j=m, i,j =m μ i μ j is the first obstruction to the integrability of μm. We consider now how to extend an infinitesimal deformation to a deformation of order 2. Suppose m = 2 and μ t = μ 0 + tμ 1 + t 2 μ 2 . The deformation equation of the truncated deformation of order 2 is equivalent to the finite system: Then μ 1 μ 1 is the first obstruction to integrate μ 1 and μ 1 μ 1 ∈ Z 3 (A, A). The elements μ 1 μ 1 which are coboundaries permit to extend the deformation of order one to a deformation of order 2. But the elements of H 3 (A, A) give the obstruction to the integrations of μ 1 .

Equivalent and trivial deformations
In this section, we characterize equivalent as well as trivial deformations of left alternative algebras.

Definition 27.
Let (A, μ 0 ) be a left alternative algebra and let (A t , μ t ) and (A t , μ t ) be two left alternative deformations of A, where μ t = i≥0 t i μ i and μ t = i≥0 t i μ i , with μ 0 = μ 0 . We say that the two deformations are equivalent if there exists a formal isomorphism Φ t : A

[[t]] → A[[t]], that is a K[[t]]-linear map that may be written in the form
where Φ i ∈ End K (A) and Φ 0 = id are such that the following relations hold:

viewed as a left alternative algebra on A[[t]]).
In the following, we discuss the equivalence of two deformations. Condition (5.3) may be written as Identifying the coefficients, we obtain that the constant coefficients are identical, that is, For the coefficients of t one finds Since Φ 0 = id, it follows that Consequently, The second-order conditions of the equivalence between two deformations of a left alternative algebra are given by (5.5), which may be written as In general, if the deformations μ t and μ t of μ 0 are equivalent, then μ 1 = μ 1 +δ 1 f 1 . Therefore, we have the following proposition.
Proposition 28. The integrability of μ 1 depends only on its cohomology class.

Proposition 30. Let (A, μ 0 ) be a left alternative algebra. There is, over K[[t]]/t 2 , a one-to-one correspondence between the elements of H 2 (A, A) and the infinitesimal deformation of A defined by
Proof. The deformation equation is equivalent to δ 2 μ 1 = 0, that is μ 1 ∈ Z 2 (A, A). Theorem 31. Let (A, μ 0 ) be a left alternative algebra and let μ t be a one parameter family of deformation of μ 0 . Then μ t is equivalent to μ t = μ 0 + t p μ p + t p+1 μ p+1 + · · · , where μ p ∈ Z 2 (A, A) and μ p / ∈ B 2 (A, A).

Remark 33.
A left alternative algebra for which every formal deformation is equivalent to a trivial deformation is called rigid. The previous corollary provides a sufficient condition for a left alternative algebra to be rigid. In general, this condition is not necessary.