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Journal of Generalized Lie Theory and Applications
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Cohomology and Formal Deformations of Alternative Algebras

Mohamed Elhamdadi1 and Abdenacer Makhlouf2*

1Department of Mathematics and Statistics, University of South Florida, Tampa, Fl 33620, USA

2Department of Mathematics, Computer Science and Applications, Haute Alsace University, 68093 Mulhouse Cedex, France

*Corresponding Author:
Abdenacer Makhlouf
Department of Mathematics
Computer Science and Applications
Haute Alsace University, 68093 Mulhouse Cedex, France
E-mail:
[email protected]

Received date 27 August 2010; Accepted date 26 January 2011

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Abstract

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.

1 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.

2 Preliminaries

Throughout this paper, K is an algebraically closed field of characteristic 0.

2.1 Definitions

Definition 1 (see [29]). A left alternative K-algebra (resp. right alternative K-algebra) (A, μ) is a vector space A over K and a bilinear multiplication μ satisfying the left alternative identity, that is, for all x, y ∈ A,

image            (2.1)

 

and respectively the right alternative identity, that is

image           (2.2)

 

An alternative algebra is one which is both left and right alternative algebra.

Lemma 2. Let as denote the associator, which is a trilinear map defined by

image

An algebra is alternative if and only if the associator imageis an alternating function of its arguments, that is

image

This lemma implies then that the following identities are satisfied:

image

By linearization, we have the following characterization of left (resp. right) alternative algebras, which will be used in the sequel.

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

image       (2.3)

 

respectively,

image               (2.4)

 

holds.

Remark 4. When considering multiplication as a linear maps image the condition (2.3) (resp. (2.4)) may be written

image              (2.5)

 

respectively

image             (2.6)

 

where id stands for the identity map and σ1 and σ2 stand for transpositions generating the permutation group S3 which are extended to trilinear maps defined by

image

for all image

In terms of associators, the identities (2.3) (resp. (2.4)) are equivalent to

image        (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

image

The dual operad of alternative algebras is defined by the associativity and the identity

image

2.2 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. imagethe 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 {e0, e1, e2, e3}, one algebra is given by the following multiplication (the unspecified products are zeros):

image

The other algebra is given by

image

Example 7 (octonions). The octonions were discovered in 1843 by John T. Graves who called them Octaves and independently by Arthur Cayley in 1845. The octonions algebra which is also called Cayley Octaves or Cayley algebra is an 8-dimensional algebra defined with respect to a basis {u, e1, e2, e3, e4, e5, e6, e7}, where u is the identity for the multiplication, by the following multiplication table. The table describes multiplying the ith row elements by the jth column elements.

s
  U e1 e2 e3 e4 e5 e6 e7
U u e1 e2 e3 e4 e5 e6 e7
e1 e1 -u e4 e7 -e2 e6 -e5 -e3
e2 e2 -e4 -u e5 e1 -e3 e7 -e6
e3 e3 -e7 -e5 -u e6 e2 -e4 e1
e4 e4 e2 -e1 -e6 -u e7 e3 -e5
e5 e5 -e6 e3 -e2 -e7 -u e1 e4
e6 e6 e5 -e7 e4 -e3 -e1 -u e2
e7 e7 e3 e6 -e1 e5 -e4 -e2 -u

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.

3 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:

image

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, image being analytic mappings. The tangent space TeM is equipped with a skew-symmetric bracket image satisfying Malcev’s identity, that is,

image            (3.4)

 

for all image and where J corresponds to Jacobi’s identity, that is,

image

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,

image

The Moufang identities (3.1), (3.2), (3.3) are expressed in terms of associator by

image

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.

The alternative algebras are connected to Jordan algebras as follows. Given an alternative algebra (A, μ), then image a Jordan algebra, that is, the commutative multiplication μ+ satisfies the identity imageFor more nonassociative algebras theory, we refer to [21,22,33,36,37].

4 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 image We denote byimage the group of all p-cochains.

4.1 First differential map

Let id denotes the identity map on image we define the first differentialimage image

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).

4.2 Second differential map

Let image we define the second differentialimage by

image        (4.1)

 

image

Remark 14. The left alternative algebra 2-differential defined in (4.1) may be written using the Hochschild differential image as

image

Proposition 15. The composite δ2 ◦ δ1 is zero.

Proof. Let image Then

image

In order to simplify the notation, the multiplication is denoted by concatenation of terms and the tensor product is removed. Then, we have

image

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)).

Example 16. Let image denote the associative algebra of 2 by 2 matrices over the field K, considered as left alternative algebra of dimension 4. Let e1, e2, e3 and e4 be a basis of A. The second cohomology group image is three-dimensional and generated with respect to the canonical basis by [f1], [f2] and [f3], where

image

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 image is non-trivial. But the exact structure of this group was not known. We completely determine the structure by giving the dimension and generators.

4.3 Third differential map and beyond

Let image we define the third differentialimage image

that is, for all image image

Remark 18. The third differential image f a left alternative algebra A may be written using the third Hochschild cohomology differential image as

image

Proposition 19. The composite image is zero.

Proof. Let image Then, by substitutingimage with image in the previous formula and rearranging the terms we get

image

Let image be the fourth Hochschild cohomology differential. We define the fourth differentialimage for a left alternative algebra A as

image

where σ is the extended map, which we still denote by σ, for a permutation image defined by

image

By direct calculation, we can prove the following proposition.

Proposition 20. The composite image is zero.

One may complete the complex by considering image 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 image be a left alternative algebra. Let image be the power series ring in one variable t and coefficients in image and let A[[t]] be the set of formal power series whose coefficients are elements of A (note that A[[t]] is obtained by extending the coefficients domain of A from image Then, A[[t]] is a image-module. When A is finite-dimensional, we have image One notes that A is a submodule of A[[t]]. Given a image -bilinear mapimage it admits naturally an extension to a image -bilinear mapimage that is, ifimage image

Definition 21. Let image be a left alternative algebra. A formal left alternative deformation of A is given by the image -bilinear map image where each μi is a image -bilinear map image (extended to be image-bilinear), such that for x, y, z ∈ A, the following formal left alternativity condition holds:

image (5.1)

 

Deformation equation and obstructions

The first problem is to give conditions about μi such that the deformation μtis alternative. Expanding the left-hand side of (5.1) and collecting the coefficients of tkyield

image

This infinite system, called the deformation equation, gives the necessary and sufficient conditions for the left alternativity of μt. It may be written as

image (5.2)

 

The first equation (k = 0) is the left alternativity condition for μ0. The second shows that μ1 must be a 2-cocycle for the alternative algebra cohomology defined above image More generally, suppose that μp is the first non-zero coefficient after μ0 in the deformation μt. This μp is called the infinitesimal of μt.

Theorem 22. The map μp is a 2-cocycle of the left alternative algebras cohomology of A with coefficient in itself.

Proof. In (5.2) make the following substitutions: k = p and image

Definition 23. The 2-cocycle μp is said integrable if it is the first non-zero term, after μ0, 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

image

Suppose that the truncated deformation

image

satisfies the deformation equation. The truncated deformation is extended to a deformation of order m, that is,

image

satisfying the deformation equation if

image

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

image

Then the obstruction may be written as

image

A straightforward computation gives the following.

Theorem 24. The obstructions are left alternative 3-cocycles

Remark 25. (1) The cohomology class of the element image 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 image The deformation equation of the truncated deformation of order 2 is equivalent to the finite system:

image

Then image is the first obstruction to integrateimage The elements image which are coboundaries permit to extend the deformation of order one to a deformation of order 2. But the elements of image give the obstruction to the integrations of μ1.

(2) If μm is integrable, then the cohomological class of image must be 0. In the previous example, μ1 is integrable, implies that image which means that the cohomology class of image vanishes.

Corollary 26. If image then all obstructions vanish and everyimage is integrable.

Equivalent and trivial deformations

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

Definition 27. Let image be a left alternative algebra and letimage be two left alternative deformations of A, where image We say that the two deformations are equivalent if there exists a formal isomorphism imagelinear map that may be written in the form

image

where image = id are such that the following relations hold:

image (5.3)

 

A deformation At of A0 is said to be trivial if and only if At is equivalent to A0 (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

image (5.4)

 

Equation (5.4) is equivalent to

image

or

image

Identifying the coefficients, we obtain that the constant coefficients are identical, that is,

image

For the coefficients of t one finds

image

Since Φ0 = id, it follows that

image

Consequently,

image (5.5)

 

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

image (5.6)

 

In general, if the deformations image are equivalent, then image Therefore, we have the following proposition.

Proposition 28. The integrability of μ1 depends only on its cohomology class.

Recall that two elements are cohomologous if their difference is a coboundary. The equation image implies that

image

If image then

image

Thus, if two integrable 2-cocycles are cohomologous, then the corresponding deformations are equivalent.

Remark 29. Elements of image give the infinitesimal deformationsimage

Proposition 30. Let image be a left alternative algebra. There is, overimage a one-to-one correspondence between the elements of imageand the infinitesimal deformation of A defined by

image

Proof. The deformation equation is equivalent to image

Theorem 31. Let image be a left alternative algebra and let μt be a one parameter family of deformation of μ0. Then μt is equivalent to

image image

Proof. Suppose now that image is a one-parameter family of deformation of μ0 for which image The deformation equation impliesimageIf further imageimage then setting the morphismimagewe have, for all x, y ∈ A,

image

And again image

Corollary 32. If image=0, then all deformations of A are equivalent to a trivial deformation.

In fact, assume that there exists a non trivial deformation of μ0. Following the previous theorem, this deformation is equivalent to

image

where image But this is impossible because image=0.

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.

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