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Journal of Generalized Lie Theory and Applications
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On Representations of Bol Algebras

Ndoune N1* and Bouetou Bouetou T2

1Department of Mathematics, University of Sherbrooke, Sherbrooke, (Québec), J1K 2R1, Canada

2Department of Mathematics and Computer Science, Polytechnic National High School of Yaounde, and Dâ €™ Excellence Centre Africain technologies €™ Information and Communication (CETIC) the Abdus Salam International Centre for Theoretical Physics (ICTP), POBox: 8390 Yaounde, Cameroon, France

Corresponding Author:
Ndoune N
Department of Mathematics, University of Sherbrooke, Sherbrooke
(Québec), J1K 2R1, Canada
Tel: +18198217000
E-mail: [email protected]

Received date: December 02, 2015; Accepted date: December 13, 2015; Published date: December 15, 2015

Citation: Ndoune N, Bouetou TB (2015) On Representations of Bol Algebras. J Generalized Lie Theory Appl S2:005. doi: 10.4172/2469-9837.1000S2-005

Copyright: © 2015 Ndoune N, 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

In this paper, we introduce the notion of representation of Bol algebra. We prove an analogue of the classical Engel’s theorem and the extension of Ado-Iwasawa theorem for Bol Algebras. We study the category of representations of Bol algebras and show that it is a tensor category. In the case of regular representations of Bol algebras, we give a complete classification of them for all two-dimensional Bol algebras.

Keywords

Bol algebra; Lie triple System; Non-associative algebras; Jordan superalgebras; Nilpotent representation

Introduction

It is well known that the algebraic systems which characterize locally a totally geodesic subspace is a Lie triple system [1-3]. A Bol algebra is realized by equipping Lie triple System with an additional binary skew operation which satisfies a pseudo-differentiation property [4,5]. A morphism of Bol algebras is a linear map which preserves the ternary and the binary operations. More generally, the algebraic structures which characterize locally Bol loops are Bol algebras [6]. Until now, the representations of these algebras have not been studied. Since the representations of Lie algebras and Lie groups have natural connection with particulars physics, we claim that the representations of Bol algebras should lead with the physical applications. More precisely, in physics the representations of Bol algebras will be useful for the description of invariant properties of physical systems. and the concomitant conservation laws as a result. In literature of Mostovoy and Pérez-Izquierdo [7], it is shown that, Malcev algebras and Lie triple systems are particular subclasses of Bol algebras. The representations of Malcev algebras can be found studies of Kuz’min [8], and those of Lie triple systems were constructed by Hodge and Parshall [9], Bertrand, et al. [10]. Now, there already exists some representations of other classes of non-associative algebras; the case of alternative algebras was constructed by Schafer [11], the one of Leibniz algebras by Kolesnikov [12] and for Jordan superalgebras, the representations was given by Consuelo and Zelmanov [13].

Let image be a Bol algebra over a field K of characteristic zero, a representation of Bol algebra image on a K-vector space V is a triplet of maps image which respect some conditions which will be given later in the paper.

Our first main result is the following.

Theorem 1.1. Let image be a finite dimensional Bol algebra over a field K and image consist of nilpotent representations of Bol algebra image in a finite dimensional space V. Then there exists a vector image such thatimage for allimage

We agree that the image of any vector v of V by the operator image is given by image whereimage.

We define also the regular representations and the adjoint representations of Bol algebras. As an easy consequence, we show that if any representation of Bol algebra is nilpotent, then its adjoint representation is also nilpotent.

We are also interested by the question of the extension theorem of Ado-Iwasawa for Bol algebras. Pérez-Izquierdo established the existence of a Poincaré-Birkhoff-Witt type basis for a universal envelope of Bol algebra [5]. The above result allows us to interest ourselves to an extension of Ado-Iwasawa theorem for Bol algebra. let A be an alternative algebra, the the generalized right alternative nucleus is the algebra RNalt (A) defined by image We then give our second theorem.

Theorem 1.2. Let image be a finite-dimensional right Bol algebra over a field of characteristic different to 2 and 3. Then there exists a unital finite-dimensional algebra A and a monomorphism of Bol algebras imageimage

The analogue of our second result above was established for Malcev algebras framed by Pérez-Izqquierdo and Shestakov [14]. The collection of all representations of Bol algebra and the morphisms between them form a category, named the category of representations of Bol algebras image One can view a representation of Bol algebra as a B-module analogously as in literature of Consuelo and Zelmanov [13] in the case of Jordan superalgebras. One can understand also the representations of Bol algebras in term of matrices with sweet properties. The investigation between the category imageand the category of left image -modules, where image is the universal enveloping algebra of image, endowed with its bialgebra structure, leads us to our third main theorem.

Theorem 1.3. The category of representations of Bol algebra image is equivalent to the category of representations of its universal enveloping algebra image

The paper is organized as follows: We introduce in section 2 the notion of representations of Bol algebra. In section 3 we establish the Engel’s theorem for Bol algebras. In section 4 an extension of Ado- Iwasawa theorem to Bol algebras is proved. Finally in section 5, we present the category of representations of Bol algebras and show that it is equivalent to the category of left modules under its universal enveloping algebra. As immediate consequence, we show the category image is a tensor category. We end the section by given a complete classification of regular representations of two-dimensional Bol algebras.

Bol Algebras and their Representations

Bol algebras were introduced in differential geometry to study smooth Bol loops [6,15,16]. A right loop is a set image together with a binary operation image such that for any b in image the right multiplication operator image is bijective, and there exists an element image such that ε .b = b for any b in image The dual definition gives rise to a left Bol loop. In case that image is both left and right loop then it is called a loop with identity element ε.

A right smooth loop image is a right loop equipped with a structure of smooth manifold, that is the map image are smooth, [15,16]. Since groups are particular loops, so the Lie groups are particular cases of smooth loops. In scientific literature, many classes of loops are known: homogeneous loops, Moufang loops, Bol loops, Kikkawa loops among others.

A right Bol loop image is a right loop that satisfies the right Bol identity

image

for all a, x, y in image Similarly, a left Bol loop satisfies the identity image

As in the case of Lie groups where the tangent space at each point is equipped with Lie algebra structure, the tangent space at each point of Bol loop is equipped with the structure of Bol algebra.

Definition 2.1. A vector space image over a field K is called Bol algebra if it is equipped with a trilinear operation [−;−,−] and a skew-symmetric operation x . y satisfying the following identities:

(i) image

(ii) image

(iii) image

(iv) image for all x, y, z, α and β in image.

In other words, a Bol algebra is a Lie triple system image with an additional bilinear skew-symmetric operation x . y such that, the derivation image on a ternary operation is a pseudo differentiation with components α, β on a binary operation, that is; for all x, y and z in image, we have

image

Dα,β is a differentiation on ternary operation [−;−,−] that is;

image

In fact, the Bol algebra defined above is called right Bol algebra. In particular, any Lie triple system may be regarded as Bol algebra with the trivial multiplication x . y=0, for allimage

Bol algebras can be realized as the tangent algebras of Bol loops with the right Bol identity, and they allow embedding in Lie algebras [6,15].

Definition 2.2. A linear map image between two Bol algebras is called morphism of Bol algebras if it is preserve the ternary and the binary operations.

The subspace S of Bol algebra image is a sub-Bol algebra if the inclusionimage is a morphism of Bol algebras.

Definition 2.3. Let image be a Bol algebra over a field K, a pseudo-differentiation is a linear map image for which, there exists z ∈ B(a companion of D) such that image the companion is not necessarily unique.

The set of all companions of D is denoted Com(D). The map image is a pseudo-differentiation with companion α . β, called inner pseudo-differentiation of image. The pseudo-differentiations of B form a Lie algebra, denoted by pder image under the natural product image The algebra ipder image generate byimage is a Lie subalgebra of pder image, called the Lie algebra of inner pseudo differentiations of B. The enlarged algebra of pseudo-differentiations of image is defined asimage and the enlarged algebra of inner pseudo-differentiation is defined as image

It is showed in [4,5] that, those algebras defined below are the Lie algebras with the bracketsimage

The direct sum image is a Lie algebra with the operation image for all x, y, a, b in B.The Lie algebra (L,[,]) is called the standard enveloping Lie algebra of Bol algebra image.

The map image is a linear map of B. We denote by image the Lie algebra generate byimage with bracketsimage We get an other Lie algebra image which is a subalgebra of the Lie algebra generated by linear maps of image

If the subspace image satisfies the stronger conditionimage then image is an ideal of B. An ideal image of B automatically satisfies image

For more understanding of Bol algebras and Bol loops, it is important to investigate about their representations. We defined a representation of Bol algebra as follows.

Definition 2.4. If image is a Bol algebra over a field K and V a vector field over K, the pair image with the skew-symmetric bilinear map image and the linear mapimage is said to be a representation of Bol algebra image in V if there exists a bilinear operationimage such that the following statements are satisfied:

(R1) image

(R2) image

(R3) image for all x, y, a, b in image.

The operation Δ is called a companion of the representation image of the Bol algebra image.

In this case we can denoted by image or simply image the representation image with companion Δ. Following the approach of Consuelo and Zelmanov for the representations of Jordan Superalgebras [2], it is equivalent to say that the vector space V is a Bol module image possesses the structure of Bol algebra such that:

(a) image is a sub-Bol algebra of EV,

(b) V is an ideal of Bol algebra EV and

(c) x . y = 0 if both x, y ∈V and [x, y, z] = 0 if any two of x, y, z lie in V.

A particular instance where V = B and we set image the pair image is a representation of Bol algebra with companion Δ(u,v) = [u,−,v] called regular representation of B.

Example 2.1. Let image be the Bol algebra with basis (e1 ,e2 ) over a field of complex numbers, wereimage andimage We recall that det(u,v) is the determinant of the pair of vectors (u,v) with image Note that this Bol algebra arise from the classification of two-dimensional Bol algebras obtained by Kuz’min and Zaidi [4]. We set

image

It is clear that (D,δ ,Δ) is a regular representation of image

Now let (ρ ,δ ,Δ) and image be two representations of Bol algebraimage a morphism of the representation (ρ ,δ ,Δ) to a representation image is a linear mapimage such thatimage and image Clearly the composition of morphisms of representations is a morphism of representations. The collection of all representations and their morphisms forms a K-linear category denoted by image and called the category of representations of Bol algebra image

We consider image the center of Bol algebra is image It is simple to see that, the kernel of the operation image given by image is the center of image

Engel’s Theorem for Bol Algebras

Before giving the Engel’s theorem, we first need to define and characterize the nilpotent representations.

A representation (ρ ,δ ,Δ) of Bol algebra image in V is nilpotent if for all x, y, z∈image , ρ (x, y),δ (x) and Δ(x, y) are nilpotent endomorphisms; that is if there is a positive integer n such that (ρ ,δ ,Δ)n = 0. Let (ρ ,δ ,Δ) be a representation of image in V. we define the triplet (adρ ,adδ ,adΔ ) as follows:image

Proposition 3.1. With the above notations, the pair (adρ ,adδ ) is a representation of Bol algebra image in a vector space V with companion adΔ.

Proof. The objective is to show that (R1), (R2) and (R3) are satisfied. Let image We have

image

Then (R2) holds. In other hand we have

image
image

Therefore we have the desire equality image This shows that (R1) is satisfied. Finally, we have for all f ∈EndV,

Thus image and the desire conclusion follows, that is (R3) is verified.

Definition 3.1. The representation image is called the adjoint representation of image

Now we give the link between nilpotent representation and adjoint representation. The above result arises to the representations of Lie algebras.

lemma 3.1. Let image be a representation of Bol algebra on the vector space V. If imageis nilpotent, then its adjoint representation is also nilpotent.

Proof. Let imagebe a nilpotent representation of Bol algebra, and image its adjoint representation. Then there exists a positive integer p such that (ρ )p = 0 , (δ )p = 0 and (Δ)p = 0 . If σ is one of the map ρ, δ, or Δ it is clear that image where lσ and hσ are nilpotent. we haveimage Hence the result.

Now we are in position to prove our first main theorem.

Theorem 3.1. Let image be a finite dimensional Bol algebra over a field K and image consists of nilpotent representations of Bol algebra image in a finite dimensional space V. Then there exists a vector v ∈V3, v ≠ 0 such that (ρ ,δ ,Δ)(v) = 0 for all image

Proof. We agree that image whereimage that is we identify image for all a, b in image. It is clear that image is a subspace of (Env)3 and we can define on it the following bracket imageimage is a Lie algebra.

The proof of the theorem goes by induction on image When imageimage is generated by a single nilpotent representation then the claim is immediate.

Suppose now that the claim is true for all subalgebras of nilpotent representations spaces of dimension less than image

Since, image we have a proper Lie subalgebra image We can choose L to be a maximal subalgebra. We show before continuing that, L has a codimension one in image and L is an ideal.

L acts via the adjoint operator on image and L. In the latter case, since .we know by Engel’s theorem apply for L, that there exists a nonzero elementimage such thatimageimage for image We know that image thenimage It follows thatimage Moreoverimage These imply that image is a Lie subalgebra of image, and contains L as an ideal. By maximality of L, it follows that image so we are done.

Now we define the vector space image Letimage and image then image for allimage Other we have

image

and image Since L is an ideal, we have also image

Now we have image for some (ρ ,δ ,Δ)∈L. We know that (ρ ,δ ,Δ) is a nilpotent operator on W, so image Let v = (v1 ,v2 ,v3 )∈ker(ρ ,δ ,Δ)W such that v ≠ 0; then any element of L and r annihilates v.

An Extension of Ado-Iwasawa Theorem to Bol Algebras

Let L be a finite-dimensional Lie algebra over a field K. The classical Ado-Iwasawa theorem asserts the existence of a finite-dimensional L-module which gives a faithful representation of L. However, Filippov proved [17] showed that this theorem does not hold for Malcev algebras, that is homogeneous Bol algebras. Thus it is not hold for general Bol algebras.

For the Lie algeras, the Poincaré-Birkhoff-Witt theorem says that any Lie algebra L is a subalgebra of A for some unital associative algebra A. In the case that L is finite dimensional, the Ado-Iwasawa theorem says that A can be taken finite dimensional too. This extension of Ado-Iwasawa theorem was established for the Malcev algebras by Pérez-Izqquierdo and Shestakov [14]. There is a version of the Poincaré-Birkhoff-Witt theorem for Bol algebra proved by Kuz’min and Zaidi [4]. Now let image be a Bol algebra [14] that there is an alternative algebra A and an injective map image whereimage is the generalized right alternative nucleus. In this section we prove that if image is a finite-dimensional Bol algebra then A can be taken finite dimension too. Our second main result is the following.

Theorem 4.1. Let image be a finite-dimensional right Bol algebra over a field of characteristic ≠ 2,3. Then there exists a unital finite-dimensional algebra A and a monomorphism of Bol algebra image

Proof. Let image be a Bol algebra, according to Pérez-Izquierdo [5], there exists a linear map image such that j(a.b) = ab − ba and j(a,b,c) = (ab)c − (ac)b −[b,c]a , where image is the universal enveloping algebra of image. Since image is closed under the binary product [−,−] given by the commutators and the ternary operation [a,b,c] = (ab)c − (ac)b −[b,c]a for all a,b,c in image. By the methods of Pérez-Izquierdo [5], image with the binary and ternary operations defined above has the structure of Bol algebra. Thus j is a monomorphism of Bol algebras. Let image be the Lie enveloping algebra of image. Then image is theimage andimage as vector space. According to Pérez-Izquierdo and Shestakov [14], there exists a two side ideal image of finite codimension. Then image is a unital finite-dimensional algebra and there exists an injective map image The injective map j induces a monomorphism of Bol algebras image

The Category of Representations of Bol Algebra

We give a relation between the category of representation of Bol algebra image and the category of representations of its universal enveloping algebra. As immediate consequence, we show that the representation category of a Bol algebra is monoidal, or tensor category. We recall that the category of representations of Bol algebras is image, and the one of finite dimensional representations of Bol algebra is image. Let image be a bialgebra, Mod(A) means the category of left A-modules (ie., representations of A). If U, V are left A-modules, then the tensor product becomes a left A-module with multiplication rule image for all a∈ A , u∈U and v∈V . The field K is also a left A-module by image The category of left A-modules is equivalent to the category of (A, A)-bimodules. Any (A, A)-bimodule can be considered as left module over image , where imageis define on the same space as A, by new multiplication imageWe know in virtue of Pérez-Izqquierdo [5] that for a given Bol algebra image there exists a universal enveloping image endowed with the structure of bialgebra, that is image is a bialgebra. Analogously we denote Rep image the category of representation of the bialgebra image. Now we state an equivalent characterization of the representation category image We prove our third main result.

Theorem 5.1. The category of representations of Bol algebra image is equivalent to the category of representations of its universal enveloping algebra image

Proof. We recall that image is the category of modules over the Bol algebra imageFollowing the consideration of Consuelo and Zelmanov [13], apply for the modules over Bol algebras, every B-module has the form image, where V is a vector space over a field K and EV possesses the structure of Bol algebra such that:

(a) imageis a sub-Bol algebra of EV,

(b) V is an ideal of Bol algebra EV and

(c) x . y =0 if both x, yV and [x, y, z] = 0 if any two of x, y, z lie in V.

We define the multiplication image by image. We consider the following mapping defined from image to Mod image define on the objets by image The map F is naturally extended on the morphisms. If U and V are the images of EU and EV under F, in virtue of Pérez-Izqquierdo [5] there exits a map image with imageThis implies that imageis a image -module.

Conversely, let V be a image-module, in virtue of Pérez- Izquierdo [5] there exist an injective map image. We define the multiplication image by image. Then V has the structure of module. We set now the mapping G from Mod image to image by image It remains to define the image of image. Let EU and EV be two modules over image, We set image We define the binary operation by image; image and a ternary by image; imageand image for all a, b in image, u in Vand v in V. We assume also that the restrictions of image and image on image correspond respectively to the binary and ternary operations of B; and x . y =0 if both image and image if any two of x,y,z lie in image

It remains to show thatimage is a Bol algebra, that is the conditions (i) - (iv) hold. By the definition, the condition (i) is satisfied. Now let image in image; u in U and v in V. We have

image

this shows that (ii) is true.

Now let us show that (iii) holds. We have

image

One can show that the above equality holds for any image stands for image That is (iii) holds.

Finally, we have

image

Thus image One can show this equality for any y, α, β stands for This image completes the proof.

Definition 5.1. A monoidal (tensor) category image is a category image equipped with tensor functor image with a fix objet 1 (called the unit of a tensor category), image;image,image are natural isomorphisms such that the associativity and unitary constraints hold, or equivalently the pentagon and the triangle diagrams are commutative [18-20].

We can now give a special characterization of the category of representations of Bol algebra as a consequence of the above proposition.

Corollary 5.1. Every category of representations of Bol algebras is a monoidal category.

Proof. It was proved by Kassel [20] that image is bialgebra if and only if the category Mod(A) is monoidal category. In virtue of Theorem 5.0.6, the category of representations of Bol algebra is equivalent to the category of representations of its enveloping algebra endowed with bialgebra structure. Hence the category Rep(B) is monoidal.

More recently it was proved by Huang and Torecillas [21], that the path coalgebra KQ of a given quiver Q always admits a bialgebra structure. So the monoidal category arising from this quiver bialgebra is the category of representations of the bialgebra KQ. This leads to the following conjecture.

Conjecture 5.1. Find necessary and sufficient conditions for the existence of quiver Q such that the monoidal category arising from quiver bialgebra KQ is the category of representations of a Bol algebra over algebraically closed field K.

A monoidal category is said to be finite, if it is equivalent to the category of finite dimensional comodules over the finite dimensional coalgebra. Thus the category Rep(B) of finite dimensional representations is finite monoidal category. This is a particular case of tensor categories of Etingof et al. [19]. The particular case where Q is a quiver without loops and 2-cyles should leads to strong relation between Bol algebras and cluster algebras of Fomin and Zelevinsky [22,23] for more details. In the same vein, it has been shown in literature of Schauenburg [24] that if A is a finite dimensional bialgebra, then A is Hopf algebra if and only if the category of finitely generated A-modules is rigid, that is finitely generate modules admit dual objets. This allows us to the following conjecture.

Conjecture 5.2. Find necessary and sufficient conditions for a finite dimensional Bol algebra to have Hopf algebra as universal enveloping algebra.

Representations of Free Bol Algebra Bol[X] of Finite Dimension

Let imagewe construct the set of binary-ternary monomials BT[X], and we assume that BT[X] is closed under image and imageLet imagebe the space spanned by X. We define the multiplication by the following rules: if image and imagein BT[X], then image, image. The free Bol algebra Bol[X] is the free binary-ternary algebra BT[X] satisfying the identities (i) - (iv). The Bol types of degree m are always to construct a product of degree m in Bol[X]. For general construction and more details of the free Bol algebra Bol[X] [25,26]. In studies of Peresi [26] it has been shown that any multilinear identity f of degree m can be written as a linear combination of multilinear monomials. We denote the Bol types of degree m by B1, B2, …, Bb(m), that is image where fk is a linear combination of polynomial having Bol type k. Therefore the author regards f as an element of b(m) copies of image, where image is group algebra of the group of permutation Sm. Applying the representation image, (σ partition of m) of Sm to f we obtain the representation matrix of f in partition σ: image

Now let V be finite dimensional space, dim(V) = s and image is a Bol algebras of dimension n. Give a representation (ρ ,δ ,Δ) of image over the space V is equivalent to give the matrix image, where image,imageare s × n s × n matrices and imageis also a s × n matrix. Hence the block matrix image is a (3n)× s matrix.

In the special case where image, image and image, with Bol types B1, B2, …, Bb(m) the representation matrix image of f corresponds to the matrix δ f, that is the expression image. At this specific case mentioned by Peresi and Jacobson [26,27], the representation of element f is understood as a the representation of Bol algebra Bol[X] given by the matrix image.

Actually we recall the classification theorem of Kuz’min and Zaidi for two-dimensional Bol algebras [4] which states as follows.

Theorem 5.2. (Kuz'min-Zaidi). Every Bol algebra B of dimension two over image has a canonical basis (e1,e2) in which its multiplication table is one of the following:

I. image where image, (−1,0) , (1,0) , (1,−1) , (1,1) , (−1,−1)

II. imagewhere image

Now we are in position to prove our classification result for regular representations of the two-dimensional Bol algebras.

Theorem 5.3. Every regular representation of two-dimensional Bol algebra B over K is up to equivalence of matrices given by one of the following matrices:

(i) image

(ii) image

(iii) image

Proof. In virtue of classification theorem of Kuz’min and Zaidi [4], every Bol algebra of dimension two is of type (I) or of type (II) by using the items of their theorem.

We suppose in the first case that our Bol algebra is of type (I), that is B has a canonical basis (e1,e2) in which its multiplication table is given by image, where

(ε12 ) = (0,0) , (−1,0) , (1,0) , (1,−1) , (1,1) , (−1,−1).

Let u and v be the two vectors of B, with image and image We have image Since image, we have

image

We have also

image

Thus image

Now we compute the matrix of Δ(u,v) as follows. We have

image

and

image

hence image Because [e1 ,e2] = 0, we have δ (u) = 0. Therefore the bloc matrix image corresponds to the matrix image

The second case corresponds to Bol algebra of type (I), that is B has a canonical basis image in which its multiplication table is given by image, where image

If image where image we use the analogous methods as at the first case to get image image and imageHence the bloc matrix image corresponds to the matrix image

Finally, for image and imageimagewe have image this end the proof.

Acknowledgement

We aim at sending our sincere gratitude to the reviewers for their reports on our paper and their availability. The second author thanks the IHES for hospitality through the Launsbery Foundation and, the IMU for travel support throughout a grant from the Simons Foundation during the writing of this paper.

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