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Journal of Generalized Lie Theory and Applications
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Moufang loops and generalized Lie-Cartan theorem1

Eugen PAAL

Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia
E-mail: [email protected]

Received July 05, 2007 Revised January 03, 2008

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Abstract

Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation are found. In particular, the Lie algebra of the multiplication group of the birepresentation is explicitly given.

Introduction

Continuous symmetries generated by the Lie transformation groups are widely exploited in modern mathematics and physics. Nevertheless, it may happen that the group theoretical methods are too rigid and one has to extend these beyond the Lie groups and algebras. From this point of view it is interesting to extend the group theoretical symmetry methods by using the Moufang loops and Mal’tsev algebras. The latter are known as minimal non-associative generalizations of the group and Lie algebra concepts, respectively.

In this paper, the generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation were found. In particular, the Lie algebra of the multiplication group of the birepresentation is explicitly given.

Based on this theorem, various applications are possible. In particular, it was recently shown [6] how the Moufang-Noether current algebras may be constructed so that the corresponding Noether charge algebra turns out to be a birepresentation of the tangent Mal’tsev algebra of an analytic Moufang loop.

Moufang loops

A Moufang loop [2, 1, 3] is a quasigroup G with the unit element e ∈ G and the Moufang identity

image

Here the multiplication is denoted by juxtaposition. In general, the multiplication need not be associative: image ha for some triple of elementsimage The inverse element g−1 of g is defined by

image

Analytic Moufang loops and Mal’tsev algebras

Following the concept of the Lie group, the notion of an analytic Moufang loop can be easily formulated.

A Moufang loop G is said [4] to be analytic if G is also a real analytic manifold and main operations - multiplication and inversion map imageare analytic mappings.

Let the local coordinates of g from the vicinity of e ∈ G be denoted by imagedim G). The tangent space of G at e ∈ G is denoted by Te(G).

As in the case of the Lie groups, the structure constants cijk of an analytic Moufang loop are defined by

image

For any image their (tangent) product [x, y] ∈ Te(G) is defined in component form by

image

The tangent space Te(G) being equipped with such an anti-commutative multiplication is called the tangent algebra of the analytic Moufang loop G. We shall use the notation image for the tangent algebra of G.

The tangent algebra of G need not be a Lie algebra. There may exist a triple x, y, z ∈ Te(G) that does not satisfy the Jacobi identity:

image

Instead, for all image one has a more general Mal’tsev identity [4]

image

Anti-commutative algebras with this identity are called the Mal’tsev algebras.

Birepresentation of a Moufang loop

Consider a pair (S, T) of the maps image of a Moufang loop G into GLn. The pair (S, T) is called [5] a (linear) birepresentation of G (in GLn) if the following conditions hold true:

image

The birepresentation (S, T) is called associative, if the following simultaneous relations are satisfied:

image

In general, birepresentations need not be associative even for groups.

Generators and structure functions

The generators of a birepresentation (S, T) are defined as follows:

image

The auxiliary functions of G are defined as

image

The structure functions image of G are defined by the generalized Maurer-Cartan equations

image

One can check the initial conditions

image

The algebra with structure functions image may be called the derivative of the tangent algebra Τ of G. It follows from the Belousov theory [1] of derivative quasigroups and loops that the derivative of the tangent Mal’tsev algebra of G is a Mal’tsev algebra as well. In a sense, the derivative Mal’tsev algebra is a deformation of ¡ with the deformation parameter g ∈ G.

Generalized Lie equations (GLE)

Define the derivative generators of (S, T) as follows:

image

By direct calculations, one can check that

image

In what follows, the commutator of linear operators A,B is denoted as image

Theorem 1 (generalized Lie equations). Let (S, T) be a differentiable birepresentation of an analytic Moufang loop G. Then the birepresentation matrices Sg, Tg (g ∈ G) satisfy the system of simultaneus differential equations

image

Proof. Differentiate the defining relations of a birepresentation (S, T) with respect to g and take g = e.

Generalized Lie-Cartan theorem

Theorem 2 (generalized Lie-Cartan theorem). The integrability conditions of GLE read as commutation relations

image

Proof. Differentiate the GLE and use

image

Corollary 3. If (S, T) is an associative birepresentation of a Lie group G, then one gets the well known Lie algebra commutation relations (Lie-Cartan theorem)

image

Yamagutian and Yamaguti brackets

Let g ∈ G and x, y ∈ Te(G). Denote

image

Define the Yamagutian Yg by

image

By direct calulations one can check that the Yamagutian obeys the constraints

image

The Yamaguti brackets [·, ·, ·]g are defined [8, 9] in Te(G) by

image

Closure of integrability conditions

It turns out that the commutation relations for derivative generators of the birepresentation (S, T) can be presented in a closed Lie algebra form.

The integrability conditions of GLE can be written as follows:

image (1)

image (2)

image(3)

One can prove the reductivity conditions [5]

image (4)

image (5)

By using the above reductivity conditions it can be shown by direct calculations that the Yamagutian obeys the Lie algebra

image (6)

Dimension of the Lie algebra (1) − (6) does not exceed 2r + r(r − 1)/2. The Jacobi identities are guaranteed by the defining identities of the Lie [7] and general Lie triple systems [8, 9] associated with the derivative Mal’tsev algebra of Te(G) of G.

The commutation relations of form (1) − (6) are well known from the theory of alternative algebras [10]. By taking g = e, the commutation relations (1) − (6) give the Lie algebra of the multiplication group of the birepresentation (S, T) of G. For field theoretical applications see [6].

Acknowledgment

The research was in part supported by the Estonian Science Foundation, Grant 6912.

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