Medical, Pharma, Engineering, Science, Technology and Business

Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia

**E-mail:** [email protected]

**Received** July 05, 2007 **Revised** January 03, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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.

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

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

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 are analytic mappings.

Let the local coordinates of g from the vicinity of e ∈ G be denoted by dim *G*). The tangent space of *G* at e ∈ G is denoted by T_{e}(G).

As in the case of the Lie groups, the structure constants c^{i}_{jk} of an analytic Moufang loop are
defined by

For any their (tangent) product [x, y] ∈ T_{e}(G) is defined in component form by

The tangent space T_{e}(*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 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:

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

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 of a Moufang loop *G* into *GL _{n}*. The pair
(

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

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

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

The *auxiliary functions* of *G* are defined as

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

One can check the initial conditions

The algebra with structure functions 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.

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

By direct calculations, one can check that

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

**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*

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

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

**Proof.** Differentiate the GLE and use

**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)*

Let *g ∈ G* and *x, y ∈ Te(G).* Denote

Define the *Yamagutian* Y_{g} by

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

The *Yamaguti brackets* [·, ·, ·]*g* are defined [8, 9] in T_{e}(G) by

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:

(1)

(2)

(3)

One can prove the *reductivity conditions* [5]

(4)

(5)

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

(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 T_{e}(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].

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

- V. D. Belousov.
*Foundations of the Theory of Quasiqroups and Loops.*Nauka, Moscow, 1967 (in Russian). - R. Moufang. Zur Struktur von Alternativk¨orpern. Math. Ann. B110 (1935), 416-430.
- H. Pflugfelder.
*Quasigroups and Loops: Introduction.*Heldermann Verlag, Berlin, 1990. - A. I. Mal’tsev. Analytical loops. Matem. Sbornik. 36 (1955), 569-576 (in Russian).
- E Paal. Continuous Moufang transformations. Acta Appl. Math. 50 (1998), 77-91.
- E. Paal. Note on Moufang-Noether currents. Czech. J. Phys. 56 (2006), 1257-1262.
- O. Loos. ¨Uber eine Beziehung zwischen Malcev-Algebren und Lie-Tripelsystemen. Pacific J. Math. 18 (1966), 553-562.
- K. Yamaguti. Note on Malcev algebras. Kumamoto J. Sci. A5 (1962), 203-207.
- K. Yamaguti. On the theory of Malcev algebras. Kumamoto J. Sci. A6 (1963), 9-45.
- R. D. Schafer.
*An Introduction to Nonassociative Algebras.*Academic Press, New York, 1966.

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