The linear deformation of discrete groups
by
Petr Yagodovsky
Moscow State University

The report informs about the construction of deformation of discrete groups in the class of multivalued groups.

Defenition ([]).  The set S with maping \mu: S×S --> (S)ⁿ (here (S)ⁿ is the symmetric power of S) is called the n-valued group if S and \mu obey the following laws. (1) The law of assoctativety: for allx, y, z in S \mu(x, \mu(y, z)) = \mu(\mu(x, y), z). (2) The law of groups unit: existse in S  for allx in S \mu(e, x) = \mu(x, e) = (x, ... , x). (3) The law of inverse element existence. for allx in S  existsx' in S C^e_{x, x'} > 0 and C^e_{x', x} > 0. Here C^z_{x, y} > 0 is the multiplicity of z in \mu(x, y). The mapping \mu is called the production.

Defenition ([]).  Let A be an algebra with a fixed basis and S be some n-valued group. Let elements of this basis are one to one correspond to elements of S. We will denote as [x\tilde] the basis element of A corresponding to x in S. Let the structural constants c^[z\tilde]_{[x\tilde], [y\tilde]} be equal to [ 1/n]C^z_{x, y} > 0. Then the algebra is called the group algebra of the n-valued group S.

The main construction ([]). Let us consider V = R^N(N-1)2 with coordinates c^k_{i, l} . Let us define the manifold of associative algebras dimension N with a unit and a fixed basis M^N_ass as the surface in V corresponding to solutions of equations of associativety. Let A be a group algebra of some n-valued group S and x be the point on the surface M^N_ass which correspondents to the algebra A. Let us consider the set of lines, which is tangential to M^N_ass in the point x. There are a lot of examples, when some of such lines are lines on the surface M^N_ass.

Defenition . Let A and B be two group algebras of multivalued groups S and C. Let x and y be the points on the surface M^N_ass, which correspondent to A and B. Let the line which passes through the points x and y in the space V lie on M^N_ass. Then the group C is called the linear deformation of the group S.

Let G be a discrete group in the usual sense. Then a linear deformation of the group G may be written on the form:

i*j= å r in G  Lr(i, j)(r·i·j)    i, j in G,

where L = \sum_{r in G}L^r is a 2-cocycle of G with the coefficients in the ad-representation of G.

Is the case of abelian group G, there is the special class of cocycles L(i, j) = (1-L^r(i, j))e-L^r(i, j)r. Linear deformations corresponding to the cocycles are called regular deformations. It turns, for for alli in N there is the cocycle L_i of the group Z_i, such that for any regular deformation of a finite abelian group G defining by a pair (L, r) there are the integer number i and projection p_i: G --> Z_i, with L = p_i^*L_i, and either p_i(r) = 0 or p_i(r) = 1. If G is Z then pairs (L, r) (where L = p_i^*L_i and either p(r) = 0 or p(r) = 1) defines periodic regular deformations. Any regular deformations of group Z which is not periodic may be build by appicating the special construction to periodic regular deformations.

Linear deformations of the groups Z₃, Z₄ and Z₂\otimesZ₂ are classified. It is interesting, that Z₄ is the linear deformation of Z₂\otimesZ₂. The following diagrams describe the sets of deformations of Z₄ (see a)) and Z₂\otimesZ₂ (see b)).

Picture Omitted

References

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V. M Buchshtaber, E. G. Rees, Multivalued groups, their representations and Hopf algebras, Transformation Groups, vol 2(4), 1997, p. 325-349;

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Date received: June 16, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-41.