INFINITE-DIMENSIONAL REPRESENTATIONS OF DISCRETE GROUPS, AND HIGHER SIGNATURES

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

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DETERMINANTAL IDENTITIES ON INTEGRABLE MAPPINGS

International Journal of Modern Physics B ◽

10.1142/s0217979294000889 ◽

1994 ◽

Vol 08 (16) ◽

pp. 2157-2201 ◽

Cited By ~ 16

Author(s):

S. BOUKRAA ◽

J-M. MAILLARD ◽

G. ROLLET

Keyword(s):

Elliptic Curves ◽

Common Factor ◽

Discrete Groups ◽

Infinite Dimensional ◽

Lattice Statistical Mechanics ◽

The Matrix ◽

Q Matrix ◽

Canonical Polynomials ◽

Integrable Mappings ◽

Determinantal Identities

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q×q matrices: the inversion of the q×q matrix and an (involutive) permutation of the entries of the matrix. In a case where the permutation is a particular elementary transposition of two entries, it is shown that the iteration of this group of birational transformations yield algebraic elliptic curves in the parameter space associated with the (homogeneous) entries of the matrix. It is also shown that the successive iterated matrices do have remarkable factorization properties which yield introducing a series of canonical polynomials corresponding to the greatest common factor in the entries. These polynomials do satisfy a simple nonlinear recurrence which also yields algebraic elliptic curves, associated with biquadratic relations. In fact, these polynomials not only satisfy one recurrence but a whole hierarchy of recurrences. Remarkably these recurrences are universal: they are independent of q, the size of the matrices. This study provides examples of infinite dimensional integrable mappings.

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ON VARIETIES OF ABELIAN TOPOLOGICAL GROUPS WITH COPRODUCTS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000654 ◽

2016 ◽

Vol 95 (1) ◽

pp. 54-65

Author(s):

SAAK S. GABRIYELYAN ◽

SIDNEY A. MORRIS

Keyword(s):

Hilbert Space ◽

Topological Group ◽

Hilbert Spaces ◽

Discrete Groups ◽

Topological Groups ◽

Infinite Dimensional ◽

Infinite Dimensional Hilbert Space ◽

Dimensional Hilbert Space ◽

Open Question ◽

Product Of Groups

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

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