11,505 research outputs found
Additive combinatorics methods in associative algebras
We adapt methods coming from additive combinatorics in groups to the study of
linear span in associative unital algebras. In particular, we establish for
these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on
sumsets and Tao's theorem on sets of small doubling. In passing we classify the
finite-dimensional algebras over infinite fields with finitely many
subalgebras. These algebras play a crucial role in our linear version of
Diderrich-Kneser's theorem. We also explain how the original theorems for
groups we linearize can be easily deduced from our results applied to group
algebras. Finally, we give lower bounds for the Minkowski product of two
subsets in finite monoids by using their associated monoid algebras.Comment: In this second version, we clarify and extend the domain of validity
of Diderrich-Kneser's theorem for associative algebras. We simplify the
proofs and we also add a section on Kneser's and Hamidoune's theorem in
monoi
The yoga of commutators
In the present paper we discuss some recent versions of localisation methods
for calculations in the groups of points of algebraic-like and classical-like
groups. Namely, we describe relative localisation, universal localisation, and
enhanced versions of localisation-completion. Apart from the general strategic
description of these methods, we state some typical technical results of the
conjugation calculus and the commutator calculus. Also, we state several recent
results obtained therewith, such as relative standard commutator formulae,
bounded width of commutators, with respect to the elementary generators, and
nilpotent filtrations of congruence subgroups. Overall, this shows that
localisation methods can be much more efficient, than expected
Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC
In this paper, we derive explicit product formulas and positive convolution
structures for three continuous classes of Heckman-Opdam hypergeometric
functions of type . For specific discrete series of multiplicities these
hypergeometric functions occur as the spherical functions of non-compact
Grassmann manifolds over one of the (skew) fields We write the product formula of these spherical
functions in an explicit form which allows analytic continuation with respect
to the parameters. In each of the three cases, we obtain a series of hypergroup
algebras which include the commutative convolution algebras of -biinvariant
functions on
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