Isomorphism of group algebras

Удк 512.742 Some Isomorphism Results on Commutative Group Algebras

Throughout the present paper, suppose FG is the group algebra of an abelian group G written multiplicatively (and possibly mixed) over a field F of positive characteristic p. As usual, FG denotes the group algebra of G over F with unit group U(FG) and normalized unit group V (FG); note that the direct decomposition U(FG) = V (FG) × F ∗ holds where F ∗ is the multiplicative group of F and thus t...

On Isomorphism of Simplicial Complexes and Their Related Algebras

The Unit Group of Small Group Algebras and the Minimum Counterexample to the Isomorphism Problem

Let KG denote the group algebra of the group G over the field K and let U(KG) denote its group of units. Here without the use of a computer we give presentations for the unit groups of all group algebras KG, where the size of KG is less than 1024. As a consequence we find the minimum counterexample to the Isomorphism Problem for group algebras. AMS Subject Classification: 16S34, 16U60, 20C05

Algorithms based on *-algebras, and their applications to isomorphism of polynomials with one secret, group isomorphism, and polynomial identity testing

We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices (B1, . . . , Bm) and (C1, . . . , Cm), whether there exists an invertible matrix A such that for every i ∈ {1, . . . ,m}, ABiA = Ci. We show that this problem can be solved in randomized polynomial time over finite fields of o...

The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ∼= U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H.