M. Dokuchaev

Associativity of crossed products by partial actions, enveloping actions and partial representations

Given a partial action a of a group G on an associative algebra A, we consider the crossed product A × α G. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of A × α G obtained in the context of C*-algebras. In particular, we prove that A × α G is associative, provided that A is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.

Globalization of twisted partial actions

Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish a criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.

On finite degree partial representations of groups

Abstract We establish a one-to-one correspondence between the irreducible finite degree partial representations of a group G and the (usual) irreducible representations of certain ideals of a groupoid algebra constructed from G. We derive a structural result about the irreducible partial representations on finite dimensional vector spaces and give the description “up to usual representations” of the irreducible partial representations of abelian groups of degrees ⩽4. We treat simultaneously irreducible and indecomposable partial representations.

ISOMORPHISMS OF PARTIAL GROUP RINGS

We consider the isomorphism problem for partial group rings

Imaginary Verma Modules for the Extended Affine Lie Algebra sl 2(ℂ q )

R_{\hbox{\scriptsize\it par}}G

Twisted partial actions and extensions of semilattices of groups by groups

and show that, in the modular case, if

FINITE CONJUGACY IN ALGEBRAS AND ORDERS

\textit{char}(R)\,{=}\,p

Recent developments around partial actions

and

Isomorphisms of Partial Group Rings

R_{\hbox{\scriptsize\it par}}G_1\,{\cong}\, R_{\hbox{\scriptsize\it par}}G_2

Crossed products by twisted partial actions and graded algebras

then the corresponding group rings of the Sylow