Trace methods in twisted group algebras

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

Let G be a finite simple group and k be an algebraically closed field of prime characteristic dividing the order of G. We show that for all 2-cocycles α ∈ Z2(G;k×), the first Hochschild cohomology group of the twisted group algebra HH1(kαG) is nonzero.

Semiprimitivity of group algebras of locally finite groups. II

We study deformation of algebras with coaction symmetry of reduced algebras of discrete groups, where the deformation parameter is givenby a continuous family of group 2 -cocycles. When the group satisfies the Baum–Connes conjecture with coefficients, we obtain an isomorphism of K-groups of the deformed algebras. This extends both the \theta -deformation of Rieffel on \mathbb{T}^n -actions, and a recent result of Echterhoff, Lück, Phillips, and Walters on the K-groups on the twisted group algebras.

Twisted Dual-Group Algebras: Equivariant Deformations of C0(G)

E. Effros and F. Hahn have conjectured that if ( G , Z ) (G,Z) is a second countable locally compact transformation group, with G G amenable, then every primitive ideal of the associated C ∗ {C^\ast } -algebra arises as the kernel of an irreducible representation induced from a stability subgroup. Results of Effros and Hahn concerning this conjecture are extended to include the twisted group algebra L 1 ( G , A ; T , α ) {L^1}(G,A;T,\alpha ) , where A A is a separable type I C ∗ {C^\ast } -algebra.

We establish a duality relation between one of the twisted group algebras of the hyperoctahedral groupf H_k and a Lie superalgebra q(n_0) \oplus q(n_1) for any integers k and n_0, n_1, where q(n_0) and q(n_1) denote the ``queer'' Liesuperalgebras. Note that this twisted group algebra \B'_k belongs to a different cocycle from the one \B_k used by A. N. Sergeev in [8] and by the present author in [11]. We will use the supertensor product \C_k \otimes \B'_k of the 2^k-dimensional Clifford algebra \C_k and \B'_k, as an intermediary for establishing our duality. We show that the algebra \C_k \otimes B'_k and q(n_0) \oplus q(n_1) act on the k-fold tensor product W=V^{\otimes k} of the natural representation V of q(n_0+n_1) ``as mutual centralizers of each other'' (Theorem 4.1). Moreover, we show that \B'_k and q(n_0) \oplus q(n_1) act on a subspace W' of W ``as mutual centralizers of each other'' (Theorem 4.2). This duality relation gives a formula for character values of simple B'_k-modules. This formula is di fferent from a formula (Theorem D) obtained by J. R. Stembridge (cf. [10, Lem 7.5]).

By applying the idea of viewing the octonions as an associative algebra in certain tensor categories, or more precisely as a twisted group algebra by a 2-cochain, we show that the octonions form an Azumaya algebra in some suitable braided linear Gr-categories.

The aim of this work is the study of the primary decomposition (p-decomposition for short) in some classes of rings. We show that every perfect hypercentral ring is p-decomposable. On the other hand, if K is a field of characteristic p, and G is a finite group, then the crossed product is p-decomposable if only if the twisted group algebra is p-decomposable, where N is the kernel of the action of G on K. As a consequence, the skew group ring is p-decomposable if and only if N is p-nilpotent.

Regarding the Cayley-Dickson algebras as twisted group algebras, this paper reveals some basic periodic properties of these twists.

Let $G$ be a noncyclic abelian $p$-group and $K$ be an infinite field of finite characteristic $p$. For every $2$-cocycle $\lambda \in Z^2(G,K^*)$ such that the twisted group algebra $K^\lambda G$ is of infinite representation type, we find natural number

A strongly regular ring R is one in which for all x ∈ R, there is an a ∈ R with x = x2a. Equivalently, for all x there is an a with x = ax2. Such a ring is regular, duo, biregular, and a left and right V-ring. Moreover since R is reduced, all nilpotent elements are central (vacuously) and so all idempotent elements are central.

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

Let $R$ be any ring with identity. An element $a \in R$ is called nil-clean, if $a=e+n$ where $e$ is an idempotent element and $n$ is a nil-potent element. In this paper we give necessary and sufficient conditions for a $2\times 2$ matrix over an integral domain $R$ to be nil-clean.

We say that an algebra is zero-product balanced if ab⊗c and a⊗bc agree modulo tensors of elements with zero-product. This is closely related to but more general than the notion of a zero-product determined algebra introduced and developed by Brešar, Villena and others. Every surjective, zero-product preserving map from a zero-product balanced algebra is automatically a weighted epimorphism, and this implies that zero-product balanced algebras are determined by their linear and zero-product structure. Further, the commutator subspace of a zero-product balanced algebra can be described in terms of square-zero elements.We show that a commutative, reduced algebra is zero-product balanced if and only if it is generated by idempotents. It follows that every commutative, zero-product balanced algebra is spanned by nilpotent and idempotent elements.