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1. CMB 2011 (vol 55 pp. 271)
On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras Let $F$ be an algebraically closed field of characteristic zero, and
let $A$ be an associative unitary $F$-algebra graded by a group of
prime order. We prove that if $A$ is finite dimensional then the
graded exponent of $A$ exists and is an integer.
Keywords:exponent, polynomial identities, graded algebras Categories:16R50, 16R10, 16W50 |
2. CMB 2011 (vol 55 pp. 208)
Abelian Gradings on Upper Block Triangular Matrices Let $G$ be an arbitrary finite abelian group. We describe all
possible $G$-gradings on upper block triangular matrix algebras
over an algebraically closed field of characteristic zero.
Keywords:gradings, upper block triangular matrices Category:16W50 |
3. CMB 2008 (vol 51 pp. 460)
On Primitive Ideals in Graded Rings Let $R=\bigoplus_{i=1}^{\infty}R_{i}$ be a graded nil ring. It is shown
that primitive ideals in $R$ are homogeneous. Let
$A=\bigoplus_{i=1}^{\infty}A_{i}$ be a graded non-PI just-infinite
dimensional algebra and let $I$ be a prime ideal in $A$. It is shown
that either $I=\{0\}$ or $I=A$. Moreover, $A$ is either primitive or
Jacobson radical.
Categories:16D60, 16W50 |
4. CMB 2002 (vol 45 pp. 451)
Coordinatization Theorems For Graded Algebras In this paper we study simple associative algebras with finite
$\mathbb{Z}$-gradings. This is done using a simple algebra $F_g$
that has been constructed in Morita theory from a bilinear form
$g\colon U\times V\to A$ over a simple algebra $A$. We show that
finite $\mathbb{Z}$-gradings on $F_g$ are in one to one
correspondence with certain decompositions of the pair $(U,V)$. We
also show that any simple algebra $R$ with finite
$\mathbb{Z}$-grading is graded isomorphic to $F_g$ for some
bilinear from $g\colon U\times V \to A$, where the grading on $F_g$
is determined by a decomposition of $(U,V)$ and the coordinate
algebra $A$ is chosen as a simple ideal of the zero component $R_0$
of $R$. In order to prove these results we first prove similar
results for simple algebras with Peirce gradings.
Category:16W50 |
5. CMB 2002 (vol 45 pp. 499)
Group Gradings on Matrix Algebras Let $\Phi$ be an algebraically closed field of characteristic zero,
$G$ a finite, not necessarily abelian, group. Given a $G$-grading on
the full matrix algebra $A = M_n(\Phi)$, we decompose $A$ as the
tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p
(\Phi)$ being a graded division algebra, while the grading of $C\cong
M_q (\Phi)$ is determined by that of the vector space $\Phi^n$. Now
the grading of $A$ is recovered from those of $A$ and $B$ using a
canonical ``induction'' procedure.
Category:16W50 |
6. CMB 2002 (vol 45 pp. 711)
Classification of Quantum Tori with Involution Quantum tori with graded involution appear as coordinate algebras of
extended affine Lie algebras of type $\rmA_1$, $\rmC$ and $\BC$.
We classify them in the category of algebras with involution. From
this, we obtain precise information on the root systems of extended
affine Lie algebras of type $\rmC$.
Category:16W50 |