Simple Associative Algebras with FiniteZ-Grading

Finite Z-gradings of simple associative algebras

“…Indeed, consider a = k,l a kl ∈ A. For any b = r,s b r s ∈ A, The following result completes [8,Theorem 4.6] in the sense that Smirnov's result shows that any grading of a simple Z-graded algebra A is induced by a complete orthogonal system of submodules, and we prove that the grading is, in fact, induced by a complete system of orthogonal idempotents lying in the (graded) maximal left quotient algebra of A. Proof.…”

“…If L is as in case I then, by (2.5), A has a nontrivial 3-grading, which is inherited by L. If L is as in case II, we cannot assure the existence of a nontrivial 3-grading for the associative algebra A, preserved by the involution (see [8,Example in pg. 182]).…”

“…Proof. Clearly, A simple implies right faithful and A = id(A −n ); moreover, in the proof of Theorem 4.6 in [8] it is said that A = A 0 A A 0 , hence we may apply (2.8). The uniqueness of H was obtained in the proof of [8, Theorem 4.6] too.…”

“…Smirnov shows in [8,Theorem 5.4] that a Lie algebra satisfying the conditions in Zelmanov's Theorem has a nontrivial 5-grading. This result is obtained as a consequence of the description of finite Z-gradings of simple associative algebras.…”

Конечномерные Простые Градуированные Алгебры

“…В работе [8] автор по-казал, что конечные Z-градуировки на простых (не обязательно конечномер-ных) ассоциативных алгебрах могут быть получены из пирсовских разложений. В работе [9] были описаны все градуировки матричной алгебры группами без кручения.…”