On symmetric, smooth and Calabi–Yau algebras

Abstract

One possible definition for a Calabi-Yau algebra is a symmetric smooth PI algebra. Our main purpose here is to prove some necessary and sufficient criteria for verifying the (local) symmetric property, in smooth PI algebras. Many known smooth PI algebras are shown to have this property. In particular quantum enveloping algebras of complex semi-simple Lie algebras, in the root of unity case and the enveloping algebra of sl(n) with (p, n) = 1, in characteristic p, are typical examples. A surprising result is that the inj.dim T , is finite, where T is the trace ring of m, n × n generic matrices over a field of zero characteristic.

Cherednik, Hecke and quantum algebras as free Frobenius and Calabi–Yau extensions

“…There is some overlap between this paper and [2], a preliminary version of which we received while this paper was being written. The methods used in the two papers are completely different, and indeed complementary.…”

“…We would like to thank Ami Braun for telling us about the results in [2] and for supplying us with a preliminary version of his paper. The third author acknowledges the support of the EP-SRC grant number GR/S14900/01.…”

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

“…Combining Theorem 2 with Braun's work [5] we obtain assertion (ii) of the following corollary. For a definition of Calabi-Yau algebra, see [5,Sect. 1].…”

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

“…There are a few approaches to topics with this name. I follow [IR08,Bra07]; see also [Gin06,Boc09]. I will always assume that the base ring is local, which eases the exposition considerably.…”