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

Abstract

We show how the existence of a PBW-basis and a large enough central subalgebra can be used to deduce that an algebra is Frobenius. We apply this to rational Cherednik algebras, Hecke algebras, quantised universal enveloping algebras, quantum Borels and quantised function algebras. In particular, we give a positive answer to [R. Rouquier, Representations of rational Cherednik algebras, in: Infinite-Dimensional Aspects of Representation Theory and Applications, Amer. Math. Soc., 2005, pp. 103-131] stating that the restricted rational Cherednik algebra at the value t = 0 is symmetric.

PBW Theorems and Frobenius Structures for Quantum Matrices

“…In this note we prove a PBW-like theorem for O q (G), and we show that -when G is Mat n or GL n -it yields explicit bases of O ε (G) over O (G). As a direct application, we prove that O ε (GL n ) and O ε (M n ) are free Frobenius extensions over O(GL n ) and O(M n ), thus extending some results of [5]. …”

“…This result is used in [5] to show that many families of algebras -in particular, some related to O ε (G), where G is a (complex, connected, simply-connected) semisimple affine algebraic group -are indeed free Frobenius extensions. But the authors could not prove the same for O ε (G), as they did not know an explicit O(G)-basis of O ε (G).…”

“…Following [5], we say that a ring R is a free Frobenius extension over a subring S, if R is a free S-module of finite rank, and there is an isomorphism F: R −→ Hom S (R, S) of R − S-bi-modules. Then F provides a non-degenerate associative S-bilinear form ‫:ނ‬ R × R −→ S, via ‫(ނ‬r, t) = F(t)(r).…”

“…The same we do for O ‫ޚ‬ ε ε (GL n ) as well. Everything follows from the ideas and methods in [5], now applied to the explicit bases given by Theorem 2.2. …”

Ring theoretical properties of affine cellular algebras

“…This seems to be the case for some affine Hecke algebras. For instance, it has been argued in [3,5.1] using [13, 3.11] that the centre of an (extended) affine Hecke algebra is a polynomial ring in finitely many variables. Small, Stafford and Warfield proved that any semiprime affine k-algebra A of GKdimension one is Noetherian and finitely generated over its centre (see [19]).…”