Mathematics – Quantum Algebra
Scientific paper
2006-10-23
Glasgow Mathematical Journal 49 (2007), 479-488
Mathematics
Quantum Algebra
AMS-TeX file, 10 pages. This paper is a natural evolution - now in *final version* - of preprint math.QA/0608016, which has be
Scientific paper
10.1017/S0017089507003813
Let G be either of Mat(n), GL(n) or SL(n), let O_q(G) be the quantum function algebra - over Z[q,q^{-1}] - associated to G, and let O_e(G) be the specialisation of O_q(G) at a root of unity, of odd order l. Then O_e(G) is a module over the corresponding classical function algebra O(G) via the quantum Frobenius morphism, which embeds O(G) as a central subbialgebra of O_e(G). In this note we prove a PBW-like theorem for O_q(G) - more or less known in literature, but not in this form (to the best of the author's knowledge) - and we show that it yields explicit bases of O_e(G) over O(G) when G is Mat(n) or GL(n): in particular, O_e(G) is free of rank l^{dim(G)}. Also, we apply the latter result to prove that O_e(G) is a free Frobenius extensions over O(G), and to compute explicitly the corresponding Nakayama automorphism, again for G being Mat(n) or GL(n) . This extends previous results by Brown, Gordon and Stroppel (see [BG], [BGS2]).
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