PBW Theorems and Frobenius Structures for Quantum Matrices

Abstract

Abstract. Let G ∈ {Mat n ‫,)ރ(‬ GL n ‫,)ރ(‬ SL n ‫,})ރ(‬ let O q (G) be the quantum function algebra -over ‫[ޚ‬q, q −1 ] -associated to G, and let O ε (G) be the specialisation of the latter at a root of unity ε, whose order is odd. There is a quantum Frobenius [4] that (the complexification of) such a module is free, with rank dim (G) . 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].