Weakly multiplicative coactions of quantized function algebras

Abstract

A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the subalgebra of coinvariants) is obtained for such coactions of a cosemisimple Hopf algebra. This is applied for two coactions α,β : A→ A⊗ O , where A is the coordinate algebra of the quantum matrix space associated with the quantized coordinate algebra O of a classical group, and α, β are quantum analogues of the conjugation action on matrices. Provided that O is cosemisimple and coquasitriangular, the α-coinvariants and the β-coinvariants form two finitely generated, commutative, graded subalgebras of A , having the same Hilbert series. Consequently, the cocommutative elements and the S 2-cocommutative elements in O form finitely generated subalgebras. A Hopf algebra monomorphism from the quantum general linear group to Laurent polynomials over the quantum special linear group is found and used to explain the strong relationship between the corepresentation (and coinvariant) theories of these quantum groups.