Orbits for the adjoint coaction on quantum matrices

Abstract

Conjugation coactions of the quantum general linear group on the algebra of quantum matrices have been introduced in an earlier paper and the coinvariants have been determined. In this paper the notion of orbit is considered via co-orbit maps associated with C-points of the space of quantum matrices, mapping the coordinate ring of quantum matrices into the coordinate ring of the quantum general linear group. The co-orbit maps are calculated explicitly for 2 × 2 quantum matrices. For quantum matrices of arbitrary size, it is shown that when the deformation parameter is transcendental over the base field, then the kernel of the co-orbit map associated with a C-point ξ is a right ideal generated by coinvariants, provided that the classical adjoint orbit of ξ is maximal. If ξ is diagonal with pairwise different eigenvalues, then the image of the co-orbit map coincides with the subalgebra of coinvariants with respect to the left coaction of the diagonal quantum subgroup of the quantum general linear group.

A Quantum Homogeneous Space of Nilpotent Matrices

“…The only thing we have to prove is that (i) holds as well, namely that F q (O cl ξ ) is a free K-module. [4]. (To be more precise, the statement of [4] is about F C(q),q (GL(n, C)), instead of F C(q),q (G).…”

“…The following result shows that our F q (O cl ξ ) is a non-embedded quantum homogeneous space for most ξ ∈ P . This indicates that the use of the interplay between L q (M ) and F q (M ) due to [2], [21] combined with the 'orbit map' of [4] can not be replaced by an orbit map going directly from L q (M ) to F q (G) (like in [8]). Proposition 4.2 Assume that for ξ ∈ P , one of the following holds:…”

“…The main technical problem is to prove that the resulting quotient is a free K-module. This is handled by applying results of [4], that are based on the fact that sufficiently many characters of F q (M ) are available to hit the orbit of each ξ ∈ P .…”

Representation rings of quantum groups

“…For example, [11] and [7] (see also the references therein) make use of adjoint invariants (central elements) of the reflection equation algebra to study quantizations of coadjoint orbits of SL(N). (Staying in the framework of quantum matrices, related results were obtained in [4].) See also [17] and [16] for discussion of other versions of the reflection equation algebra.…”

Weakly multiplicative coactions of quantized function algebras

“…When β is also an algebra homomorphism, A β-co-O is automatically a subalgebra of A. However, it has recently become apparent that progress can be made even when β is not an algebra homomorphism, see [4,5,6,7], for example, where examples of such coactions are studied, motivated by seeking quantum versions of results concerning the classical invariant theory of the general linear and special linear groups. In these quantum cases, at the outset, it is not even clear that the set of coinvariants forms a subalgebra of A.…”