Conjugation Coinvariants of Quantum Matrices

Abstract

A quantum deformation of the classical conjugation action of GL(N, C) on the space of N × N matrices M(N, C) is defined via a coaction of the quantum general linear group O(GL q (N, C)) on the algebra of quantum matrices O(M q (N, C)). The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of O(M q (N, C)) generated by (weighted) sums of principal quantum minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction.

Orbits for the adjoint coaction on quantum matrices

“…Note that σ 1 = x 11 + · · · + x N N and that σ N = det q . It is shown in [4] that the σ i are α-coinvariants that pairwise commute, and if q is not a root of unity, …”

“…. ,τ N are the basic coinvariants for β introduced in [4]. When ξ is diagonal, the image of β ξ is contained in the quantum quotient space O(D \ GL q ), the subalgebra of coinvariants of the left coaction of the diagonal quantum subgroup O(D) on O(GL q ) (note that in the classical case q = 1, D is the stabilizer of ξ, provided that ξ has pairwise different diagonal entries).…”

“…When q is specialized to 1, the kernel of β ξ is the vanishing ideal of the closure of the orbit of ξ, so the image of β ξ can be identified with the coordinate ring of the closure of the orbit of ξ. The q-deformed map β ξ is not an algebra homomorphism since β is not an algebra homomorphism (see [4]). However, it turns out that the kernel of β ξ contains the right ideal of O(M q ) generated by the elements τ i − τ i (ξ), i = 1, .…”

“…Namely, if ξ is diagonal, and ξ is not a scalar multiple of any of the non-negative powers of diag(q 2 , 1), then β ξ is surjective onto O(D \ GL q ). Moreover, for such ξ, the kernel of β ξ is the right ideal of O(M q ) generated by the elements τ 1 − τ 1 (ξ), τ 2 − τ 2 (ξ), where τ 1 , τ 2 are the basic β-coinvariants from [4]; that is, τ 1 = q −2 x 11 + q −4 x 22 , a weighted trace, and τ 2 = x 11 x 22 − qx 12 x 21 , the quantum determinant. When ξ is a scalar multiple of some non-negative power of diag(q 2 , 1), then the kernel of β ξ is larger, and the image is finite dimensional.…”

“…These coactions can be viewed as quantum analogues of the adjoint action of GL(N, C) on its Lie algebra M(N, C). The coinvariants with respect to these coactions were described in [4].…”