Let K be a (algebraically closed ) field. A morphism A ⟼ g −1 Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational GL q × GL q -modules is a highest weight category.
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Translated from Algebra i Logika, Vol. 48, No. 4, pp. 425-442, July-August, 2009.
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Antonov, V.V., Zubkov, A.N. Coinvariants for a coadjoint action of quantum matrices. Algebra Logic 48, 239–249 (2009). https://doi.org/10.1007/s10469-009-9060-2
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DOI: https://doi.org/10.1007/s10469-009-9060-2