The first Hochschild cohomology group of quantum matrices and the quantum special linear group

Abstract

Abstract. We calculate the first Hochschild cohomology group of quantum matrices, the quantum general linear group and the quantum special linear group in the generic case when the deformation parameter is not a root of unity. As a corollary, we obtain information about twisted Hochschild homology of these algebras.

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

“…HH 1 (U q (sl + 4 )) is free of rank 3 over the center of U q (sl + 4 ). As an application, using the methods of [3] and [89], we confirmed the Andruskiewitsch-Dumas conjecture for U q (sl + 4 ).…”

Poisson deleting derivations algorithm and Poisson spectrum

“…(5) 5) and the relation (13), one can deduce that Suppose that there exists (i 1 , i 3 , i 4 , i 5 , i 6 ) ∈ J such that b (i 1 ,i 3 ,i 4 ,i 5 ,i 6 ) = 0. Let (w 1 , w 3 , w 4 , w 5 , w 6 ) ∈ J be the greatest element (in the lexicographic order on N 3 × Z 2 ) of J such that b (w 1 ,w 3 ,w 4 ,w 5 ,w 6 ) = 0.…”

“…In the noncommutative world, the knowledge of the derivations of twisted group algebras, studied by Osborn and Passman [19], has helped in studying the derivations of other non-commutative algebras such as the quantum second Weyl algebra (see [15]), quantum matrices (see [13]), generalized Weyl algebras (see [11]) and some specific examples of quantum enveloping algebras (see [14], [20], and [21]). In view of this, we also study the Poisson derivations of the Poisson analogue of the twisted group algebras-called Poisson group algebras-and apply the results to study the Poisson derivations of a semiclassical limit A α,β of A α,β .…”