Automorphisms and derivations of Uq(sl4+)

Abstract

We compute the automorphism group of the q-enveloping algebra U q (sl + 4 ) of the nilpotent Lie algebra of strictly upper triangular matrices of size 4. The result obtained gives a positive answer to a conjecture of Andruskiewitsch and Dumas. We also compute the derivations of this algebra and then show that the Hochschild cohomology group of degree 1 of this algebra is a free (left) module of rank 3 (which is the rank of the Lie algebra sl 4 ) over the center of U q (sl + 4 ).

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

“…This conjecture was first shown to hold for g + = sl + 3 in [5,29] and for g + = so + 5 in [88]. In [90] we computed the Lie algebra of derivations of U q (sl + 4 ) and showed that the first Hochschild cohomology group HH 1 (U q (sl + 4 )) is a free module of rank 3 over the center of U q (sl + 4 ). To do this, we first apply the deleting derivations algorithm of Cauchon [34] so that, after suitably localizing, we can embed U q (sl + 4 ) in a quantum torus T .…”

Poisson deleting derivations algorithm and Poisson spectrum

“…Now, z i 1 1 t ξ+1 3 t i 4 4 t i 5 5 t i 6 6 ∈ Span(P 5 ) when ξ = 0, 1. For ξ = 2, one can easily verify that z i 1 1 t 3 3 t i 4 4 t i 5 5 t i 6 6 ∈ Span(P 5 ) by using the relation in (14). Therefore, by the principle of mathematical induction, P 5 spans R 5 .…”

“…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 .…”