Derivations et automorphismes de quelques algebras quantiques

Automorphisms of Generalized Down-Up Algebras

“…We start by computing some examples, which have appeared elsewhere in the literature, namely [2], [3] and [9].…”

“…Despite these and other successful results on the description of automorphism groups of quantum algebras, e.g. [2,3,9,14,19,20,21], there is yet much to be done. For example, regarding the quantized enveloping algebras U q (g + ), where g is a finite-dimensional complex simple Lie algebra and g + is a maximal nilpotent subalgebra of g, there is a conjecture of Andruskiewitsch and Dumas [4] describing the automorphism group of U q (g + ) as a semidirect product of a torus of rank equal to the rank of g by a finite group corresponding to the automorphisms of the Dynkin diagram of g. So far, only particular cases of this conjecture have been verified, for g of rank at most 3 [9,3,19,21].…”

“…The problem of describing the automorphism group seems to be considerably simpler when a deformation is introduced. Indeed, the automorphism group of the quantized enveloping algebra U q (sl 2 ) was computed in [2], and in [9,3] the authors independently described the group of automorphisms of the quantum Heisenberg algebra; in all cases it was assumed that the deformation parameter is not a root of unity. Despite these and other successful results on the description of automorphism groups of quantum algebras, e.g.…”

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

“…In this paper, we begin to study the Hochschild cohomology of the algebra of quantum matrices, O q .M n /, in the generic case where q is not a root of unity. To be more specific, we calculate the first Hochschild cohomology, HH 1 .O q .M n //, of O q .M n /: in other words, we calculate the derivations of O q .M n /. Once this has been done, we are also able to calculate HH 1 for the quantum general linear group, O q .GL n /, and the quantum special linear group, O q .SL n /.…”

Noetherian Algebras of Quantum Differential Operators

“…Operators related to quantum differential operators, such as derivations, skew derivations, and generalized derivations, on the algebras that we consider have been well studied, see [1,2,3,4,16] and references therein.…”