Rigidité des plongements des quotients primitifs minimaux de U_g(sl(2)) dans l’algebre quantique de Weyl-Hayashi

Abstract

Ce travail est consacré aux groupes d’automorphismes de certaines algebres quantiques de dimension 2 ou 3. Dans la théorie classique des algebres enveloppantes, si désigne l’algèbre de Lie de Heisenberg de dimension 3, U() admet l’algèbre de Weyl A1 comme seul quotient primitif de dimension 2, avec les propriétés suivantes: d’ une part tout automorphisme de A1 se relève en un automor-phisme de U(), d’autre part U() admet des automorphismes non modérés (cf. [A1], [Di1], [ML]). On retrouve la même situation pour les quotients primitifs minimaux de U(sl(2)) paramétrés par C (cf. [Di2], [Jo]). En outre, dans ce cas, on dispose des plongements de Conze de ces quotients dans A1 (cf. [Di2], [Co], [Ro]); bien que les groupes d’automorphismes soient comparables, la restriction correspondant à un tel plongement est seulement définie sur le sous-groupe des automorphismes triangulaires de A1 (cf. annexe).

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

“…Ever since it was introduced by Maltisiniotis in [26], the multiparameter quantized Weyl algebra A q,Γ n (K) has been extensively studied in the literature [2,14,17,12,20,32]. When n = 1, A q,Γ n (K) is the rank-one quantized Weyl algebra A q 1 (K).…”

“…Recall that A q 1 (K) is a K−algebra generated by x, y subject to the relation xy − qyx = 1, whose prime ideals of A q 1 (K) were classified in [17]. The automorphism group of A q 1 (K) was completely determined in [2]. There has been further research on the Maltisiniotis multiparameter quantized Weyl algebras A q,Γ n (K) of higher ranks.…”

Automorphisms of Generalized Down-Up Algebras

“…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.…”

Isomorphisms Between Quantum Generalized Weyl Algebras

“…Then the algebra A q (a) is a particular subalgebra of the primitive quotient B q ′ ,λ of the quantum enveloping algebra U q ′ (sl (2)). More precisely (see [2] for details), A q (a) = D q ′ ,λ is the canonical image in B q ′ ,λ of ad-locally finite elements in U q ′ (sl (2)). …”

“…The above theorem covers for example the cases of the quantum Weyl algebra A q 1 , k-generated by variables x, y such that xy −qyx = 1, and the algebras D q,λ defined in [2], that is, the quotients of the ad-locally finite elements F q of U q (sl 2 ) by the ideal generated by (Ω − λ). The algebra D q,λ verifies that Aut(D q,λ ) ∼ = Aut(F q ).…”