The discriminant controls automorphism groups of noncommutative algebras

Ce travail est consacre aux groupes d’automorphismes de certaines algebres quantiques de dimension 2 ou 3. Dans la theorie classique des algebres enveloppantes, si designe l’algebre de Lie de Heisenberg de dimension 3, U() admet l’algebre de Weyl A1 comme seul quotient primitif de dimension 2, avec les proprietes suivantes: d’ une part tout automorphisme de A1 se releve en un automor-phisme de U(), d’autre part U() admet des automorphismes non moderes (cf. [A1], [Di1], [ML]). On retrouve la meme situation pour les quotients primitifs minimaux de U(sl(2)) parametres 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 a un tel plongement est seulement definie sur le sous-groupe des automorphismes triangulaires de A1 (cf. annexe).

/pdf/rigidite-des-plongements-des-quotients-primitifs-minimaux-de-9kxq45p11j.pdf

Rigidité des plongements des quotients primitifs minimaux de $U_q({\rm sl}(2))$ dans l'algèbre quantique de Weyl-Hayashi

The discriminant criterion and automorphism groups of quantized algebras

Let R be an iterated Ore extension k[y][x;σ,δ] of the complex number field k, with δ a k-automorphism of k[y] and δ a u-derivation of k[y] vanishing on k. We suppose that the center of R is k. Up to a change of variables, any finite group G of k-automorphisms of R acts linearly on kx⊕D1ky. When the quotient division ring D of R is isomorphic to the Weyl skewfield D1(k)1 , then DG⋍D1 (k). In any other noncommutative case, D is isomorphic to the quantum Weyl skewfield Dq 1(k) for some q∊k∗ not a root of one, and DG⋍Ds 1(k) with s = q‖G‖.

Invariants du corps de weyl sous l'action de groupes finis

Let $\mathbb{K}$
 be a field and q be a nonzero element of $\mathbb{K}$
 that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 of quantum matrices. Next, we describe all height one primes of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 whenever ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 has only finitely many height one primes. Finally, we compute the automorphism group of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 . Then, by using the preferred basis of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 and PBW bases, we prove that the automorphism group of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$
 is isomorphic to the torus ${\left( {\mathbb{K}*} \right)}^{{m + n - 1}} $
 when m ≠ n and (m,n) ≠ (1, 3),(3, 1).

/pdf/primitive-ideals-and-automorphisms-of-quantum-matrices-2acl774p5z.pdf

Primitive ideals and automorphisms of quantum matrices

(1992). Derivations et automorphismes de quelques algebras quantiques. Communications in Algebra: Vol. 20, No. 6, pp. 1787-1802.

J. Alev

Papers

Derivations et automorphismes de quelques algebras quantiques