PRIMITIVE IDEALS AND AUTOMORPHISM GROUP OF $U_{q}^{+}(B_{2})$

Abstract

Let g be a complex simple Lie algebra of type B 2 and q be a nonzero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of Gelfand-Kirillov dimension 2 of the positive part U + (g) of the enveloping algebra of g are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part U + q (g) of the quantized enveloping algebra of g and then we study the simple factor algebras of Gelfand-Kirillov dimension 2 of U + q (g). In particular, we show that the centers of such simple factor algebras are reduced to the ground field C and we compute their group of invertible elements. These computations allow us to prove that the automorphism group of U + q (g) is isomorphic to the torus (C * ) 2 , as conjectured by Andruskiewitsch and Dumas.

Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

“…Then the element ω X (θ) is found by applying ω X to the equality (20) and using the equality ω X (t) = t where…”

Endomorphisms of Quantum Generalized Weyl Algebras

“…Let K be a field and q ∈ K * = K \ {0} not a root of unity. In [10], primitive factor algebras of Gelfand-Kirillov dimension 2 of the positive part U + q (so 5 ) of the quantized enveloping algebra U q (so 5 ) were classified and can be thought of as quantum analogues of the first Weyl algebra. Among those are A.P.…”

“…S. Launois is grateful for the financial support of EPSRC first grant EP/I018549/1. the algebras A α,q with α = 0, where A α,q is the K-algebra generated by e 1 , e 2 If α = 0, then A α,q is simple (see [10]). Note that setting q = 1 and α = 1 we indeed get an algebra isomorphic to the first Weyl algebra A 1 (K).…”

“…, q 2 ) (see [10,Proposition 3.10] As a consequence, every endomorphism of a simple quantum generalized Weyl algebra is an automorphism, so that Theorem 1.1 is just a consequence of Theorem. Previously, the fact that every endomorphism of a quantum generalized Weyl algebra A(a(h), q) is an automorphism, has only been established, by Richard, in the case where a(h) is a (Laurent) monomial [11].…”

Torsion simple modules over the quantum spatial ageing algebra

“…The prime or/and primitive ideals of various quantum algebras (and their classification) are considered in [15,16,21,24,25,26,27,34,37,39,43]. The automorphism group of some (quantum) algebras are considered in [1,2,13,22,32,35,36,44].…”