Abstract
We prove that every endomorphism of a simple quantum generalized Weyl algebra $A$ over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of $A$. Our main result applies to minimal primitive factors of the quantized enveloping algebra of $U_q(\mathfrak{sl}_2)$ and certain minimal primitive quotients of the positive part of $U_q(\mathfrak{so}_5)$.
Highlights
At the end of his foundational paper on the Weyl algebra [6], Dixmier posed the famous conjecture that any algebra endomorphism of the Weyl algebras is an automorphism
The main aim of this paper is to exhibit algebras related to the first Weyl algebra, and which possess the property that every endomorphism is an automorphism
The structure of the primitive factor algebras of the enveloping algebra U (n) of n are well known: a theorem of Dixmier asserts that these factor algebras are isomorphic to Weyl algebras [7, 4.7.9]
Summary
At the end of his foundational paper on the Weyl algebra [6], Dixmier posed the famous conjecture that any algebra endomorphism of the Weyl algebras is an automorphism. In this letter we classify all endomorphisms of quantum generalized Weyl algebras and obtain the following result which is a consequence of Section 2, Corollaries 3.2 and 4.2 and Proposition 5.1. 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]. When a(h) is invertible in K[h±1], it is easy to check that A(a(h), q) is the K-algebra generated by h±1 and x±1 subject to the relations h±1h∓1 = 1, x±1x∓1 = 1 and xh = qhx In this case, it was proved by Richard [11] that every endomorphism is an automorphism.
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