Automorphism groups of some algebras

Abstract

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum nplane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W (m, m + n).

Automorphisms and representations of quasi Laurent polynomial algebras

“…By restricting the maps (3.2) to the centre Z of L q [H], and recalling that Z is the algebra of Laurent polynomials in m = n − r variables, we easily recovered the following well-known result (see, e.g., [26]).…”

“…The QPA and QLPA corresponding to H = 0 1 −1 0 , denoted by by A q (2) and L q (2) respectively, attracted much research [10,16,19,26,29] because of their important roles in the theory of quantum groups. The algebra L q (2) is also known as the quantum torus of rank 2, which appears in many contexts.…”

“…The main results are given in Theorem 3.1 and Theorem 3.2, which solve the problem of automorphism groups of QLPAs in full generality. We remark that the automorphism group of L q (2) was determined in [19], and also in [26] where the automorphism groups were described for a class of quantum algebras including the quantum torus of rank 2.…”

The Laurent Extension of Quantum Plane: a Complete List of U_q(sl₂)-Symmetries

“…(see also [2,7]). We also rephrase this via introducing the associated action of SL(2, Z) by group automorphisms of (C\{0}) 2 k m l n…”