Abstract
Our aim in this paper is to investigate symmetry and reversibility pro-perties for quantum algebras and skew PBW extensions. Under certainconditions we prove that these properties transfer from a ring of coeffi-cients to a quantum algebra or skew PBW extension over this ring. In thisway we generalize several results established in the literature and consideralgebras which have not been studied before. We illustrate our results withremarkable examples of theoretical physics
Highlights
Two of the three concepts of interest in this paper are symmetry and reversibility
We characterize reversibility and symmetry properties over quantum algebras and skew PBW extensions (Theorem 3.1), and we introduce our definitions of right Σ-reversible and right Σ-symmetric (Definition 3.1) which extends notions (6) and (7) mentioned above
The results presented (Theorems 3.2, 3.3, 3.4) are new for skew PBW extensions and quantum algebras, and all of them extend similar results for Ore extensions appearing in [3],[30],[13],[17],[16] and [31]
Summary
Two of the three concepts of interest in this paper are symmetry and reversibility. A ring B is called symmetric, if abc ⇒ acb = 0, for every elements a, b, c ∈ B. We characterize reversibility and symmetry properties over quantum algebras and skew PBW extensions (Theorem 3.1), and we introduce our definitions of right Left) Σ-symmetric (Definition 3.1) which extends notions (6) and (7) mentioned above In this way, the results presented (Theorems 3.2, 3.3, 3.4) are new for skew PBW extensions and quantum algebras, and all of them extend similar results for Ore extensions appearing in [3],[30],[13],[17],[16] and [31]. C will denote the field of complex numbers and the letter k will denote any field
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