arXiv subjects
Publications and source records attributed to Rabigul Tuniyaz.
Let $M_q(n)$ be the standard quantized matrix algebra (introduced by Faddeev, Reshetikhin, and Takhtajan). It is shown that $M_q(n)$ is Auslander regular, Cohen-Macaulay, Artin-Schelter regular, and a maximal order in its quotient division algebra.
Let $D_q(n)$ be the quantized matrix algebra introduced by Dipper and Donkin. It is shown that some structural properties of $D_q(n)$ and their modules may be established and realized by means of Gröbner-Shirshov basis theory.
Let $M_q(n)$ be the standard quantized matrix algebra, introduced by Faddeev, Reshetikhin, and Takhtajan. It is shown, by constructing an appropriate monomial ordering $\prec$ on its PBW $K$-basis ${\cal B}$ , that $M_q(n)$ is a solvable polynomial algebra. Consequently, further structural properties of $M_q(n)$ and their modules may be established and realized in a constructive-computational way.
Let $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) be the $(+)$-part (resp. $(-)$-part) of the Drinfeld-Jimbo quantum group of type $A_N$ over a field $K$. With respect to Jimbo relations and the PBW $K$-basis ${\cal B}$ of $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) established by Yamane, it is shown, by constructing an appropriate monomial ordering $\prec$ on ${\cal B}$, that $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) is a solvable polynomial algebra. Consequently, further structural properties of $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) and their modules may be established and realized in a constructive-computational way.
The so called generalized down-up algebras are revisited from a viewpoint of Gröbner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $λω\ne 0$), and by means of homogeneous Gröbner defining relations, the associated graded structures of generalized down-up algebras, namely the associated graded algebras, Rees algebras, and the homogenized algebras of generalized down-up algebras, are explored comprehensively.