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Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type $A_{n}$

Rabigul Tunyaz

College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046 P. R. China

E-mail Address: rabigul802@sina.com

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and

Abdukadir Obul

College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046 P. R. China

E-mail Address: abdu@vip.sina.com

Corresponding author.

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https://doi.org/10.1142/S0219498818500354Cited by:0 (Source: Crossref)

Abstract

In this paper, first, we give a Gröbner–Shirshov pair of finite-dimensional irreducible module $V_{q} (λ)$ over $U_{q} (A_{n}),$ the quantized enveloping algebra of type $A_{n},$ by using the double free module method and the known Gröbner–Shirshov basis of $U_{q} (A_{n}) .$ Then, by specializing a suitable version of $U_{q} (A_{n})$ at $q = 1,$ we get a Gröbner–Shirshov basis of $U (A_{n}),$ and get a Gröbner–Shirshov pair for the finite-dimensional irreducible module $V (λ)$ over $U (A_{n})$ .

Communicated by L. A. Bokut

Keywords:

AMSC: 16S15, 13P10, 17B37

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