Gröbner-Shirshov bases of irreducible modules of the quantum group of type G ₂

Abstract

First, the authors give a Gröbner-Shirshov basis of the finite-dimensional irreducible module V _q(λ) of the Drinfeld-Jimbo quantum group U _q(G ₂) by using the double free module method and the known Gröbner-Shirshov basis of Uq(G2). Then, by specializing a suitable version of U _q(G ₂) at q = 1, they get a Gröbner-Shirshov basis of the universal enveloping algebra U(G ₂) of the simple Lie algebra of type G ₂ and the finite-dimensional irreducible U(G ₂)-module V (λ).

Keywords

Quantum group / Gröbner-Shirshov basis / Double free module / Indecomposable module / Highest weight module

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