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Gröbner-Shirshov pair of irreducible modules over quantum group \(U_q(D_4)\). (Chinese. English summary) Zbl 1474.17016

Summary: We first compute a Gröbner-Shirshov pair of the irreducible module \(V_q(\lambda)\) over the quantum group \(U_q(D_4)\) by using the known Gröbner-Shirshov basis of the quantum group \(U_q(D_4)\) and the double free module method of Chibrikov, then define a suitable version \(U'_q(D_4)\) of \(U_q(D_4)\) and by specializing \(U'_q(D_4)\) at \(q = 1\), we get a Gröbner-Shirshov pair of the irreducible module \(V (\lambda)\) over the universal enveloping algebra \(U (D_4)\) of simple Lie algebra of type \(D_4\).

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17A61 Gröbner-Shirshov bases in nonassociative algebras
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