On the center of the quantized enveloping algebra of a simple Lie algebra

2016/07/04 by Li, Libin, Xia, Limeng, Zhang, Yinhuo · 1 citation
#16N20 #19A22 #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1607.00802

Abstract

Let \frakg be a finite dimensional simple complex Lie algebra and U=U_q(\frakg) the quantized enveloping algebra (in the sense of Jantzen) with q being generic. In this paper, we show that the center Z(U_q(\frakg)) of the quantum group U_q(\frakg) is isomorphic to a monoid algebra, and that Z(U_q(\frakg)) is a polynomial algebra if and only if \frakg is of type A₁, B_n, C_n, D_2k+2, E₇, E₈, F₄ or G₂. Moreover, in case \frakg is of type D_n with n odd, then Z(U_q(\frakg)) is isomorphic to a quotient algebra of a polynomial algebra in n+1 variables with one relation; in case \frakg is of type E₆, then Z(U_q(\frakg)) is isomorphic to a quotient algebra of a polynomial algebra in fourteen variables with eight relations; in case \frakg is of type A_n, then Z(U_q(\frakg)) is isomorphic to a quotient algebra of a polynomial algebra described by n-sequences.

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