Bidiagonal pairs, the Lie algebra sl_2, and the quantum group U_q(sl_2)

Details Bidiagonal pairs, the Lie algebra sl_2, and the quantum group U_q(sl_2) Bidiagonal pairs, the Lie algebra sl_2, and the quantum group U_q(sl_2)

2011-08-04

arxiv.org/abs/1108.1219v1

Mathematics

Representation Theory

41 pages, submitted to the Journal of Algebra and its Applications

Scientific paper

Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. We associate to each bidiagonal pair a sequence of scalars called a parameter array. We present a classification of bidiagonal pairs up to isomorphism using this concept of a parameter array. The statement of this classification theorem does not explicitly mention the Lie algebra sl_2 or the quantum group U_q(sl_2). However, its proof makes use of the finite dimensional representation theory of sl_2 and U_q(sl_2). In addition to the classification theorem we present four theorems which make explicit the relationship between bidiagonal pairs and sl_2, U_q(sl_2)-modules.

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