2023/04/11 by Nomura, Kazumasa, Terwilliger, Paul · 1 citation
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2304.04965
Let \F denote a field, and let V denote a vector space over \F with finite positive dimension. A Leonard pair on V is an ordered pair of diagonalizable \F-linear maps A: V → V and A^* : V → V that each act on an eigenbasis for the other in an irreducible tridiagonal fashion. Let A,A^* denote a Leonard pair on V. Let \vi\i=0d denote an eigenbasis for A^* on which A acts in an irreducible tridiagonal fashion. For 0 ≤ i ≤ d define an \F-linear map E^*i : V → V such that E^*i vi = vi and E^*i vj = 0 if j ≠ i (0 ≤ j ≤ d). The map F = ∑i=0d E^*i A E^*i is called the flat part of A. The Leonard pair A,A^* is bipartite whenever F=0. The Leonard pair A,A^* is said to be near-bipartite whenever the pair A-F, A^* is a Leonard pair on V. In this case, the Leonard pair A-F, A^* is bipartite, and called the bipartite contraction of A,A^*. Let B,B^* denote a bipartite Leonard pair on V. By a near-bipartite expansion of B,B^* we mean a near-bipartite Leonard pair on V with bipartite contraction B,B^*. In the present paper we have three goals. Assuming \F is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over \F; (ii) for each near-bipartite Leonard pair over \F we describe its bipartite contraction; (iii) for each bipartite Leonard pair over \F we describe its near-bipartite expansions.