Near-bipartite Leonard pairs

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 ᵢ\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. Our classification (i) is summarized as follows. We identify two families of Leonard pairs, said to have Krawtchouk type and dual q-Krawtchouk type. A Leonard pair of dual q-Krawtchouk type is said to be reinforced whenever q²ⁱ ≠ -1 for 1 ≤ i ≤ d-1. A Leonard pair A,A^* is said to be essentially bipartite whenever the flat part of A is a scalar multiple of the identity. Assuming F is algebraically closed, we show that a Leonard pair A,A^* over F with d ≥ 3 is near-bipartite if and only if at least one of the following holds: (i) A,A^* is essentially bipartite; (ii) A,A^* has reinforced dual q-Krawtchouk type; and (iii) A,A^* has Krawtchouk type.