Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition

Let K denote a field and let d denote a nonnegative integer. Let A denote a K-algebra isomorphic to Matd+1(K). An element of A is called multiplicity-free whenever its eigenvalues are mutually distinct and contained in K. Let A and A denote multiplicity-free elements in A. Let {Ei} d i=0 (resp. {E ∗ i } d i=0) denote an ordering of the primitive idempotents of A (resp. A.) For 0 ≤ i ≤ d let θi (resp. θ ∗ i ) denote the eigenvalue of A (resp. A) for Ei (resp. E ∗ i .) Let V denote an irreducible left A-module. By a decomposition of V we mean a sequence {Ui} d i=0 consisting of 1dimensional subspaces of V such that V = ∑d i=0 Ui. A decomposition {Ui} d i=0 of V is said to be split (with respect to the orderings {Ei} d i=0,{E ∗ i } d i=0) whenever both (i) (A − θiI)Ui = Ui+1 (0 ≤ i ≤ d − 1), (A − θdI)Ud = 0; and (ii) (A ∗ − θ i I)Ui = Ui−1 (1 ≤ i ≤ d), (A − θ 0I)U0 = 0. We show there exists at most one decomposition of V which is split with respect to {Ei} d i=0, {E ∗ i } d i=0. We show the following are equivalent: (i) there exists a decomposition of V which is split with respect to {Ei} d i=0, {E ∗ i } d i=0; (ii) both E i AE ∗ j = { 0, if i− j > 1; 6= 0, if i− j = 1 EiA Ej = { 0, if j − i > 1; 6= 0, if j − i = 1 for 0 ≤ i, j ≤ d. We call the sequence (A;A; {Ei} d i=0; {E ∗ i } d i=0) a Leonard system whenever both E i AE ∗ j = { 0, if |i− j| > 1; 6= 0, if |i− j| = 1 EiA Ej = { 0, if |i− j| > 1; 6= 0, if |i− j| = 1 for 0 ≤ i, j ≤ d. We show (A;A; {Ei} d i=0; {E ∗ i } d i=0) is a Leonard system if and only if both (i) there exists a decomposition of V which is split with respect to {Ei} d i=0, {E ∗ i } d i=0; (ii) there exists a decomposition of V which is split with respect to {Ed−i} d i=0, {E ∗ i } d i=0. We also show (A;A ; {Ei} d i=0; {E ∗ i } d i=0) is a Leonard system if and only if both (i) there exists a decomposition of V which is split with respect to {Ei} d i=0, {E ∗ i } d i=0; (ii) there exists an antiautomorphism † of A such that A † = A and A = A.