An LR pair that can be extended to an LR triple

الأصل · EN

Fix an integer d ≥ 0, a field F, and a vector space V over F with dimension d+1. By a decomposition of V we mean a sequence {Vᵢ}ᵢ₌₀ᵈ of 1-dimensional F-subspaces of V such that V = ∑ᵢ₌₀ᵈ Vᵢ (direct sum). Consider F-linear transformations A, B from V to V. Then A,B is called an LR pair whenever there exists a decomposition {Vᵢ}ᵢ₌₀ᵈ of V such that A Vᵢ = Vᵢ₋₁ and B Vᵢ = Vᵢ₊₁ for 0 ≤ i ≤ d, where V₋₁=0 and Vd₊₁=0. By an LR triple we mean a 3-tuple A,B,C of F-linear transformations from V to V such that any two of them form an LR pair. In the present paper, we consider how an LR pair A,B can be extended to an LR triple A,B,C.