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Let D denote a positive integer and let Q _D denote the graph of the D-dimensional hypercube. Let X denote the vertex set of Q _D and let A Ĩ Mat _X(\mathbb C) A\in {\rm Mat}_{X}({\mathbb{C}}) denote the adjacency matrix of Q _D . Fix x\in  X and let A ^* Ĩ Mat _X(\mathbb C) A^{*}\in {\rm Mat}_{X}({\mathbb{C}}) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat _X(\mathbb C) {\rm Mat}_{X}({\mathbb{C}}) generated by A, A ^*. We refer to T as the Terwilliger algebra of Q _D with respect to x. The matrices A and A ^* are related by the fact that 2 i A= A ^* A ^ϵ   -  A ^ϵ  A ^* and 2 i A ^*= A ^ϵ  A -  AA ^ϵ  , where 2 i A ^ϵ  = AA ^* -  A ^* A and i ²= - 1.

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