The tetrahedron algebra and the hypercube

• Main Author: Penaflor, Randy Abo
• Format: text
• Language: English
• Published: Animo Repository 2012
• Subjects: Algebra; Hypercube; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/957; https://animorepository.dlsu.edu.ph/context/etd_doctoral/article/1956/viewcontent/CDTG005113_P.pdf
• Institution: De La Salle University

Let QD denote the graph of the D-dimensional hypercube where D is a positive integer. Let X denote the vertex set of QD. Let MatX(C) denote the C-algebra of matrices with entries in C and whose rows and columns are indexed by X. Let A denote the adjacency matrix of QD and let 0 > 1 > > D denote the eigenvalues of A. Fix a vertex x 2 X. For 0 i D, let Ei (resp. E i = E i (x)) denote the primitive idempotent (ith dual idempotent with respect to x) of QD. Let A = A (x) denote the dual adjacency matrix of QD and let 0 > 1 > > D denote the dual eigenvalues of A . Let V = CX. For 0 i D, we de ne Ui := (E 0V + E 1V + + E i V ) \ (EiV + Ei+1V + + EDV ) where EiV (resp. E i V ) is the eigenspace associated with the eigenvalue i (resp. dual eigenvalue i ). We show that V = U0 + U1 + + UD (direct sum). We give a basis for Ui (0 i D). We give the action of A and A on this basis. We de ne B := A + A 1 2 (AA A A). We show that for 0 i D, Ui is the eigenspace of B associated with the eigenvalue i. We display a Lie algebra isomorphism from a Lie subalgebra of MatX(C) to sl2(C), where sl2(C) denotes the Lie subalgebra of Mat2(C) consisting of 2 2 matrices whose trace is 0. Using this, we display an action of the tetrahedron Lie algebra on V.