## Dissertations

#### Title

The tetrahedron algebra and the hypercube

5-2012

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

Mathematics

#### College

College of Science

#### Department/Unit

Mathematics and Statistics Department

Arlene A. Pascasio

#### Defense Panel Chair

Ederlina G. Nocon

Isagani B. Jos
Edmundo R. Perez

#### Abstract/Summary

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.

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English

Electronic

MS WORD

TG05113

#### Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

#### Physical Description

63 leaves : ill. ; 1 computer optical disc

#### Keywords

Algebra; Hypercube