The Generalized Terwilliger Algebra and its Finite-dimensional Modules when d=2

Abstract

Terwilliger [J. Algebraic Combin. 1 (1992), considered the -algebra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M * . This algebra is now known as the Terwilliger algebra and is usually denoted by T . Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T . These relations are often called the triple product relations. They determine much of the structure of T , though not all of it in general. To illuminate the role these relations play, the current author introduced [J. Algebra 233 (2000), 213-252] a generalization of T . To go from T to , we replace M and M * with a pair of dual character algebras C and C * . The dimensions of C and C * are equal; let d + 1 denote this common dimension. Intuitively, is the associative -algebra with identity generated by C and C * subject to the analogues of Terwilliger's triple product relations. is infinite-dimensional and noncommutative in general. In this paper we study and its finite-dimensional modules when d = 2 and has no "extra" vanishing intersection numbers or dual intersection numbers. In this case we show is -algebra isomorphic to M 3 ⊕ , where M 3 denotes the -algebra consisting of all 3-by-3 matrices with entries in and denotes the associative -algebra with identity generated by the symbols e and f subject to the relations e 2 = e and f 2 = f . We find a basis for and we determine the center of . We classify the finite-dimensional indecomposable -modules up to isomorphism. There are four such -modules in every odd dimension, and in every even dimension these modules are parameterized by a single complex number. We also classify the finite-dimensional irreducible -modules up to isomorphism. Using our results concerning , we find a basis for , we describe the center of , and we classify both the finite-dimensional indecomposable and the finite-dimensional irreducible -modules up to isomorphism.  2002 Elsevier Science (USA)

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

“…Now consider the left-hand side of (20). Replacing g i in this expression using (19), and eliminating λp i , λg i−2 in the result using ( 12), (22), respectively, we find…”

“…Since T is semi-simple, each T-module is a direct sum of irreducible T-modules. Describing the irreducible T-modules is an active area of research [4][5][6][7][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23][24]26,[28][29][30][31][32][33][34][35][36].…”

Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra

“…We call T the Terwilliger algebra (or subconstituent algebra) of Γ with respect to x. We refer the reader to [1,3,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,24] for more information on the Terwilliger algebra. We will use the following facts.…”

The Terwilliger algebra of a distance-regular graph of negative type

“…Therefore each T -module is a direct sum of irreducible T -modules. Describing the irreducible T -modules is an active area of research [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], [21,26,28,31].…”