The Terwilliger Algebras Of Graphs (?)(2D,2) And Some Special Linear Transformations

Terwilliger algebras of distance regular graphs are important problems in algebraic combinatorics.Leonard pairs and Leonard triple are powerful tools to study the ter-williger algebra.This paper consists two parts:1.the structures of Terwilliger algebras of (?)(2D,2),the folded graph of the hypercube (?)(2D,2);2.linear transformations which commute with normalized Leonard triple on V that has Bannai/Ito type.The conclusions are as follows.Let X be vertex set of (?)(2D,2)graph.Fix a vertex x ? X.Let T = T(x)denote the Terwilliger algebra of (?)(2D,2)corresponding to x.Let U(sl2)be the universal enveloping algebra of sl2.First,we get the central elements of T.Then we display a C-algebra homomorphism v:U(sl2)? T using the theory of Leonard pairs.Finally,we show that T is generated by the image of v and some central elements of T.Let K denote a field,and let V denote a vector space over K with finite positive dimension.Let(A,A*,A?)denote a normalized Leonard triple on V that has Bannai//to type.We show that there exist invertible W,W*,W? in V such that(i)A commutes with W and(W-1 A*W-A?);(ii)A*commutes with W*and((W*)-1 A?W*-A);(iii)A? commutes with W? and((W?)1AW?-A*).First,we define A = W W-1A*W-A? and similarly define A*,A?.Then we show that A is a polynomial of A.Moreover each of W,W*,W? is unique up to multiplication by a nonzero scalar in K.