Representation Theory Of Motzkin Algebras Of Type D

Motzkin algebras were first introduced by Georgia Benkart and Tom Halverson in 2014.They are diagram algebras and a generalization of Temperley-Lieb algebras of type A.Temperley-Lieb algebras are finite dimensional associative algebras,which are closely related to topological quantum field theory,statistical physics,functional theory,algebraic combinatorics and Lie theory.Until now,they also have many important applications.In this paper,we will generalize Motzkin algebras and define a class of new diagram algebras.These algebras can also be regarded as a generalization of Temperley-Lieb algebras of type D,which we call Motzkin algebras of type D.We will focus on the study of type D Motzkin algebras by the diagrammatic approach.More exactly,we first give a diagrammatic definition,and prove that every base element in Motzkin algebra of type D can be written as a finite product of several kinds of diagrams.Secondly,we prove that these algebras are cellular by constructing cellular bases for them.As byproducts,we get a dimension formula and prove that a Motzkin algebra of type D is quasi-heredity algebra when the parameter is non-zero.Finally,we study Gram matrices of several modules by restriction.