# Combinatorics Seminar 2015

Date: Fri, Sep 11, 2015

Time: 15:00 - 16:00

Venue: Middle Meeting Room

Title: The TD-algebra at $q^2=-1$

Abstract:

The tridiagonal algebra (TD-algebra), which is denoted by $\mathcal {A}_q$ with $q$ the ground parameter, was introduced in the course of studying Terwilliger algebras for (P and Q)-polynomial association schemes. Finite-dimensional irreducible $\mathcal {A}_q$-modules play a key role in the classification project of (P and Q)-polynomial association schemes.

When standardized by affine transformations, the TD-algebra falls into three types: type I if $q^2 \ne \pm 1$, type II if $q^2 = 1$, type III if $q^2= -1$. Finite-dimensional irreducible $\mathcal {A}_q$-modules for type III behave quite differently from those for other types. In this talk, I will determine finite-dimensional irreducible $\mathcal {A}_q$-modules for type III up to isomorphisms by (i) embedding $\mathcal {A}_q$ into the augmented TD-algebra $\mathcal {T}_q$, (ii) defining two kinds of Drinfel'd polynomials for $\mathcal {T}_q$-modules that are constructed as a tensor product of certain sort of evaluation modules and (iii) finding the zeros of the Drinfel'd polynomials. It should be noted that only one kind of Drinfel'd polynomials appear in the case of type I, type II.

Apparently, the augmented TD-algebra $\mathcal {T}_q$ at $q^2=-1$ is related to the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ at $q^2=-1$, but it seems that when $q^2=-1$, $\mathcal {T}_q$ cannot be embedded into $U_q(\widehat{\mathfrak{sl}}_2)$. A question is left: whether there exists a bigger algebra equipped with coproduct in which $\mathcal {T}_q$ at $q^2=-1$ can be embedded.

Slides: View slides

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