[Paper Review] Tridiagonal pairs of $q$-Racah type, the double lowering operator $\psi$, and the quantum algebra $U_q(\mathfrak{sl}_2)$

Sarah Bockting-Conrad|arXiv (Cornell University)|Jul 28, 2013

Algebraic structures and combinatorial models12 references3 citations

TL;DR

This paper establishes a novel connection between tridiagonal pairs of $q$-Racah type and the quantum algebra $U_q(\mathfrak{sl}_2)$ by introducing a double lowering operator $\psi$. Using $\psi$, $K$, and $B$, the authors construct two isomorphic $U_q(\mathfrak{sl}_2)$-module structures on the underlying vector space, show that the Casimir element acts as a scalar in both, and derive a quadratic relation between $K$ and $B$ via rational expressions in $\psi$, $K$, and $B$. The key contribution is a unified algebraic framework linking $q$-Racah tridiagonal pairs to quantum groups through $\psi$.

ABSTRACT

Let \K denote an algebraically closed field and let V denote a vector space over \K with finite positive dimension. We consider an ordered pair of linear transformations A:V o V,A*:V o V that satisfy the following conditions:(i)Each of A,A* is diagonalizable;(ii)there exists an ordering {V_i}_{i=0}^d of the eigenspaces of A such that A*V_i\subseteq V_{i-1}+V_i+V_{i+1} for 0\leq i\leq d, where V_{-1}=0 and V_{d+1}=0;(iii)there exists an ordering {V*_i}_{i=0}^\delta of the eigenspaces of A* such that A V*_i\subseteq V*_{i-1}+V*_i+V*_{i+1} for 0\leq i\leq\delta, where V*_{-1}=0 and V*_{\delta+1}=0;(iv)there does not exist a subspace W of V such that AW\subseteq W,A*W\subseteq W,W eq 0,W eq V. We call such a pair a tridiagonal pair on V. It is known that d=\delta; to avoid trivialities assume d\geq 1. We assume that A,A* belongs to a family of tridiagonal pairs said to have q-Racah type. This is the most general type of tridiagonal pair. Let {U_i}_{i=0}^d and {U_i^\Downarrow}_{i=0}^d denote the first and second split decompositions of V. In an earlier paper we introduced the double lowering operator \psi:V o V. One feature of \psi is that both \psi U_i\subseteq U_{i-1} and \psi U_i^\Downarrow\subseteq U_{i-1}^\Downarrow for 0\leq i\leq d. Define linear transformations K:V o V and B:V o V such that (K-q^{d-2i}I)U_i=0 and (B-q^{d-2i}I)U_i^\Downarrow=0 for 0\leq i\leq d. Our results are summarized as follows. Using \psi,K,B we obtain two actions of Uq(sl2) on V. For each of these Uq(sl2)-module structures, the Chevalley generator e acts as a scalar multiple of \psi. For each of the Uq(sl2)-module structures, we compute the action of the Casimir element on V. We show that these two actions agree. Using this fact, we express \psi as a rational function of K^{\pm 1},B^{\pm 1} in several ways. Eliminating \psi from these equations we find that K,B are related by a quadratic equation.

Motivation & Objective

• To establish a new realization of $U_q(\mathfrak{sl}_2)$-module structures on the vector space of a tridiagonal pair of $q$-Racah type.
• To define and analyze the double lowering operator $\psi$ and its action on the split decompositions $\{U_i\}$ and $\{U^{\Leftarrow}_i\}$.
• To relate the operators $K$ and $B$, defined via eigenvalue conditions on the split decompositions, through rational expressions involving $\psi$.
• To derive a quadratic relation between $K$ and $B$ by eliminating $\psi$ from the rational expressions.
• To show that the two $U_q(\mathfrak{sl}_2)$-module structures induced by $\psi$ are isomorphic and agree on the Casimir element.

Proposed method

• Define the double lowering operator $\psi$ such that $\psi U_i \subseteq U_{i-1}$ and $\psi U^{\Leftarrow}_i \subseteq U^{\Leftarrow}_{i-1}$, with $U_{-1} = U^{\Leftarrow}_{-1} = 0$.
• Introduce operators $K$ and $B$ such that $K$ acts as scalar $q^{d-2i}$ on $U_i$, and $B$ acts as scalar $q^{d-2i}$ on $U^{\Leftarrow}_i$.
• Construct two $U_q(\mathfrak{sl}_2)$-module structures on $V$ using $\psi$ as a scalar multiple of the Chevalley generator $e$.
• Use the relations $\psi = \frac{I - BK^{-1}}{q(aI - a^{-1}BK^{-1})}$ and similar expressions to express $\psi$ rationally in terms of $K$ and $B$.
• Eliminate $\psi$ from the rational expressions to derive the quadratic relation $aK^2 - \frac{aq^{-1} + a^{-1}q}{q - q^{-1}} KB - \frac{aq + a^{-1}q^{-1}}{q - q^{-1}} BK + a^{-1}B^2 = 0$.
• Verify the isomorphism of the two $U_q(\mathfrak{sl}_2)$-module structures by showing they agree on the Casimir element.

Experimental results

Research questions

• RQ1 How can the double lowering operator $\psi$ be used to construct $U_q(\mathfrak{sl}_2)$-module structures on the vector space of a $q$-Racah tridiagonal pair?
• RQ2 What is the relationship between the operators $K$, $B$, and $\psi$ in the context of the split decompositions $\{U_i\}$ and $\{U^{\Leftarrow}_i\}$?
• RQ3 Can the operators $K$ and $B$ be related via a quadratic equation derived from rational expressions in $\psi$?
• RQ4 Do the two $U_q(\mathfrak{sl}_2)$-module structures induced by $\psi$ agree on the Casimir element?
• RQ5 How can the operators $R$, $K$, $B$, and $\psi$ be mutually expressed in both directions between the two module structures?

Key findings

• The double lowering operator $\psi$ acts as a scalar multiple of the Chevalley generator $e$ in both $U_q(\mathfrak{sl}_2)$-module structures on $V$.
• The two $U_q(\mathfrak{sl}_2)$-module structures induced by $\psi$ are isomorphic and agree on the action of the Casimir element.
• The operator $\psi$ is expressed rationally in four equivalent ways involving $K$, $B$, and their inverses, with invertible denominators as proven in Lemma 9.7.
• By equating two rational expressions for $\psi$, the authors derive the quadratic relation $aK^2 - \frac{aq^{-1} + a^{-1}q}{q - q^{-1}} KB - \frac{aq + a^{-1}q^{-1}}{q - q^{-1}} BK + a^{-1}B^2 = 0$.
• The inverse relation $aB^{-2} - \frac{aq^{-1} + a^{-1}q}{q - q^{-1}} K^{-1}B^{-1} - \frac{aq + a^{-1}q^{-1}}{q - q^{-1}} B^{-1}K^{-1} + a^{-1}K^{-2} = 0$ is obtained by inverting the original quadratic relation.
• The operators $K$ and $B$ satisfy the commutation relations $q(K - B)(aK - a^{-1}B) = q^{-1}(aK - a^{-1}B)(K - B)$, which reflect the quantum group structure.