Related papers: Leonard triples of $q$-Racah type

The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $A$, $A^*$ and two relations, called the $q$-Dolan/Grady relations. Recently P. Baseilhac and S. Kolb found an automorphism $L$ of $\mathcal O_q$, that fixes $A$ and sends…

Let $\mathbb{k}$ be a field, and let $\Lambda$ be a (not necessarily finite dimensional) $\mathbb{k}$-algebra. Let $V$ be a left $\Lambda$-module such that is finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weak…

Let $\mathbf{k}$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra, and let $\widehat{\Lambda}$ be the repetitive algebra of $\Lambda$. For the stable category of finitely generated left…

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the…

Fix an algebraically closed field $\mathbb{F}$ and an integer $n \geq 1$. Let $\text{Mat}_n(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $n \times n$ matrices that have all entries in $\mathbb{F}$. We consider a pair of…

Given coprime positive integers $d',d''$, B\'ezout's Lemma tells us that there are integers $u,v$ so that $d'u-d''v=1$. We show that, interchanging $d'$ and $d''$ if necessary, we may choose $u$ and $v$ to be Loeschian numbers, i.e., of the…

We prove that the subvariety of $SL(2)\times SL(2)$ given by the matrix equation $w(X,Y)=\alpha$, where $w$ is a word in two letters, is closely related to an explicit smooth conic bundle over the associated `trace surface' in the…

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic $p \ge 0$ with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (i) There exists a…

We generalize Bonahon-Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, given a non-zero complex parameter $q=e^{2 \pi i \hbar}$, we associate to each isotopy class of framed…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. By a Leonard pair on $V$ we mean an ordered pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following two…

Let $\F$ denote a field, and let $V$ denote a vector space over $\F$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\F$-linear maps $A: V \to V$ and $A^* : V \to V$ that each act on an eigenbasis…

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite…

We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The…

The positive part $U^+_q$ of $U_q(\hat{\mathfrak{sl}}_2)$ has a presentation with two generators $W_0$, $W_1$ and two relations called the $q$-Serre relations. The algebra $U^+_q$ contains some elements, said to be alternating. There are…

Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\mathbb F_q$ and $E\subset \mathbb F_q^d$, we define $W(r)$ as the number of quadruples $(x,y,z,w)\in…

Let $\mathfrak{A}$ and $\mathfrak{B}$ be JBW$^*$-algebras whose sets of unitaries are denoted by $\mathcal{U}(\mathfrak{A})$ and $\mathcal{U}(\mathfrak{B})$, respectively. We show that $\mathcal{U}(\mathfrak{A})$ is closed for Jordan…

This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of…

Let $\mathbb F$ denote an algebraically closed field and assume that $q\in \mathbb F$ is a primitive $d^{\rm \, th}$ root of unity with $d\not=1,2,4$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb…

Assume that $\mathbb F$ is an algebraically closed field and $q$ is a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra generated by…