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The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. In 2019, Baseilhac and Kolb introduced two automorphisms of $O_q$, now called the Lusztig automorphisms. Recently, we introduced a generalization of $O_q$ called the $S_3$-symmetric $q$-Onsager algebra $\mathbb O_q$. The algebra $\mathbb O_q$ has six distinguished generators, said to be standard. The standard $\mathbb O_q$-generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy the $q$-Dolan/Grady relations. In the present paper we do the following: (i) for each standard $\mathbb O_q$-generator we construct an automorphism of $\mathbb O_q$ called a Lusztig automorphism; (ii) we describe how the six Lusztig automorphisms of $\mathbb O_q$ are related to each other; (iii) we describe what happens if a finite-dimensional irreducible $\mathbb O_q$-module is twisted by a Lusztig automorphism; (iv) we give a detailed example involving an irreducible $\mathbb O_q$-module with dimension 5., Comment: 24 pages

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 1$. For a vertex $x$ of $\Gamma$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $\Gamma$ and the dual adjacency matrix $A^*=A^*(x)$ of $\Gamma$ with respect to $x$. We introduce a $T$-module $\mathcal N = \mathcal N(x)$ called the nucleus of $\Gamma$ with respect to $x$. We describe $\mathcal N$ from various points of view. We show that all the irreducible $T$-submodules of $\mathcal N$ are thin. Under the assumption that $\Gamma$ is a nonbipartite dual polar graph, we give an explicit basis for $\mathcal N$ and the action of $A, A^*$ on this basis. The basis is in bijection with the set of elements for the projective geometry $L_D(q)$, where $GF(q)$ is the finite field used to define $\Gamma$., Comment: 37 pages, a few typos corrected

In 2023 we obtained a $Q$-polynomial structure for the projective geometry $L_N(q)$. In the present paper, we display a more general $Q$-polynomial structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a free parameter $\varphi$ that takes any positive real value. For $\varphi=1$ we recover the original $Q$-polynomial structure. We interpret the new $Q$-polynomial structure using the quantum group $U_{q^{1/2}}(\mathfrak{sl}_2)$ in the equitable presentation. We use the new $Q$-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of $Q$-polynomial distance-regular graphs., Comment: 30 pages

The tridiagonal algebra is defined by two generators and two relations, called the tridiagonal relations. Special cases of the tridiagonal algebra include the $q$-Onsager algebra, the positive part of the $q$-deformed enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$, and the enveloping algebra of the Onsager Lie algebra. In this paper, we introduce the $S_3$-symmetric tridiagonal algebra. This algebra has six generators. The generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy a pair of tridiagonal relations. For a $Q$-polynomial distance-regular graph $\Gamma$ we turn the tensor power $V^{\otimes 3}$ of the standard module $V$ into a module for an $S_3$-symmetric tridiagonal algebra. We investigate in detail the case in which $\Gamma$ is a Hamming graph. We give some conjectures and open problems., Comment: 32 pages

Let ${\mathcal X} = (X, \{R_i\}_{i=0}^d)$ denote a symmetric association scheme. Fix an ordering $\{E_i\}_{i=0}^d$ of the primitive idempotents of $\mathcal{X}$, and let $P$ (resp.\ $Q$) denote the corresponding first eigenmatrix (resp.\ second eigenmatrix) of $\mathcal X$. The scheme $\mathcal X$ is said to be formally self-dual (with respect to the ordering $\{E_i\}_{i=0}^d$) whenever $P=Q$. We define $\mathcal X$ to be numerically self-dual (with respect to the ordering $\{E_i\}_{i=0}^d$) whenever the intersection numbers and Krein parameters satisfy $p^h_{i,j} =q^h_{i,j}$ for $0 \leq h,i,j \leq d$. It is known that with respect to the ordering $\{E_i\}_{i=0}^d$, formal self-duality implies numerical self-duality. This raises the following question: is it possible that with respect to the ordering $\{E_i\}_{i=0}^d$, $\mathcal X$ is numerically self-dual but not formally self-dual? This is possible as we will show. We display an example of a symmetric association scheme and an ordering the primitive idempotents with respect to which the scheme is numerically self-dual but not formally self-dual. We have the following additional results about self-duality. Assume that $\mathcal X$ is $P$-polynomial. We show that the following are equivalent: (i) $\mathcal X$ is formally self-dual with respect to the ordering $\{E_i\}_{i=0}^d$; (ii) $\mathcal X$ is numerically self-dual with respect to the ordering $\{E_i\}_{i=0}^d$. Assume that the ordering $\{E_i\}_{i=0}^d$ is $Q$-polynomial. We show that the following are equivalent: (i) $\mathcal X$ is formally self-dual with respect to the ordering $\{E_i\}_{i=0}^d$; (ii) $\mathcal X$ is numerically self-dual with respect to the ordering $\{E_i\}_{i=0}^d$.

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. The standard module $V$ has a basis $\lbrace {\hat x} \vert x \in X\rbrace$, where ${\hat x}$ denotes column $x$ of the identity matrix $I \in {\rm Mat}_X(\mathbb C)$. Let $E$ denote a $Q$-polynomial primitive idempotent of $\Gamma$. The eigenspace $EV$ is spanned by the vectors $\lbrace E {\hat x} \vert x \in X\rbrace$. It was previously known that these vectors satisfy a condition called the balanced set condition. In this paper, we introduce a variation on the balanced set condition called the Norton-balanced condition. The Norton-balanced condition involves the Norton algebra product on $EV$. We define $\Gamma$ to be Norton-balanced whenever $\Gamma$ has a $Q$-polynomial primitive idempotent $E$ such that the set $\lbrace E {\hat x} \vert x \in X\rbrace$ is Norton-balanced. We show that $\Gamma$ is Norton-balanced in the following cases: (i) $\Gamma$ is bipartite; (ii) $\Gamma$ is almost bipartite; (iii) $\Gamma$ is dual-bipartite; (iv) $\Gamma$ is almost dual-bipartite; (v) $\Gamma$ is tight; (vi) $\Gamma$ is a Hamming graph; (vii) $\Gamma$ is a Johnson graph; (viii) $\Gamma$ is the Grassmann graph $J_q(2D,D)$; (ix) $\Gamma$ is a halved bipartite dual-polar graph; (x) $\Gamma$ is a halved Hemmeter graph; (xi) $\Gamma$ is a halved hypercube; (xii) $\Gamma$ is a folded-half hypercube; (xiii) $\Gamma$ has $q$-Racah type and affords a spin model. Some theoretical results about the Norton-balanced condition are obtained, and some open problems are given., Comment: 79 pages. Added a coauthor and made some small adjustments

Let $\Gamma$ denote a distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. We assume that $\Gamma$ is formally self-dual and $q$-Racah type. We also assume that for each $x \in X$ the subconstituent algebra $T=T(x)$ contains a certain central element $Z=Z(x)$. We use $Z$ to construct a spin model $\sf W$ afforded by $\Gamma$. We investigate the combinatorial implications of $Z$. We reverse the logical direction and recover $Z$ from $\sf W$. We finish with some open problems., Comment: 48 pages

The goal of this article is to display a $Q$-polynomial structure for the Attenuated Space poset $\mathcal A_q(N,M)$. The poset $\mathcal A_q(N,M)$ is briefly described as follows. Start with an $(N+M)$-dimensional vector space $H$ over a finite field with $q$ elements. Fix an $M$-dimensional subspace $h$ of $H$. The vertex set $X$ of $\mathcal A_q(N,M)$ consists of the subspaces of $H$ that have zero intersection with $h$. The partial order on $X$ is the inclusion relation. The $Q$-polynomial structure involves two matrices $A, A^* \in {\rm Mat}_X(\mathbb C)$ with the following entries. For $y, z \in X$ the matrix $A$ has $(y,z)$-entry $1$ (if $y$ covers $z$); $q^{{\rm dim}\,y}$ (if $z$ covers $y$); and 0 (if neither of $y,z$ covers the other). The matrix $A^*$ is diagonal, with $(y,y)$-entry $q^{-{\rm dim}\,y}$ for all $y\in X$. By construction, $A^*$ has $N+1$ eigenspaces. By construction, $A$ acts on these eigenspaces in a (block) tridiagonal fashion. We show that $A$ is diagonalizable, with $2N+1$ eigenspaces. We show that $A^*$ acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that $A$ is $Q$-polynomial. We show that $A, A^*$ satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra $T$ of ${\rm Mat}_X(\mathbb C)$ generated by $A, A^*$. We show that $A,A^*$ act on each irreducible $T$-module as a Leonard pair., Comment: 23 pages

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra $\mathcal A$. The algebra $\mathcal A$ is associative, noncommutative, and infinite-dimensional. It is defined by two generators $A,B$ and two relations called the down-up relations. In the present paper, we introduce the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A$. We define $\mathbb A$ by generators and relations. There are three generators $A,B,C$ and any two of these satisfy the down-up relations. We describe how $\mathbb A$ is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, the $\mathfrak{sl}_3$ loop algebra, the Kac-Moody Lie algebra $A^{(1)}_2$, the $q$-Weyl algebra, the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$, and the quantized enveloping algebra $U_q (A^{(1)}_2)$. We give some open problems and conjectures., Comment: 31 pages. In Memory of Georgia Benkart (1947--2022)

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 for the other in an irreducible tridiagonal fashion. Let $A,A^*$ denote a Leonard pair on $V$. Let $\{v_i\}_{i=0}^d$ denote an eigenbasis for $A^*$ on which $A$ acts in an irreducible tridiagonal fashion. For $0 \leq i \leq d$ define an $\F$-linear map $E^*_i : V \to V$ such that $E^*_i v_i = v_i$ and $E^*_i v_j = 0$ if $j \neq i$ $(0 \leq j \leq d)$. The map $F = \sum_{i=0}^d E^*_i A E^*_i$ is called the flat part of $A$. The Leonard pair $A,A^*$ is bipartite whenever $F=0$. The Leonard pair $A,A^*$ is said to be near-bipartite whenever the pair $A-F, A^*$ is a Leonard pair on $V$. In this case, the Leonard pair $A-F, A^*$ is bipartite, and called the bipartite contraction of $A,A^*$. Let $B,B^*$ denote a bipartite Leonard pair on $V$. By a near-bipartite expansion of $B,B^*$ we mean a near-bipartite Leonard pair on $V$ with bipartite contraction $B,B^*$. In the present paper we have three goals. Assuming $\F$ is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over $\F$; (ii) for each near-bipartite Leonard pair over $\F$ we describe its bipartite contraction; (iii) for each bipartite Leonard pair over $\F$ we describe its near-bipartite expansions.

A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero. Let $\mathbb F$ denote a field, and let $V$ denote a nonzero finite-dimensional vector space over $\mathbb F$. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*: V \to V$ that satisfy the following two conditions: (i) there exists a basis for $V$ with respect to which the matrix representing $A$ is circular bidiagonal and the matrix representing $A^*$ is diagonal; (ii) there exists a basis for $V$ with respect to which the matrix representing $A^*$ is circular bidiagonal and the matrix representing $A$ is diagonal. We call such a pair a circular bidiagonal pair on $V$. We classify the circular bidiagonal pairs up to affine equivalence. There are two infinite families of solutions, which we describe in detail., Comment: 30 pages

There is a type of distance-regular graph, said to be $Q$-polynomial. In this paper we investigate a generalized $Q$-polynomial property involving a graph that is not necessarily distance-regular. We give a detailed description of an example associated with the projective geometry $L_N(q)$., Comment: 21 pages

This survey paper contains a tutorial introduction to distance-regular graphs, with an emphasis on the subconstituent algebra and the $Q$-polynomial property., Comment: 60 pages; typos corrected

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 four kinds of alternating elements, denoted $\lbrace W_{-k}\rbrace_{k\in \mathbb N}$, $\lbrace W_{k+1}\rbrace_{k\in \mathbb N}$, $\lbrace G_{k+1}\rbrace_{k\in \mathbb N}$, $\lbrace {\tilde G}_{k+1}\rbrace_{k \in \mathbb N}$. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps $A, A^*$ on a nonzero, finite-dimensional vector space $V$, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let $A$, $A^*$ denote a tridiagonal pair on $V$. Associated with this pair are six well-known direct sum decompositions of $V$; these are the eigenspace decompositions of $A$ and $A^*$, along with four decompositions of $V$ that are often called split. In our main results, we assume that $A$, $A^*$ has $q$-Serre type. Under this assumption $A$, $A^*$ satisfy the $q$-Serre relations, and $V$ becomes an irreducible $U^+_q$-module on which $W_0=A$ and $W_1=A^*$. We describe how the alternating elements of $U^+_q$ act on the above six decompositions of $V$. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of $A$ and $A^*$ all have dimension one. In the second case $A$ and $A^*$ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual., Comment: 39 pages, 9 diagrams

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 positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A : V \to V$ and $A^* : V \to V$ that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r$. For a Leonard pair $A,A^*$ on $V$, by a companion of $A,A^*$ we mean an $\mathbb{F}$-linear map $K: V \to V$ such that $K$ is a polynomial in $A^*$ and $A-K, A^*$ is a Leonard pair on $V$. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs $A,A^*$ and $B,B^*$ on $V$, define $K = A-B$. Then $K$ is a companion of $A,A^*$. For a Leonard pair $A,A^*$ on $V$ and a companion $K$ of $A,A^*$, define $B = A-K$ and $B^* = A^*$. Then $B,B^*$ is a Leonard pair on $V$ that is compatible with $A,A^*$. Let $A,A^*$ denote a Leonard pair on $V$. We find all the Leonard pairs $B, B^*$ on $V$ that are compatible with $A,A^*$. For each solution $B, B^*$ we describe the corresponding companion $K = A-B$., Comment: 65 pages

We consider the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ and its basic module $V(\Lambda_0)$. This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe $V(\Lambda_0)$ using a $q$-shuffle algebra in the following way. Start with the free associative algebra $\mathbb V$ on two generators $x,y$. The standard basis for $\mathbb V$ consists of the words in $x,y$. In 1995 M. Rosso introduced an associative algebra structure on $\mathbb V$, called a $q$-shuffle algebra. For $u,v\in \lbrace x,y\rbrace$ their $q$-shuffle product is $u\star v = uv+q^{(u,v)}vu$, where $( u,v) =2$ (resp. $(u,v) =-2$) if $u=v$ (resp. $u\not=v$). Let $\mathbb U$ denote the subalgebra of the $q$-shuffle algebra $\mathbb V$ that is generated by $x, y$. Rosso showed that the algebra $\mathbb U$ is isomorphic to the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$. In our first main result, we turn $\mathbb U$ into a $U_q(\widehat{\mathfrak{sl}}_2)$-module. Let $\bf U$ denote the $U_q(\widehat{\mathfrak{sl}}_2)$-submodule of $\mathbb U$ generated by the empty word. In our second main result, we show that the $U_q(\widehat{\mathfrak{sl}}_2)$-modules $\bf U$ and $V(\Lambda_0)$ are isomorphic. Let $\bf V$ denote the subspace of $\mathbb V$ spanned by the words that do not begin with $y$ or $xx$. In our third main result, we show that $\bf U = \mathbb U \cap {\bf V}$., Comment: 35 pages

We consider the positive part $U^+_q$ of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation involving two generators and two relations, called the $q$-Serre relations. There is a PBW basis for $U^+_q$ due to Damiani, and a PBW basis for $U^+_q$ due to Beck. In 2019 we used Catalan words and a $q$-shuffle algebra to express the Damiani PBW basis in closed form. In this paper we use a similar approach to express the Beck PBW basis in closed form. We also consider how the Damiani PBW basis and the Beck PBW basis are related to the alternating PBW basis for $U^+_q$., Comment: 22 pages. arXiv admin note: text overlap with arXiv:1902.00721

Let $U^+_q$ denote the positive part of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Serre relations. In 1993 I. Damiani obtained a PBW basis for $U^+_q$, consisting of some elements $\lbrace E_{n \delta+ \alpha_0} \rbrace_{n=0}^\infty$, $\lbrace E_{n \delta+ \alpha_1} \rbrace_{n=0}^\infty$, $\lbrace E_{n \delta} \rbrace_{n=1}^\infty$. In 2019 we introduced the alternating central extension $\mathcal U^+_q$ of $U^+_q$. We defined $\mathcal U^+_q$ by generators and relations. The generators, said to be alternating, are denoted $\lbrace \mathcal W_{-k}\rbrace_{k=0}^\infty$, $\lbrace \mathcal W_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace \mathcal G_{k+1}\rbrace_{k=0}^\infty$, $\lbrace \mathcal {\tilde G}_{k+1}\rbrace_{k=0}^\infty$. Let $\langle \mathcal W_0, \mathcal W_1 \rangle$ denote the subalgebra of $\mathcal U^+_q$ generated by $\mathcal W_0$, $\mathcal W_1$. It is known that there exists an algebra isomorphism $U^+_q\to \langle \mathcal W_0, \mathcal W_1 \rangle$ that sends $W_0 \mapsto \mathcal W_0$ and $W_1 \mapsto \mathcal W_1$. Via this isomorphism we identify $U^+_q$ with $\langle \mathcal W_0, \mathcal W_1 \rangle$. In our main result, we express the Damiani PBW basis elements in terms of the alternating generators. We give the answer in terms of generating functions., Comment: 22 pages. arXiv admin note: text overlap with arXiv:2106.14041

The $q$-Onsager algebra $O_q$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. The alternating central extension $\mathcal O_q$ has a presentation involving the alternating generators $\lbrace \mathcal W_{-k}\rbrace_{k=0}^\infty$, $\lbrace \mathcal W_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace \mathcal G_{k+1}\rbrace_{k=0}^\infty$, $\lbrace \mathcal {\tilde G}_{k+1}\rbrace_{k=0}^\infty$ and a large number of relations. Let $\langle \mathcal W_0, \mathcal W_1 \rangle$ denote the subalgebra of $\mathcal O_q$ generated by $\mathcal W_0$, $\mathcal W_1$. It is known that there exists an algebra isomorphism $O_q \to \langle \mathcal W_0, \mathcal W_1 \rangle$ that sends $W_0\mapsto \mathcal W_0$ and $W_1 \mapsto \mathcal W_1$. It is known that the center $\mathcal Z$ of $\mathcal O_q$ is isomorphic to a polynomial algebra in countably many variables. It is known that the multiplication map $\langle \mathcal W_0, \mathcal W_1 \rangle \otimes \mathcal Z \to \mathcal O_q$, $ w \otimes z \mapsto wz$ is an isomorphism of algebras. We call this isomorphism the standard tensor product factorization of $\mathcal O_q$. In the study of $\mathcal O_q$ there are two natural points of view: we can start with the alternating generators, or we can start with the standard tensor product factorization. It is not obvious how these two points of view are related. The goal of the paper is to describe this relationship. We give seven main results; the principal one is an attractive factorization of the generating function for some algebraically independent elements that generate $\mathcal Z$., Comment: 38 pages

The Onsager Lie algebra $O$ is often used to study integrable lattice models. The universal enveloping algebra of $O$ admits a $q$-deformation $O_q$ called the $q$-Onsager algebra. Recently, an algebra $\mathcal O_q$ was introduced called the alternating central extension of $O_q$. In this paper we introduce a Lie algebra $\mathcal O$ that is roughly described by the following two analogies: (i) $\mathcal O$ is to $O$ as $\mathcal O_q$ is to $O_q$; (ii) $O_q$ is to $O$ as $\mathcal O_q$ is to $\mathcal O$. We call $\mathcal O$ the alternating central extension of $O$. This paper contains a comprehensive description of $\mathcal O$., Comment: 24 pages

The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. We investigate the alternating central extension $\mathcal O_q$ of $O_q$. The algebra $\mathcal O_q$ was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal O_q$ that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal O_q$. This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$., Comment: 26 pages. arXiv admin note: text overlap with arXiv:2103.03028

The $q$-Onsager algebra $O_q$ is presented by two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra $\mathcal A_q$ for $O_q$. Soon afterwards, Baseilhac and Shigechi gave a presentation of $\mathcal A_q$ by generators and relations. We show that these generators give a PBW basis for $\mathcal A_q$. Using this PBW basis, we show that the algebra $\mathcal A_q$ is isomorphic to $O_q \otimes \mathbb F \lbrack z_1, z_2, \ldots \rbrack$, where $\mathbb F$ is the ground field and $\lbrace z_n \rbrace_{n=1}^\infty $ are mutually commuting indeterminates. Recall the positive part $U^+_q$ of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. Our results show that $O_q$ is related to $\mathcal A_q$ in the same way that $U^+_q$ is related to the alternating central extension of $U^+_q$. For this reason, we propose to call $\mathcal A_q$ the alternating central extension of $O_q$., Comment: 58 pages

The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $W_0, W_1$ and two relations called the $q$-Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for $\mathcal O_q$ with elements denoted $\lbrace B_{n \delta+ \alpha_0} \rbrace_{n=0}^\infty, \lbrace B_{n \delta+ \alpha_1} \rbrace_{n=0}^\infty, \lbrace B_{n \delta} \rbrace_{n=1}^\infty $. In their recent study of a current algebra $\mathcal A_q$, Baseilhac and Belliard conjecture that there exist elements $\lbrace W_{-k}\rbrace_{k=0}^\infty, \lbrace W_{k+1}\rbrace_{k=0}^\infty, \lbrace G_{k+1} \rbrace_{k=0}^\infty, \lbrace {\tilde G}_{k+1} \rbrace_{k=0}^\infty$ in $\mathcal O_q$ that satisfy the defining relations for $\mathcal A_q$. In order to establish this conjecture, it is desirable to know how the elements in the second list above are related to the elements in the first list above. In the present paper, we conjecture the precise relationship and give some supporting evidence. This evidence consists of some computer checks on SageMath due to Travis Scrimshaw, and a proof of our conjecture for a homomorphic image of $\mathcal O_q$ called the universal Askey-Wilson algebra., Comment: 25 pages, updated Intro, references added

This paper concerns the positive part $U^+_q$ of the quantum group $U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U^+_q$ has a presentation involving two generators that satisfy the cubic $q$-Serre relations. We recently introduced an algebra $\mathcal U^+_q$ called the alternating central extension of $U^+_q$. We presented $\mathcal U^+_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal U^+_q$ that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal U^+_q$., Comment: 23 pages

We recently introduced the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. There is a type of idempotent system, said to be symmetric. In the present paper we classify up to isomorphism the idempotent systems and the symmetric idempotent systems. We also describe how symmetric idempotent systems are related to character algebras.

We consider the Norton algebra associated with a $Q$-polynomial primitive idempotent of the adjacency matrix for a distance-regular graph. We obtain a formula for the Norton algebra product that we find attractive., Comment: 11 pages

Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*:V\to V$ such that (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ 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 $\lbrace V^*_i\rbrace_{i=0}^{\delta}$ 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 $U$ of $V$ such that $AU\subseteq U$, $A^*U \subseteq U$, $U\not=0$, $U\not=V$. We call such a pair a tridiagonal pair on $V$. We assume that $A, A^*$ belongs to a family of tridiagonal pairs said to have $q$-Racah type. There is an infinite-dimensional algebra $\boxtimes_q$ called the $q$-tetrahedron algebra; it is generated by four copies of $U_q(\mathfrak{sl}_2)$ that are related in a certain way. Using $A, A^*$ we construct two $\boxtimes_q$-module structures on $V$. In this construction the two main ingredients are the double lowering map $\psi:V\to V$ due to Sarah Bockting-Conrad, and a certain invertible map $W:V\to V$ motivated by the spin model concept due to V. F. R. Jones., Comment: 37 pages

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 $A^*$ to a linear combination of $A^*$, $A^2A^*$, $AA^*A$, $A^*A^2$. Let $V$ denote an irreducible $\mathcal O_q$-module of finite dimension at least two, on which each of $A$, $A^*$ is diagonalizable. It is known that $A$, $A^*$ act on $V$ as a tridiagonal pair of $q$-Racah type, giving access to four familiar elements $K$, $B$, $K^\downarrow$, $B^\downarrow$ in ${\rm End}(V)$ that are used to compare the eigenspace decompositions for $A$, $A^*$ on $V$. We display an invertible $H \in {\rm End}(V)$ such that $L(X)=H^{-1} X H$ on $V$ for all $X \in \mathcal O_q$. We describe what happens when one of $K$, $B$, $K^\downarrow$, $B^\downarrow$ is conjugated by $H$. For example $H^{-1}KH=a^{-1}A-a^{-2}K^{-1}$ where $a$ is a certain scalar that is used to describe the eigenvalues of $A$ on $V$. We use the conjugation results to compare the eigenspace decompositions for $A$, $A^*$, $L^{\pm 1}(A^*)$ on $V$. In this comparison we use the notion of an equitable triple; this is a 3-tuple of elements in ${\rm End}(V)$ such that any two satisfy a $q$-Weyl relation. Our comparison involves eight equitable triples. One of them is $a A - a^2 K$, $M^{-1}$, $K$ where $M= (a K-a^{-1} B)(a-a^{-1})^{-1}$. The map $M$ appears in earlier work of S. Bockting-Conrad concerning the double lowering operator $\psi$ of a tridiagonal pair., Comment: 28 pages

In this paper we introduce the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. We focus on a family of idempotent systems, said to be symmetric. A symmetric idempotent system is an abstraction of the primary module for the subconstituent algebra of a symmetric association scheme. We describe the symmetric idempotent systems in detail. We also consider a class of symmetric idempotent systems, said to be $P$-polynomial and $Q$-polynomial. In the topic of orthogonal polynomials there is an object called a Leonard system. We show that a Leonard system is essentially the same thing as a symmetric idempotent system that is $P$-polynomial and $Q$-polynomial.

Around 2001 we classified the Leonard systems up to isomorphism. The proof was lengthy and involved considerable computation. In this paper we give a proof that is shorter and involves minimal computation. We also give a comprehensive descriptionof the intersection numbers of a Leonard system., Comment: 57 pages

Recently Sarah Bockting-Conrad introduced the double lowering operator $\psi$ for a tridiagonal pair. Motivated by $\psi$ we consider the following problem about polynomials. Let $\mathbb F$ denote an algebraically closed field. Let $x$ denote an indeterminate, and let $\mathbb F\lbrack x \rbrack$ denote the algebra consisting of the polynomials in $x$ that have all coefficients in $\mathbb F$. Let $N$ denote a positive integer or $\infty$. Let $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ denote scalars in $\mathbb F$ such that $\sum_{h=0}^{i-1} a_h \not= \sum_{h=0}^{i-1} b_h$ for $1 \leq i \leq N$. For $0 \leq i \leq N$ define polynomials $\tau_i, \eta_i \in \mathbb F\lbrack x \rbrack$ by $\tau_i = \prod_{h=0}^{i-1} (x-a_h)$ and $\eta_i = \prod_{h=0}^{i-1} (x-b_h)$. Let $V$ denote the subspace of $\mathbb F\lbrack x \rbrack$ spanned by $\lbrace x^i\rbrace_{i=0}^N$. An element $\psi \in \operatorname{End}(V)$ is called double lowering whenever $\psi \tau_i \in \mathbb F \tau_{i-1}$ and $\psi \eta_i \in \mathbb F \eta_{i-1}$ for $0 \leq i \leq N$, where $\tau_{-1}=0$ and $\eta_{-1}=0$. We give necessary and sufficient conditions on $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.

Let $\Gamma$ be a $Q$-polynomial distance-regular graph of diameter $d\geq 3$. Fix a vertex $\gamma$ of $\Gamma$ and consider the subgraph induced on the union of the last two subconstituents of $\Gamma$ with respect to $\gamma$. We prove that this subgraph is connected., Comment: Journal of Combinatorial Theory, Series A, accepted for publication, 6 pages

This paper is about the positive part $U^+_q$ of the quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. Recently we introduced a type of element in $U^+_q$, said to be alternating. Each alternating element commutes with exactly one of $A$, $B$, $qBA-q^{-1}AB$, $qAB-q^{-1}BA$; this gives four types of alternating elements. There are infinitely many alternating elements of each type, and these mutually commute. In the present paper we use the alternating elements to obtain a central extension $\mathcal U^+_q$ of $U^+_q$. We define $\mathcal U^+_q$ by generators and relations. These generators, said to be alternating, are in bijection with the alternating elements of $U^+_q$. We display a surjective algebra homomorphism $\mathcal U^+_q \to U^+_q$ that sends each alternating generator of $\mathcal U^+_q$ to the corresponding alternating element in $U^+_q$. We adjust this homomorphism to obtain an algebra isomorphism $\mathcal U_q^+ \to U^+_q \otimes \mathbb F \lbrack z_1, z_2,\ldots\rbrack$ where $\mathbb F$ is the ground field and $\lbrace z_n\rbrace_{n=1}^\infty$ are mutually commuting indeterminates. We show that the alternating generators form a PBW basis for $\mathcal U_q^+$. We discuss how $\mathcal U^+_q$ is related to the work of Baseilhac, Koizumi, Shigechi concerning the $q$-Onsager algebra and integrable lattice models., Comment: 26 pages

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard pair, said to have spin. The notion of a spin model was introduced by V.F.R. Jones to construct link invariants. A spin model is a symmetric matrix over $\mathbb{C}$ that satisfies two conditions, called the type II and type III conditions. It is known that a spin model $\sf W$ is contained in a certain finite-dimensional algebra $N({\sf W})$, called the Nomura algebra. It often happens that a spin model $\sf W$ satisfies ${\sf W} \in {\sf M} \subseteq N({\sf W})$, where $\sf M$ is the Bose-Mesner algebra of a distance-regular graph $\Gamma$; in this case we say that $\Gamma$ affords $\sf W$. If $\Gamma$ affords a spin model, then each irreducible module for every Terwilliger algebra of $\Gamma$ takes a certain form, recently described by Caughman, Curtin, Nomura, and Wolff. In the present paper we show that the converse is true; if each irreducible module for every Terwilliger algebra of $\Gamma$ takes this form, then $\Gamma$ affords a spin model. We explicitly construct this spin model when $\Gamma$ has $q$-Racah type. The proof of our main result relies heavily on the theory of spin Leonard pairs.

The positive part $U^+_q$ of $U_q(\widehat{\mathfrak{sl}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. We introduce a PBW basis for $U^+_q$, said to be alternating. Each element of this PBW basis commutes with exactly one of $A$, $B$, $qAB-q^{-1}BA$. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a $q$-shuffle algebra associated with affine $\mathfrak{sl}_2$. We show how the alternating PBW basis is related to the PBW basis for $U^+_q$ found by Damiani in 1993., Comment: 37 pages

Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms $T_0$, $T_1$ of the $q$-Onsager algebra $\mathcal O_q$, that are roughly analogous to the Lusztig automorphisms of $U_q(\widehat{\mathfrak{sl}}_2)$. We use $T_0, T_1$ and a certain antiautomorphism of $\mathcal O_q$ to obtain an action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on $\mathcal O_q$ as a group of (auto/antiauto)-morphisms. The action forms a pattern much more symmetric than expected. We show that a similar phenomenon occurs for the associated current algebra $\mathcal A_q$. We give some conjectures and problems concerning $\mathcal O_q$ and $\mathcal A_q$., Comment: 15 pages

The positive part $U^+_q$ of $U_q({\widehat {\mathfrak{sl}}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. In 1993 I. Damiani obtained a PBW basis for $U^+_q$, consisting of some elements $\lbrace E_{n\delta+\alpha_0}\rbrace_{n=0}^\infty$, $\lbrace E_{n\delta+\alpha_1}\rbrace_{n=0}^\infty$, $\lbrace E_{n\delta}\rbrace_{n=1}^\infty$ that are defined recursively. Our goal is to describe these elements in closed form. We achieve this goal using Catalan words and a $q$-shuffle algebra., Comment: 13 pages

Let $\mathbb F$ denote a field, and pick a nonzero $q \in \mathbb F$ that is not a root of unity. Let $\mathbb Z_4=\mathbb Z/4 \mathbb Z$ denote the cyclic group of order 4. Define a unital associative ${\mathbb F}$-algebra $\square_q$ by generators $\lbrace x_i \rbrace_{i \in \mathbb Z_4}$ and relations $$\frac{q x_i x_{i+1}-q^{-1}x_{i+1}x_i}{q-q^{-1}} = 1,\qquad x^3_i x_{i+2} - \lbrack 3 \rbrack_q x^2_i x_{i+2} x_i + \lbrack 3 \rbrack_q x_i x_{i+2} x^2_i -x_{i+2} x^3_i = 0,$$ where $\lbrack 3 \rbrack_q = \big(q^3-q^{-3}\big)/\big(q-q^{-1}\big)$. Let $V$ denote a $\square_q$-module. A vector $\xi\in V$ is called NIL whenever $x_1 \xi = 0 $ and $x_3 \xi=0$ and $\xi \not=0$. The $\square_q$-module $V$ is called NIL whenever $V$ is generated by a NIL vector. We show that up to isomorphism there exists a unique NIL $\square_q$-module, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description an important role is played by the $q$-shuffle algebra for affine $\mathfrak{sl}_2$., Comment: 42 pages

Let $\F$ denote a field and let $V$ denote a vector space over $\F$ with finite positive dimension. Consider a pair $A,A^*$ of diagonalizable $\F$-linear maps on $V$, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of $V$ that swaps $A$ and $A^*$. Such an automorphism is unique, and called the duality $A \leftrightarrow A^*$. In the present paper we give a comprehensive description of this duality. In particular, we display an invertible $\F$-linear map $T$ on $V$ such that the map $X \mapsto T X T^{-1}$ is the duality $A \leftrightarrow A^*$. We express $T$ as a polynomial in $A$ and $A^*$. We describe how $T$ acts on $4$ flags, $12$ decompositions, and 24 bases for $V$., Comment: 25 pages

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra $\Delta_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to \Delta_q$. We apply $\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.

There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or TB. Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey-Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey-Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair., Comment: 73 pages

Let $\Gamma$ denote a finite, undirected, connected graph, with vertex set $X$. Fix a vertex $x \in X$. Associated with $x$ is a certain subalgebra $T=T(x)$ of ${\rm Mat}_X(\mathbb C)$, called the subconstituent algebra. The algebra $T$ is semisimple. Hora and Obata introduced a certain subalgebra $Q \subseteq T$, called the quantum adjacency algebra. The algebra $Q$ is semisimple. In this paper we investigate how $Q$ and $T$ are related. In many cases $Q=T$, but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible $T$-modules. We show that the following are equivalent: (i) $Q \neq T$; (ii) there exists a pair of quasi-isomorphic irreducible $T$-modules that have different endpoints. To illustrate this result we consider two examples. The first example concerns the Hamming graphs. The second example concerns the bipartite dual polar graphs. We show that for the first example $Q=T$, and for the second example $Q \neq T$., Comment: 15 pages

For an integer $n\geq 1$, we display a poset $\Phi_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat \Phi_n$ is obtained from the $n$-cube by adding some edges. We show that the dihedral group $D_{2n}$ acts on $\widehat \Phi_n$ as a group of automorphisms., Comment: 6 pages