arXiv subjects
Publications and source records attributed to Tatsuro Ito.
Let $Γ^{(x_0)}$ be a finite rooted tree, for which $Γ$ is the underlying tree and $x_0$ the root. Let $T$ be the Terwilliger algebra of $Γ$ with respect to $x_0$. We study the structure of the principal $T$-module. As a result, it is shown that $T$ recognizes the isomorphism class of $Γ^{(x_0)}$.
In [Kyushu J. Math. 64 (2010), 81-144, arXiv:0904.2889], it is discussed that a certain subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ controls the second kind TD-algebra of type I (the degenerate $q$-Onsager algebra). The subalgebra, which we denote by $U'_q(\widehat{\mathfrak{sl}}_2)$, is generated by $e_0^+$, $e_1^\pm$, $k_i^{\pm1}$ $(i=0,1)$ with $e^-_0$ missing from the Chevalley generators $e_i^\pm$, $k_i^{\pm1}$ $(i=0,1)$ of $U_q(\widehat{\mathfrak{sl}}_2)$. In this paper, we determine the finite-dimensional irreducible representations of $U'_q(\widehat{\mathfrak{sl}}_2)$. Intertwiners are also determined.
Let $\mathbb F$ denote an algebraically closed field, and fix a nonzero $q \in \mathbb F$ that is not a root of unity. We consider the $q$-tetrahedron algebra $\boxtimes_q$ over $\mathbb F$. It is known that each finite-dimensional irreducible $\boxtimes_q$-module of type 1 is a tensor product of evaluation modules. This paper contains a comprehensive description of the evaluation modules for $\boxtimes_q$. This description includes the following topics. Given an evaluation module $V$ for $\boxtimes_q$, we display 24 bases for $V$ that we find attractive. For each basis we give the matrices that represent the $\boxtimes_q$-generators. We give the transition matrices between certain pairs of bases among the 24. It is known that the cyclic group $\Z_4$ acts on $\boxtimes_q$ as a group of automorphisms. We describe what happens when $V$ is twisted via an element of $\Z_4$. We discuss how evaluation modules for $\boxtimes_q$ are related to Leonard pairs of $q$-Racah type.
We consider the double affine Hecke algebra $H=H(k_0,k_1,k^\vee_0,k^\vee_1;q)$ associated with the root system $(C^\vee_1,C_1)$. We display three elements $x$, $y$, $z$ in $H$ that satisfy essentially the $Z_3$-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra $\hat H$ that is more general than $H$, called the universal double affine Hecke algebra of type $(C_1^\vee,C_1)$. An advantage of $\hat H$ over $H$ is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism ${\hat H} \to H$. We define some elements $x$, $y$, $z$ in $\hat H$ that get mapped to their counterparts in $H$ by this homomorphism. We give an action of Artin's braid group $B_3$ on $\hat H$ that acts nicely on the elements $x$, $y$, $z$; one generator sends $x\mapsto y\mapsto z \mapsto x$ and another generator interchanges $x$, $y$. Using the $B_3$ action we show that the elements $x$, $y$, $z$ in $\hat H$ satisfy three equations that resemble the $Z_3$-symmetric Askey-Wilson relations. Applying the homomorphism ${\hat H}\to H$ we find that the elements $x$, $y$, $z$ in $H$ satisfy similar relations.
Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (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 $\lbrace V^*_i\rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$ and for $ 0 \leq i \leq d$ the dimensions of $V_i,V_{d-i},V^*_i, V^*_{d-i}$ coincide. The pair $A,A^*$ is called {\it sharp} whenever ${\rm dim} V_0=1$. It is known that if $F$ is algebraically closed then $A,A^*$ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the $μ$-conjecture.
For a field $F$ and an integer $d\geq 1$, we consider the universal associative $F$-algebra $A$ generated by two sets of $d+1$ mutually orthogonal idempotents. We display four bases for the $F$-vector space $A$ that we find attractive. We determine how these bases are related to each other. We describe how the multiplication in $A$ looks with respect to our bases. Using our bases we obtain an infinite nested sequence of 2-sided ideals for $A$. Using our bases we obtain an infinite exact sequence involving a certain $F$-linear map $\partial: A \to A$. We obtain several results concerning the kernel of $\partial$; for instance this kernel is a subalgebra of $A$ that is free of rank $d$.
Motivated by investigations of the tridiagonal pairs of linear transformations, we introduce the augmented tridiagonal algebra ${\mathcal T}_q$. This is an infinite-dimensional associative ${\mathbb C}$-algebra with 1. We classify the finite-dimensional irreducible representations of ${\mathcal T}_q$. All such representations are explicitly constructed via embeddings of ${\mathcal T}_q$ into the $U_q(sl_2)$-loop algebra. As an application, tridiagonal pairs over ${\mathbb C}$ are classified in the case where $q$ is not a root of unity.
This article gives a summary of the finite-dimesional irreducible representations of the $q$-Onsager algebra, which are treated in detail in our paper `The augmented tridiagonal algebra'.
We introduce the notion of a {\it mock tridiagonal system}. This is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain non-vanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism.
Let $\F$ denote a field and let $V$ denote a vector space over $\F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (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 $\lbrace V^*_i\rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$, and for $0 \leq i \leq d$ the dimensions of $V_i, V^*_i, V_{d-i}, V^*_{d-i}$ coincide. Denote this common dimension by $ρ_i$ and call $A,A^*$ {\it sharp} whenever $ρ_0=1$. Let $T$ denote the $\F$-subalgebra of ${\rm End}_\F(V)$ generated by $A,A^*$. We show: (i) the center $Z(T)$ is a field whose dimension over $\F$ is $ρ_0$; (ii) the field $Z(T)$ is isomorphic to each of $E_0TE_0$, $E_dTE_d$, $E^*_0TE^*_0$, $E^*_dTE^*_d$, where $E_i$ (resp. $E^*_i$) is the primitive idempotent of $A$ (resp. $A^*$) associated with $V_i$ (resp. $V^*_i$); (iii) with respect to the $Z(T)$-vector space $V$ the pair $A,A^*$ is a sharp tridiagonal pair.
Let $K$ denote an algebraically closed field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (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 $\lbrace V^*_i\rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$. For $0 \leq i \leq d$ let $θ_i$ (resp. $θ^*_i$) denote the eigenvalue of $A$ (resp. $A^*$) associated with $V_i$ (resp. $V^*_i$). The pair $A,A^*$ is said to have {\it $q$-Racah type} whenever $θ_i = a + b q^{2i-d}+ c q^{d-2i}$ and $θ^*_i = a^* + b^*q^{2i-d}+c^*q^{d-2i}$ for $0 \leq i \leq d$, where $q, a,b,c,a^*,b^*,c^*$ are scalars in $K$ with $q,b,c,b^*,c^*$ nonzero and $q^2 \not\in \lbrace 1,-1\rbrace$. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over $K$ that have $q$-Racah type. Our proof involves the representation theory of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$.
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ 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 $\{V^*_i\}{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$ and for $0 \leq i \leq d$ the dimensions of $V_i$, $V_{d-i}$, $V^*_i$, $V^*_{d-i}$ coincide. The pair $A,A^*$ is called {\it sharp} whenever $\dim V_0=1$. It is known that if $K$ is algebraically closed then $A,A^*$ is sharp. Assuming $A,A^*$ is sharp, we use the data $Φ=(A; \{V_i\}{i=0}^d; A^*; \{V^*_i\}{i=0}^d)$ to define a polynomial $P$ in one variable and degree at most $d$. We show that $P$ remains invariant if $Φ$ is replaced by $(A;\{V_{d-i}\}{i=0}^d; A^*; \{V^*_i\}{i=0}^d)$ or $(A;\{V_i\}{i=0}^d; A^*; \{V^*_{d-i}\}{i=0}^d)$ or $(A^*; \{V^*_i\}{i=0}^d; A; \{V_i\}{i=0}^d)$. We call $P$ the {\it Drinfel'd polynomial} of $A,A^*$. We explain how $P$ is related to the classical Drinfel'd polynomial from the theory of Lie algebras and quantum groups. We expect that the roots of $P$ will be useful in a future classification of the sharp tridiagonal pairs. We compute the roots of $P$ for the case in which $V_i$ and $V^*_i$ have dimension 1 for $0 \leq i \leq d$.
In this paper we discuss a relationship between the following two algebras: (i) the subconstituent algebra $T$ of a distance-regular graph that has $q$-Racah type; (ii) the $q$-tetrahedron algebra $\boxtimes_q$ which is a $q$-deformation of the three-point $sl_2$ loop algebra. Assuming that every irreducible $T$-module is thin, we display an algebra homomorphism from $\boxtimes_q$ into $T$ and show that $T$ is generated by the image together with the center $Z(T)$.
Recently Brian Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra $\boxtimes$ by generators and relations, and displayed an isomorphism from $\boxtimes$ to the three-point $sl_2$ loop algebra. In this paper we describe the finite-dimensional irreducible $\boxtimes$-modules from multiple points of view.
Let $K$ denote an algebraically closed field with characteristic 0 and let $V$ denote a vector space over $K$ with finite positive dimension. Let $A,A^*$ denote a tridiagonal pair on $V$ with diameter $d$. We say that $A,A^*$ has Krawtchouk type whenever the sequence $\lbrace d-2i\rbrace_{i=0}^d$ is a standard ordering of the eigenvalues of $A$ and a standard ordering of the eigenvalues of $A^*$. Assume $A,A^*$ has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form $< , >$ on $V$ such that $ = < u,Av>$ and $ = < u,A^*v>$ for $u,v\in V$. We show that the following tridiagonal pairs are isomorphic: (i) $A,A^*$; (ii) $-A,-A^*$; (iii) $A^*,A$; (iv) $-A^*,-A$. We give a number of related results and conjectures.
Let $Γ$ denote a distance-regular graph with classical parameters $(D,b,α,β)$ and $b\not=1$, $α=b-1$. The condition on $α$ implies that $Γ$ is formally self-dual. For $b=q^2$ we use the adjacency matrix and dual adjacency matrix to obtain an action of the $q$-tetrahedron algebra $\boxtimes_q$ on the standard module of $Γ$. We describe four algebra homomorphisms into $\boxtimes_q$ from the quantum affine algebra $U_q({\hat{\mathfrak{sl}}_2})$; using these we pull back the above $\boxtimes_q$-action to obtain four actions of $U_q({\hat{\mathfrak{sl}}_2})$ on the standard module of $Γ$.
As part of our study of the $q$-tetrahedron algebra $\boxtimes_q$ we introduce the notion of a $q$-inverting pair. Roughly speaking, this is a pair of invertible semisimple linear transformations on a finite-dimensional vector space, each of which acts on the eigenspaces of the other according to a certain rule. Our main result is a bijection between the following two sets: (i) the isomorphism classes of finite-dimensional irreducible $\boxtimes_q$-modules of type 1; (ii) the isomorphism classes of $q$-inverting pairs.
Recently B. Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra via generators and relations. To obtain this presentation they defined an algebra $\boxtimes$ by generators and relations, and displayed an isomorphism from $\boxtimes$ to the three-point $sl_2$ loop algebra. We introduce a quantum analog of $\boxtimes$ which we call $\boxtimes_q$. We define $\boxtimes_q$ via generators and relations. We show how $\boxtimes_q$ is related to the quantum group $U_q(sl_2)$, the $U_q(sl_2)$ loop algebra, and the positive part of $U_q(\hat{sl_2})$. We describe the finite dimensional irreducible $\boxtimes_q$-modules under the assumption that $q$ is not a root of 1, and the underlying field is algebraically closed.