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 conditions:
(i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.
(ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.
Let $v^*_0, ..., v^*_d$ (resp. $v_0, ..., v_d$) denote a basis for $V$ that satisfies (i) (resp. (ii)). For $0 \leq i \leq d$ let $a_i$ denote the coefficient of $v^*_i$, when we write $A v^*_i$ as a linear combination of $v^*_0, ..., v^*_d$, and let $a^*_i$ denote the coefficient of $v_i$, when we write $A^* v_i$ as a linear combination of $v_0..., v_d$. In this paper we show $a_0=a_d$ if and only if $a^*_0=a^*_d$. Moreover we show that for $d \geq 1$ the following are equivalent: (i) $a_0=a_d$ and $a_1=a_{d-1}$; (ii) $a^*_0=a^*_d$ and $a^*_1=a^*_{d-1}$; (iii) $a_i=a_{d-i}$ and $a^*_i=a^*_{d-i}$ for $0 \leq i \leq d$. We say $A$, $A^*$ is balanced whenever (i)--(iii) hold. We say $A$, $A^*$ is essentially bipartite (resp. essentially dual bipartite}) whenever $a_i$ (resp. $a^*_i$) is independent of $i$ for $0 \leq i \leq d$. Observe that if $A$, $A^*$ is essentially bipartite or dual bipartite, then $A$, $A^*$ is balanced. For $d \neq 2$ we show that if $A$, $A^*$ is balanced then $A$, $A^*$ is essentially bipartite or dual bipartite.
Let $Φ$, $Φ'$ be Leonard systems over a field $\mathbb{K}$, and $V$, $V'$ the vector spaces underlying $Φ$, $Φ'$, respectively. In this paper, we introduce and discuss a balanced bilinear form on $V\times V'$. Such a form naturally arises in the study of $Q$-polynomial distance-regular graphs. We characterize a balanced bilinear form from several points of view.
Let $V$ denote a vector space with finite positive dimension, and let $(A,B)$ denote a Leonard pair on $V$. As is known, the linear transformations $A,B$ satisfy the Askey-Wilson relations A^2B -bABA +BA^2 -g(AB+BA) -rB = hA^2 +wA +eI, B^2A -bBAB +AB^2 -h(AB+BA) -sA = gB^2 +wB +fI, for some scalars $b,g,h,r,s,w,e,f$. The scalar sequence is unique if the dimension of $V$ is at least 4. If $c,c*,t,t*$ are scalars and $t,t*$ are not zero, then $(tA+c,t*B+c*)$ is a Leonard pair on $V$ as well. These affine transformations can be used to bring the Leonard pair or its Askey-Wilson relations into a c…
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 $A^*:V\to V$ which satisfy the following two properties: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Referring to the above…
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 (i), (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\em Leonard pair} on $V$. It is known that there exists a …
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 $B:V\to V$ which satisfy both (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $B$ is diagonal; (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $B$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We introduce two canonical for…
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…
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 (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Let $x$, $c$, $x^*$, $c…
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 diagonalizable matrices in $\text{Mat}_{n}(\mathbb{F})$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}_{n}(\mathbb{F})$. In the present paper, we find all Leonard pairs $A,A^*$ in $\text{Mat}_{n}(\mathbb{F})$ such that each of $A$ and $A^*$ is irreducible tridiagonal wi…
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 matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In this paper, we characterize the Leonard pairs using the noti…
Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $\text{Mat}_{d+1}(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $(d+1) \times (d+1)$ matrices that have all entries in $\mathbb{F}$. We consider a pair of diagonalizable matrices $A,A^*$ in $\text{Mat}_{d+1}(\mathbb{F})$, each acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}_{d+1}(\mathbb{F})$. For a Leonard pair $A,A^*$ there is a nonzero scalar $q$ that is used to describe the eigenvalues of $A$ and $A^*$. In the pres…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^*: V\rightarrow V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Very roughly speaking, a Leonard pair is a line…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^{*}: V\rightarrow V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^{*}$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{*}$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In the literature, there are some paramet…
Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars $(\{þ_i\}_{i=0}^d, \{þ^*_i\}_{i=0}^d, \{\vphi_i\}_{i=1}^d, \{ϕ_i\}_{i=1}^d)$, called its parameter array. The scalars $\{þ…
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 matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Arlene Pascasio recently obtained a characterization of the $Q$…
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 (i), (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\em Leonard pair} on $V$. In the literature on Leonard pai…
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 …
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 No…
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 both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Referring to the above Leonar…
Let $V$ denote a vector space with finite positive dimension. We consider a 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 matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\em Leonard pair} on $V$. Let $v_0,v_1,...,v_d$ (resp. $w_0,w_1,...,w_d$) denote a basis…
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 (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\it Leonard pair} on $V$. Let $\cal X$ denote the se…
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 (i), (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\em Leonard pair} on $V$. In this paper we investigate th…
Let $V$ denote a vector space with finite positive dimension. We consider a 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 matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In an earlier paper we described 24 special bases for $V$. One featu…
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 basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Let $A,A^*$ den…
Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $V$ be a vector space over $\mathbb{F}$ with dimension $d+1$. A Leonard pair on $V$ is an ordered pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let $\{v_i\}_{i=0}^d$ (resp.\ $\{v^*_i\}_{i=0}^d$) be such an eigenbasis for $A$ (resp.\ $A^*$). For $0 \leq i \leq d$ define a linear transformation $E_i : V \to V$ such that $E_i v_i=v_i$ and $E_i v_j =0$ if $j \neq i$ $(0 \leq j \leq d)$. Define $E^*_i : …
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 both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. Refining this notion a bit, we in…
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 basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We give an overview o…
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 basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. The structure o…
Let $\mathcal{C}$ be a triangulated category. We first introduce the notion of balanced pairs in $\mathcal{C}$, and then establish the bijective correspondence between balanced pairs and proper classes $ξ$ with enough $ξ$-projectives and enough $ξ$-injectives. Assume that $ξ:=ξ_{\mathcal{X}}=ξ^{\mathcal{Y}}$ is the proper class induced by a balanced pair $(\mathcal{X},\mathcal{Y})$. We prove that $(\mathcal{C}, \mathbb{E}_ξ, \mathfrak{s}_ξ)$ is an extriangulated category. Moreover, it is proved that $(\mathcal{C}, \mathbb{E}_ξ, \mathfrak{s}_ξ)$ is a triangulated category if and only if $\mathc…
We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent homotopy categories of complexes. As an application, we prove that for a left-Gorenstein ring, there exists a triangle-equivalence between the homotopy category of its Gorenstein projective modules and the homotopy category of its Gorenstein injective modules, which restricts to a triangle-equivalence between the homotopy category of projective modules and the homotopy category of injective modules. In t…
Let $(\mathscr{X}$, $\mathscr{Y})$ be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to $(\mathscr{X}$, $\mathscr{Y})$, and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the $\mathscr{X}$-resolution dimension of $\mathscr{Y}$ (resp. $\mathscr{Y}$-coresolution dimension of $\mathscr{X}$) is finite, then the bounded homotopy category of $\mathscr{Y}$ (resp. $\mathscr{X}$) is contained in that of $\mathscr{X}$ (resp. $\mathscr{Y}$). As a consequence, we get that the right $\m…
