arXiv subjects
Publications and source records attributed to Paul Terwilliger.
We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ in the equitable presentation. From this point of view, we describe the Lusztig automorphism and the corresponding Lusztig operator.
A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let $\mathbb F$ denote an algebraically closed field, and fix a nonzero $q \in \mathbb F$ that is not a root of unity. The universal double affine Hecke algebra (DAHA) $\hat{H}_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by generators $\lbrace t_i^{\pm 1}\rbrace_{i=0}^3$ and relations (i) $t_it_i^{-1}=t_i^{-1}t_i=1$; (ii) $t_i+t_i^{-1}$ is central; (iii) $t_0t_1t_2t_3 = q^{-1}$. We consider the elements $X=t_3t_0$ and $Y=t_0t_1$ of $\hat{H}_q$. Let $\mathcal V$ denote a finite-dimensional irreducible $\hat{H}_q$-module on which each of $X$, $Y$ is diagonalizable and $t_0$ has two distinct eigenvalues. Then $\mathcal V$ is a direct sum of the two eigenspaces of $t_0$. We show that the pair $X+X^{-1}$, $Y+Y^{-1}$ acts on each eigenspace as a Leonard pair, and each of these Leonard pairs falls into a class said to have $q$-Racah type. Thus from $\mathcal V$ we obtain a pair of Leonard pairs of $q$-Racah type. It is known that a Leonard pair of $q$-Racah type is determined up to isomorphism by a parameter sequence $(a,b,c,d)$ called its Huang data. Given a pair of Leonard pairs of $q$-Racah type, we find necessary and sufficient conditions on their Huang data for that pair to come from the above construction.
Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. Pick a nonzero $q \in \mathbb F$ such that $q^4 \not=1$, and let $A,B,C$ denote a Leonard triple on $V$ that has $q$-Racah type. We show that there exist invertible $W, W', W'' $ in ${\rm End}(V)$ such that (i) $A$ commutes with $W$ and $W^{-1}BW-C$; (ii) $B$ commutes with $W'$ and $(W')^{-1}CW'-A$; (iii) $C$ commutes with $W''$ and $(W'')^{-1}AW''-B$. Moreover each of $W,W', W''$ is unique up to multiplication by a nonzero scalar in $\mathbb F$. We show that the three elements $W'W, W''W', WW''$ mutually commute, and their product is a scalar multiple of the identity. A number of related results are obtained.
We describe the Bockting operator $ψ$ for a tridiagonal pair of $q$-Racah type, in terms of a certain $L$-operator for the quantum loop algebra $U_q(L({\mathfrak {sl}}_2))$.
We introduce the notion of a lowering-raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We describe the LR triples using various maps, such as the reflectors, the inverters, the unipotent maps, and the rotators. We relate the LR triples to the equitable presentation of the quantum algebra $U_q(\mathfrak{sl}_2)$ and Lie algebra $\mathfrak{sl}_2$.
The positive part $U^+_q$ of $U_q({\widehat{\mathfrak{sl}}}_2)$ has a presentation by two generators $X,Y$ that satisfy the $q$-Serre relations. The $q$-Onsager algebra $\mathcal O_q$ has a presentation by two generators $A,B$ that satisfy the $q$-Dolan/Grady relations. We give two results that describe how $U^+_q$ and $\mathcal O_q$ are related. First, we consider the filtration of $\mathcal O_q$ whose $n$th component is spanned by the products of at most $n$ generators. We show that the associated graded algebra is isomorphic to $U^+_q$. Second, we introduce an algebra $\square_q$ and show how it is related to both $U^+_q$ and $\mathcal O_q$. The algebra $\square_q$ is defined by generators and relations. The generators are $\lbrace x_i \rbrace_{i \in \mathbb Z_4}$ where $\mathbb Z_4$ is the cyclic group of order 4. For $i \in \mathbb Z_4$ the generators $x_i, x_{i+1}$ satisfy a $q$-Weyl relation, and $x_i,x_{i+2}$ satisfy the $q$-Serre relations. We show that $\square_q$ is related to $U^+_q$ in the following way. Let $ \square^{\rm even}_q$ (resp. $ \square^{\rm odd}_q$) denote the subalgebra of $ \square_q$ generated by $x_0, x_2$ (resp. $x_1, x_3$). We show that (i) there exists an algebra isomorphism $U^+_q \to \square^{\rm even}_q$ that sends $X\mapsto x_0$ and $Y\mapsto x_2$; (ii) there exists an algebra isomorphism $U^+_q \to \square^{\rm odd}_q$ that sends $X\mapsto x_1$ and $Y\mapsto x_3$; (iii) the multiplication map $\square^{\rm even}_q \otimes \square^{\rm odd}_q \to \square_q$, $u \otimes v \mapsto uv$ is an isomorphism of vector spaces. We show that $\square_q$ is related to $\mathcal O_q$ in the following way. For nonzero scalars $a,b$ there exists an injective algebra homomorphism $ \mathcal O_q \to \square_q$ that sends $A \mapsto a x_0+ a^{-1} x_1$ and $B \mapsto b x_2+ b^{-1} x_3$.
We introduce the notion of a Billiard Array. This is an equilateral triangular array of one-dimensional subspaces of a vector space $V$, subject to several conditions that specify which sums are direct. We show that the Billiard Arrays on $V$ are in bijection with the 3-tuples of totally opposite flags on $V$. We classify the Billiard Arrays up to isomorphism. We use Billiard Arrays to describe the finite-dimensional irreducible modules for the quantum algebra $U_q(\mathfrak{sl}_2)$ and the Lie algebra $\mathfrak{sl}_2$.
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.
Let $\C$ denote the field of complex numbers, and fix a nonzero $q \in \C$ such that $q^4 \ne 1$. Define a $\C$-algebra $Δ_q$ by generators and relations in the following way. The generators are $A,B,C$. The relations assert that each of $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $Δ_q$. The algebra $Δ_q$ is called the universal Askey-Wilson algebra. Let $Γ$ denote a distance-regular graph that has $q$-Racah type. Fix a vertex $x$ of $Γ$ and let $T=T(x)$ denote the corresponding subconstituent algebra. In this paper we discuss a relationship between $Δ_q$ and $T$. Assuming that every irreducible $T$-module is thin, we display a surjective $\C$-algebra homomorphism $Δ_q \to T$. This gives a $Δ_q$ action on the standard module of $T$.
We discuss a connection between the algebra $U_q({\mathfrak{sl}_2})$ and the tridiagonal pairs of $q$-Racah type. To describe the connection, let $x,y^{\pm 1},z$ denote the equitable generators for $U_q({\mathfrak{sl}_2})$. Let $U^\vee_q$ denote the subalgebra of $U_q({\mathfrak{sl}_2})$ generated by $x,y^{-1},z$. Using a tridiagonal pair of $q$-Racah type we construct two finite-dimensional $U^\vee_q$-modules. The constructions yield two nonstandard presentations of $U^\vee_q$ by generators and relations. These presentations are investigated in detail.
Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\not=1$. The universal Askey-Wilson algebra $Δ_q$ is the associative $\mathbb F$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$. The relations assert that each of $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $Δ_q$. The universal DAHA $\hat H_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by generators $\lbrace t^{\pm1}_i\rbrace_{i=0}^3$ and relations (i) $t_i t^{-1}_i=t^{-1}_i t_i=1$; (ii) $t_i+t^{-1}_i$ is central; (iii) $t_0t_1t_2t_3=q^{-1}$. We display an injection of $\mathbb F$-algebras $ψ:Δ_q\to\hat H_q$ that sends $A\mapsto t_1t_0+(t_1t_0)^{-1}$, $B\mapsto t_3t_0+(t_3t_0)^{-1}$, $C\mapsto t_2t_0+(t_2t_0)^{-1}$. For the map $ψ$ we compute the image of the three central elements mentioned above. The algebra $Δ_q$ has another central element of interest, called the Casimir element $Ω$. We compute the image of $Ω$ under $ψ$. We describe how the Artin braid group $B_3$ acts on $Δ_q$ and $\hat H_q$ as a group of automorphisms. We show that $ψ$ commutes with these $B_3$ actions. Some related results are obtained.
We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ with $q$ not a root of unity. We describe the finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules from the point of view of the equitable presentation.
Motivated by the structure of the uniform posets we introduce the notion of an augmented down-up (or ADU) algebra. We discuss how ADU algebras are related to the down-up algebras defined by Benkart and Roby. For each ADU algebra we give two presentations by generators and relations. We also display a $Z$-grading and a linear basis. In addition we show that the center is isomorphic to a polynomial algebra in two variables. We display seven families of uniform posets and show that each gives an ADU algebra module in a natural way. The main inspiration for the ADU algebra concept comes from the second author's thesis concerning a type of uniform poset constructed using a dual polar graph.
Around 1992 A. Zhedanov introduced the Askey-Wilson algebra AW(3). Recently we introduced a central extension $Δ$ of AW(3) called the universal Askey-Wilson algebra. In this paper we discuss a connection between $Δ$ and the quantum algebra $U_q(\mathfrak{sl}_2)$. Our main result is an algebra injection from $Δ$ into a relative of $U_q(\mathfrak{sl}_2)$; the relative is obtained from $U_q(\mathfrak{sl}_2)$ by adjoining three mutually commuting indeterminates. We describe the injection using the equitable presentation of $U_q(\mathfrak{sl}_2)$
A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. In the present paper we give an elementary but comprehensive account of how the following are related: (i) Krawtchouk polynomials; (ii) finite-dimensional irreducible modules for the Lie algebra ${\mathfrak{sl}_2}$; (iii) a class of Leonard pairs said to have Krawtchouk type. Along the way we obtain elementary proofs of some well-known facts about Krawtchouk polynomials, such as the three-term recurrence, the orthogonality, the difference equation, and the generating function. The paper is a tutorial meant for a graduate student or a researcher unfamiliar with the above topics.
Let $\G$ denote a bipartite distance-regular graph with vertex set $X$ and diameter $D \ge 3$. Fix $x \in X$ and let $L$ (resp. $R$) denote the corresponding lowering (resp. raising) matrix. We show that each $Q$-polynomial structure for $\G$ yields a certain linear dependency among $RL^2$, $LRL$, $L^2R$, $L$. Define a partial order $\le$ on $X$ as follows. For $y,z \in X$ let $y \le z$ whenever $\partial(x,y)+\partial(y,z)=\partial(x,z)$, where $\partial$ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.
In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $Δ$ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call $Δ$ the {\it universal Askey-Wilson algebra}. We give a faithful action of the modular group ${\rm {PSL}}_2({\mathbb Z})$ on $Δ$ as a group of automorphisms. We give a linear basis for $Δ$. We describe the center of $Δ$ and the 2-sided ideal $Δ[Δ,Δ]Δ$. We discuss how $Δ$ is related to the $q$-Onsager algebra.
Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A \in \MX$ denote the adjacency matrix of $Q_D$. A matrix $B \in \MX$ is called $A$-{\em like} whenever both (i) $BA = AB$; (ii) for all $x,y \in X$ that are not equal or adjacent, the $(x,y)$-entry of $B$ is zero. Let $\Al$ denote the subspace of $\MX$ consisting of the $A$-like elements. We decompose $\Al$ into the direct sum of its symmetric part and antisymmetric part. We give a basis for each part. The dimensions of the symmetric part and antisymmetric part are $D+1$ and ${D \choose 2}$, respectively.