相关论文: The universal additive DAHA of type $(C_1^\vee,C_1…

Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C, D$ and the…

Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal DAHA (double affine Hecke algebra) $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital…

Assume that ${\mathbb F}$ is an algebraically closed field with characteristic zero. The Racah algebra $\Re$ is the unital associative ${\mathbb F}$-algebra defined by generators and relations in the following way. The generators are $A$,…

Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote 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 defined…

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…

Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\Re$ is the unital associative $\mathbb{F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$. The…

Assume that ${\mathbb F}$ is a field with $\operatorname{char}{\mathbb F}\not=2$. The Racah algebra $\Re$ is a unital associative ${\mathbb F}$-algebra defined by generators and relations. The generators are $A$, $B$, $C$, $D$ and the…

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 $\Delta_q$ is the associative $\mathbb F$-algebra defined by generators and relations in the following way. The…

Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\mathfrak{R}$ is the unital associative $\mathbb{F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$,…

The universal character of the Racah algebra will be illustrated by showing that it is at the center of the relations between the Racah polynomials, the recoupling of three su(1,1) representations and the symmetries of the generic…

We prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible assuming that it is finite dimensional (apart from the roots of unity). We…

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a…

Hecke algebras are usually defined algebraically, via generators and relations. We give a new algebro-geometric construction of affine and double-affine Hecke algebras (the former is known as the Iwahori-Hecke algebra, and the latter was…

Analogously to the construction of Suzuki and Vazirani, we construct representations of the $GL_m$-type Double Affine Hecke Algebra at roots of unity. These representations are graded and the weight spaces for the $X$-variables are…

The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the $2$-sphere. It is further identified as the commutant of the $\mathfrak{o}(2) \oplus…

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of $2$-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a…

The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and…

We define generalized double affine Hecke algebras (GDAHA) of higher rank, attached to a non-Dynkin star-like graph D. This generalizes GDAHA of rank 1 defined in math.QA/0406480 and math.QA/0409261. If the graph is extended D4, then GDAHA…

A finite algebra $\bA=\alg{A;\cF}$ is \emph{dualizable} if there exists a discrete topological relational structure $\BA=\alg{A;\cG;\cT}$, compatible with $\cF$, such that the canonical evaluation map $e\_{\bB}\colon \bB\to \Hom(…

The universal enveloping algebra $U(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ is a unital associative algebra over $\mathbb C$ generated by $E,F,H$ subject to the relations \begin{align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H.…