Related papers: The generalized Racah algebra as a commutant

The conformal transformations with respect to the metric defining $o(n,\mbb{C})$ give rise to a nonhomogeneous polynomial representation of $o(n+2,\mbb{C})$. Using Shen's technique of mixed product, we generalize the above representation to…

We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then \[ \sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N), \] where $\rho=1/2+i\gamma$ runs…

This paper introduces and studies the Heun-Racah and Heun-Bannai-Ito algebras abstractly and establishes the relation between these new algebraic structures and generalized Heun-type operators derived from the notion of algebraic Heun…

We develop and study the generalization of rational Schur algebras to the super setting. Similar to the classical case, this provides a new method for studying rational supermodules of the general linear supergroup $GL(m|n)$. Furthermore,…

Let $k \leq n$ be nonnegative integers and let $\lambda$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes…

We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in…

The purpose of this paper is to give a reciprocity between $U_q(h)$ and $\Cal H_{n,r}$, the Hecke algebra of $(\Bbb Z / r\Bbb Z)\wr \frak S_n$ introduced by Ariki and Koike. Let $K=\Bbb Q(q,u_1,\dots ,u_r)$ be the field of rational…

We generalize the connection between 2t physics and noncommutative geometry. In particular, we apply our formalism to a target spacetime of signature (2+2). Specifically, we compute an algebra of a generalized SL(2, R)-Hamiltonian…

We study W-algebras obtained by quantum Hamiltonian reduction of $sl(Mn)$ associated to the $sl(2)$ embedding of rectangular type. The algebra can be realized as the asymptotic symmetry of higher spin gravity with $M \times M$ matrix valued…

The $n$-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of $n$-slice algebras via their $(n+1)$-preprojective algebras and the trivial…

We establish sufficient conditions, involving Rankin--Cohen (RC) brackets, under which certain combinations of meromorphic quasi-modular forms and their derivatives yield meromorphic modular forms. To achieve this, we adopt an algebraic…

We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the…

Let V be a finite dimensional representation of the connected complex reductive group H. Denote by G the derived subgroup of H and assume that the categorical quotient of V by G is one dimensional. In this situation there exists a…

Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to…

We show that the space R^n x gl(n,R) with a certain antisymmetric bracket operation contains all n-dimensional Lie algebras. The bracket does not satisfy the Jacobi identity, but it does satisfy it for subalgebras which are isotropic under…

Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n \times n$ monomial matrices whose entries are $r^{\textrm{th}}$ roots of unity and let $0 \leq k \leq n$ be an integer. Recently, Haglund, Rhoades and…

We derive sufficient conditions under which the ``second'' Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical $\cal W$-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These…

From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $\sharp$ from the universal Racah algebra $\Re$ into $U(\mathfrak{sl}_2)$. We use the Casimir elements of $\Re$ to describe the kernel of…

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and…

We show in this paper that it is possible to formulate General Relativity in a phase space coordinatized by two $SO(3)$ connections. We analyze first the Husain-Kucha\v{r} model and find a two connection description for it. Introducing a…