相关论文: The Racah algebra as a commutant and Howe duality
Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…
The polynomial algebra is a deformed SU(2) algebra. Here, we use polynomial algebra as a method to solve a series of deformed oscillators. Meanwhile, we find a series of physics systems corresponding with polynomial algebra with different…
The universal Askey-Wilson algebra $AW(3)$ can be obtained as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes3}$. We analyze the commutant of…
Let $\mathbb{S}$ denote the oscillatory module over the complex symplectic Lie algebra $\mathfrak{g}= \mathfrak{sp}(\mathbb{V}^{\mathbb{C}},\omega).$ Consider the $\mathfrak{g}$-module…
The main result of this paper is a formula for the scalar product of semiclassical eigenvectors of two integrable systems on the same symplectic manifold. An important application of this formula is the Ponzano-Regge type of asymptotic of…
We define the quadratic algebra su(2)_{\alpha} which is a one-parameter deformation of the Lie algebra su(2) extended by a parity operator. The odd-dimensional representations of su(2) (with representation label j, a positive integer) can…
It has long been appreciated that the toroidal reduction of any gravity or supergravity to two dimensions gives rise to a scalar coset theory exhibiting an infinite-dimensional global symmetry. This symmetry is an extension of the…
We interpret the Rahman polynomials in terms of the Lie algebra $sl_3(C)$. Using the parameters of the polynomials we define two Cartan subalgebras for $sl_3(C)$, denoted $H$ and $\tilde{H}$. We display an antiautomorphism $\dagger$ of…
A Lie-Rinehart algebra consists of a commutative algebra and a Lie algebra with additional structure which generalizes the mutual structure of interaction between the algebra of functions and the Lie algebra of smooth vector fields on a…
We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincar\'e duality in Lie algebra cohomology and de Rham cohomology. We show that the duality…
This paper is devoted to the study of Poisson superalgebras over fields of characteristic $2$. We investigate their representations, semidirect products, cohomology, formal deformations, and universal enveloping algebras. We also introduce…
The Jordanian deformation of $sl(2)$ bi-algebra structure is studied in view of physical applications to breaking of conformal symmetry in the high energy asymptotics of scattering. Representations are formulated in terms of polynomials,…
The notion of hidden symmetry algebra used in the context of exactly solvable systems is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By…
The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank $q$-Bannai-Ito algebra $\mathcal{A}_n^q$. Lifting…
The decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics.…
The goal of our work is to study the decomposition of the joint action of $\mathscr{G} = \text{SpO}(2n|1)$ and $\mathfrak{g}' = \mathfrak{osp}(2|2)$ on the supersymmetric algebra $\text{S} = \text{S}(\mathbb{C}^{2n|1} \otimes…
Let $(R,{}^-)$ be an arbitrary unital associative superalgebra with superinvolution over a commutative ring $\Bbbk$ with $2$ invertible. The second homology of the generalized periplectic Lie superalgebra $\mathfrak{p}_m(R,{}^-)$ for…
Gasper & Rahman's multivariate $q$-Racah polynomials are shown to arise as connection coefficients between families of multivariate $q$-Hahn or $q$-Jacobi polynomials. The families of $q$-Hahn polynomials are constructed as nested…
Given a semi-simple algebra equipped with a coproduct, the Clebsch--Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the…
Racah matrices of quantum algebras are of great interest at present time. These matrices have a relation with $\mathcal{R}$-matrices, which are much simpler than the Racah matrices themselves. This relation is known as the eigenvalue…