Yi Ming Zou
Quantum super 2-shpheres and the corresponding quantum super transformation group are introduced in analogy to the well-known quantum 2-shpheres and quantum SL(2), connection between little $t$-Jacobi polynomials and the finite dimensional representations of the quantum super group is formulated, and the Peter-Weyl theorem is obtained.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra generated by twisted primitive elements of $\mathfrak U_q(sl(2))$. The little q-Jacobi operator and a tridiagonalization of it are shown to realize the equitable embedding of the Askey-Wilson algebra into $\mathfrak U_q(sl(2))$.
This paper studies properties of q-Jacobi polynomials and their duals by means of operators of the discrete series representations for the quantum algebra U_q(su_{1,1}). Spectrum and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthogonal basis in the representation space. The initial U_q(su_{1,1})-basis and the bases of these eigenfunctions are interconnected by matrices, whose entries are expressed in terms of little and big q-Jacobi polynomials. The orthogonality by rows in these unitary connection matrices leads to the orthogonality…
The little q-Jacobi function transform depends on three parameters. An explicit expression as a sum of two very-well-poised 8W7-series is derived for the dual transmutation kernel (a kind of non-symmetric Poisson kernel) relating little q-Jacobi function transforms for different parameter sets. A product formula for the dual transmutation kernel is obtained. For the inverse transform the transmutation kernel is given as a 3\phi2-series, and a product formula as a finite sum is derived. The transmutation kernel gives rise to intertwining operators for the second order hypergeometric q-differenc…
Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$
本文研究了$U_q(sl_2)$量子群作用于张量积Verma模的 intertwining 算子，利用小$q$-雅可比多项式和$q$-变形的Rankin-Cohen算子，揭示了非交换变量与Clebsch-Gordan系数之间的联系。
We obtain a closed form expression of the universal T-matrix encapsulating the duality of the quantum superalgebra U_q[osp(1/2)] and the corresponding supergroup OSp_q(1/2). The classical q-->1 limit of this universal T-matrix yields the group element of the undeformed OSp(1/2) supergroup. The finite dimensional representations of the quantum supergroup OSp_q(1/2) are readily constructed employing the said universal T-matrix and the known finite dimensional representations of the dually related deformed U_q[osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence r…
We derive explicit dimension formulas for irreducible $M_F$-spherical $K_F$-representations where $K_F$ is the maximal compact subgroup of the general linear group $GL(d,F)$ over a local field $F$ and $M_F$ is a closed subgroup of $K_F$ such that $K_F/M_F$ realizes the Grassmannian of $n$-dimensional $F$-subspaces of $F^d$. We explore the fact that $(K_F,M_F)$ is a Gelfand pair whose associated zonal spherical functions identify with various degenerations of the multivariable little $q$-Jacobi polynomials. As a result, we are led to consider generalized dimensions defined in terms of evaluatio…
The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q^{-1}, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas inv…
Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as …
The quantum complex Grassmannian U_q/K_q of rank l is the quotient of the quantum unitary group U_q=U_q(n) by the quantum subgroup K_q=U_q(n-l)xU_q(l). We show that (U_q,K_q) is a quantum Gelfand pair and we express the zonal spherical functions, i.e. K_q-biinvariant matrix coefficients of finite- dimensional irreducible representations of U_q, as multivariable little q-Jacobi polynomials depending on one discrete parameter. Another type of biinvariant matrix coefficients is identified as multivariable big q-Jacobi polynomials. The proof is based on earlier results by Noumi, Sugitani and the f…
In this paper we present an addition to Askey's scheme of q-hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and Bessel function, and are Askey-Wilson functions, big q-Jacobi functions and little q-Jacobi functions and the corrresponding q-Bessel functions. The generalised orthogonality relations and the second order q-difference equations for these families are given. Limit transitions between these families are discussed. The quantum group theoretic interpretations are …
The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second order q-difference operator that maps polynomials of degree $n$ to polynomials of degree $n+1$. It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil…
From Koornwinder's interpretation of big $q$-Legendre polynomials as spherical elements on the quantum $SU(2)$ group an addition formula is derived for the big $q$-Legendre polynomial. The formula involves Al-Salam--Carlitz polynomials, little $q$-Jacobi polynomials and dual $q$-Krawtchouk polynomials. For the little $q$-ultraspherical polynomials a product formula in terms of a big $q$-Legendre polynomial follows by $q$-integration. The addition and product formula for the Legendre polynomials are obtained when $q$ tends to 1.
An infinite-dimensional version of Calogero-Moser operator of $BC$-type and the corresponding Jacobi symmetric functions are introduced and studied, including the analogues of Pieri formula and Okounkov's binomial formula. We use this to describe all the ideals linearly generated by the Jacobi symmetric functions and show that the deformed $BC(m,n)$ Calogero-Moser operators, introduced in our earlier work, appear here in a natural way as the restrictions of the $BC_{\infty}$ operator to the corresponding finite-dimensional subvarieties. As a corollary we have the integrability of these quantum…
Jacobi-Trudi identity and Drinfeld functor for super Yangian
本文证明超 Yangian 对应的量子 Berezinian 可表示为两个差分算子之比，并补全了其表示论中的 q-特征标理论、Jacobi-Trudi 恒等式、Drinfeld 函子等关键工具。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。