Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0309252

Transformations of elliptic hypergometric integrals

Eric M. Rains
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, we also consider their "Type II" integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, and satisfying the analogues of the Macdonald conjectures. Finally, we discuss some transformations of Type II-style integrals. In particular, we find that adding two parameters to the Type II integral gives an integral invariant under an appropriate action of the Weyl group E_7.

Explore related subjects

math.QAmath.CAmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eric M. Rains. 2005-04-20. Transformations of elliptic hypergometric integrals. https://arxiv.org/abs/math/0309252

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Nodal degeneration of chiral algebras II: Local structure and chiral Zhu algebras

Given a universal factorization algebra $\mathcal{A}$, we constructed in our previous paper a derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, together with chiral modules $\\mathfrak{Z}_{\mathcal{A}}^+$ and $\mathfrak{Z}_{\mathcal{A}}^-$ associated to a puncture, and a chiral bimodule $\mathfrak{Z}_{\mathcal{A}}$ associated to a node. Furthermore, these constructions assemble to a factorization $\mathcal{A}$-module over any family of nodal punctured curves. In this paper, we show that in the case where $\mathcal{A}$ is constructed from a quasi-conformal vertex algebra $V$, the zeroth homology algebra $H^0\mathfrak{Z}_{\mathcal{A}}$ is naturally isomorphic to Zhu's associative algebra $A(V)$, and we identify $H^0\mathfrak{Z}_{\mathcal{A}}$ with the bimodule underlying the mode-transition algebra of Damiolini-Gibney-Krashen. We also give an explicit description of the smoothing module $H^0\tilde{\mathfrak{Z}}_{\mathcal{C}}$ which describes the deformation of $H^0\mathfrak{Z}_{\mathcal{A}}$ which we attach to a smoothing family of a nodal curve. We therefore get a geometric interpretation of the Zhu algebra and the mode-transition algebra, as the integration of a factorization algebra over a certain compactification of configuration spaces of punctured nodal curves.

math.QA↗
arXiv2026

Braided Hopf algebroids and Lie algebroids

We construct braided Hopf algebroids in a braided monoidal category over a field $k$ and study its properties using graphical representation. Then we study Ehresmann-Schauenburg Hopf algebroids assocaited to braided Hopf Galois extensions. We introduce braided Lie-Rinehart algebras which generalize the definition in \cite{ALP24, ALP23, ALP25} and study its universal enveloping algebra under symmetric condition. In particular, we introduce Lie-Rinehart algebras (braided Lie-algebroids) associated with braided Hopf algebroids in the category of the module of a triangular Hopf algebra. We then focus on the case for braided Ehresmann-Schauenburg Hopf algebroids and study their right invariant vector fields, which turn out to be isomorphic to the $H$-equivariant vector fields on the quantum principal bundle (Hopf Galois extension) as Lie-Rinehart algebras. Finally, we introduce braided jet Hopf algebroids associate to braided Hopf algebroids in the case that the source and target subalgebra belong to the braided center. Moreover, we show there is a dual pairing between the $k$-order universal enveloping algebra of the braided right invariant vector fields and the $k$-th jet space of a braided Hopf algebroid and further more a skew pairing between the universal enveloping algebra of its braided right invariant vector fields and its jet Hopf algebroid under a finiteness condition.

math.QA↗
arXiv2026

Affine quantum Schur--Weyl duality

Let $\mathpzc K$ be an arbitrary commutative ring containing an invertible element $\varepsilon$. Let ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\varepsilon$, let $Ω_{\mathpzc K}^{\otimes r}$ be the affine tensor space, and let ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ on $Ω_{\mathpzc K}^{\otimes r}$ is always faithful. Assume further that $\mathpzc K$ is a field of characteristic $0$ and that $\varepsilon$ is not a root of unity. We prove that, for any $n\geq 2$, the natural algebra homomorphism $ξ_r:{\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}\rightarrow\operatorname{End}_{{\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}}(Ω_{\mathpzc K}^{\otimes r})^{\mathrm{op}}$ is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ is left and right Noetherian whenever $\mathpzc K$ is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.

math.QA↗