Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0305123

Berezin transform on the quantum unit ball

D. Shklyarov·G. Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce and study, in the framework of a theory of quantum Cartan domains, a q-analogue of the Berezin transform on the unit ball. We construct q-analogues of weighted Bergman spaces, Toeplitz operators and covariant symbol calculus. In studying the analytical properties of the Berezin transform we introduce also the q-analogue of the SU(n,1)-invariant Laplace operator (the Laplace-Beltrami operator) and present related results on harmonic analysis on the quantum ball. These are applied to obtain an analogue of one result by A.Unterberger and H.Upmeier. An explicit asymptotic formula expressing the q-Berezin transform via the q-Laplace-Beltrami operator is also derived. At the end of the paper, we give an application of our results to basic hypergeometric q-orthogonal polynomials.

Explore related subjects

math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

D. Shklyarov, G. Zhang. 2003-05-08. Berezin transform on the quantum unit ball. https://doi.org/10.1063/1.1593226

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

KEEP EXPLORING

Related papers

arXiv2026

Graded Necklace Lie Bialgebras and Batalin-Vilkovisky Formalism

An involutive Lie bialgebra induces a Batalin-Vilkovisky operator on its exterior algebra. We introduce a graded generalization of the necklace Lie bialgebra, which depends on a choice of a quiver $Q$. We relate the resulting Batalin-Vilkovisky structure to the Batalin-Vilkovisky structure coming from a degree $-1$ symplectic form on a suitably defined representation variety of the quiver $Q$. The morphism intertwining these Batalin-Vilkovisky algebras will be given by a twisted trace, recovering the usual (super)trace and the odd trace.

math.QA↗
arXiv2026

Freeness and divisibility for right $H$-simple left $H$-comodule algebras over a pointed Hopf algebra $H$

Let $H$ be a pointed Hopf algebra and let $A$ be a right $H$-simple left $H$-comodule algebra. We show that every relative $(H,A)$-Hopf module is free as an $A$-module and that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative $(H,A)$-Hopf modules to be semisimple. We also show that $A$ can be embedded into a left $H$-comodule algebra of a specific form when $H$ and $A$ are $\mathbb{N}_0$-graded. As a consequence, we prove that if $H$ is finite-dimensional and $A^{\mathrm{co} H}=\Bbbk$, then $A$ is finite-dimensional and $\dim A$ divides $\dim H$.

math.QA↗
arXiv2026

$C_2$-Cofiniteness and Rationality of the Icosahedral Orbifold $V_{L_2}^{A_5}$

Let $L_2=\mathbb{Z}α$ be the rank-one root lattice with $(α,α)=2$, and let $A_5$ act on the lattice vertex operator algebra $V_{L_2}$ through an icosahedral subgroup of $\operatorname{Aut}(V_{L_2})\cong PSL_2(\mathbb{C})$. We prove that the fixed-point vertex operator algebra $V_{L_2}^{A_5}$ is strongly rational.

math.QA↗