arXiv math/00102592000-10-27math.QAmath.SG

Non-Locality of Equivariant Star Products on $T*(RP^n)$

Ranee Brylinski

备注：: 6 pages

英文摘要：

Lecomte and Ovsienko constructed $SL_{n+1}(R)$-equivariant quantization maps $Q_λ$ for symbols of differential operators on $λ$-densities on $\RP^n$. We derive some formulas for the associated graded equivariant star products $star_λ$ on the symbol algebra $Pol(T*\RP^n)$. These give some measure of the failure of locality. Our main result expresses (for $n$ odd) the coefficients $C_p$ of $star_λ$ when $λ=\half$ in terms of some new $SL_{n+1}(C)$-invariant algebraic bidifferential operators $Z_p$ on $T*\CP^n$ and the operators $(E+\frac{n}{2}\pm s)^{-1}$ where $E$ is the fiberwise Euler vector field and $s\in\{1,2,...,[\frac{p}{2}]\}$.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

Classification of Equivariant Star Products on Symplectic Manifolds

In this note we classify invariant star products with quantum momentum maps on symplectic manifolds by means of an equivariant characteristic class taking values in the equivariant cohomology. We establish a bijection between the equivalence classes and the formal series in the second equivariant cohomology, thereby giving a refined classification which takes into account the quantum momentum map as well.

2015-07-01math-phmath.MPmath.QA

Tensor products and the Milnor-Moore theorem in the locality setup

The present exploratory paper deals with tensor products in the locality framework {developed in previous work}, a natural setting for an algebraic formulation of the locality principle in quantum field theory. Locality tensor products of locality vector spaces raise challenging questions, such as whether the locality tensor product of two locality vector spaces is a locality vector space. A related question is whether the quotient of locality vector spaces is a locality vector space, which we first reinterpret in a group theoretic language and then in terms of short exact sequences. We prove …

2022-05-29math-phmath.MPmath.QA

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.

2010-04-22math-phmath.MPmath.QA

$N$-point locality for vertex operators: normal ordered products, operator product expansions, twisted vertex algebras

In this paper we study fields satisfying $N$-point locality and their properties. We obtain residue formulae for $N$-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of $N$-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of $N$-point local fields include the vertex operators generating the boson-fermion correspondences of type B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the repr…

2013-07-18math-phmath.MPmath.QA

Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

We construct a unique G-equivariant graded star product on the algebra $S(g)/I$ of polynomial functions on the minimal nilpotent coadjoint orbit $\Omin$ of G where G is a complex simple Lie group and $g\neq\sl_2(C)$. This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to compute, for G classical, the star product of a momentum function $μ_x$ with any function f. We find $μ_x\star f=μ_xf+\half\{μ_x,f\}t+Λ^x(f)t^2$. For $\g$ different from $sp_n(\C)$, $Λ^x$ is not a differential operator. Instead $\Lamda^x$ is the left quotient of an explicit order 4 algebraic differenti…

2001-03-20math.QAmath.RTmath.SG

Equivariant Deformation Quantization for the Cotangent Bundle of a Flag Manifold

Let $\XR$ be a (generalized) flag manifold of a non-compact real semisimple Lie group $\GR$, where $\XR$ and $\GR$ have complexifications X and G. We investigate the problem of constructing a graded star product on $Pol(T^*\XR)$ which corresponds to a $\GR$-equivariant quantization of symbols into smooth differential operators acting on half-densities on $\XR$. We show that any solution is algebraic in that it restricts to a G-equivariant graded star product star on the algebraic part R of $Pol(T^*\XR)$. We construct, when R is generated by the momentum functions $μ^x$ for G, a preferred choic…

2000-10-27math.QAmath.RTmath.SG

更多相关论文收起更多论文

Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra

For a space X acted by a finite group $\G$, the product space $X^n$ affords a natural action of the wreath product $\Gn$. In this paper we study the K-groups $K_{\tG_n}(X^n)$ of $\Gn$-equivariant Clifford supermodules on $X^n$. We show that $\tFG =\bigoplus_{n\ge 0}K_{\tG_n}(X^n) \otimes \C$ is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on $\tFG$, when $\G$ is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linea…

2002-09-01hep-thmath.KTmath.QA

On $G$--equivariant modular categories

In this paper, we study $G$-equivariant tensor categories for a finite group $G$. These categories were introduced by Turaev under the name of $G$-crossed categories; the motivating example of such a category is the category of twisted modules over a vertex operator algebra $V$ with a finite group of automorphisms $G$. We discuss the notion of "orbifold quotient" of such a category (in the example above, this quotient is the category of modules over the subalgebra of invariants $V^G$). We introduce an extended Verlinde algebra for a $G$-equivariant tensor category and give a simple description…

2004-01-12math.CTmath.QA

Framed discs operads and the equivariant recognition principle

The framed n-discs operad fD_n is studied as semidirect product of SO(n) and the little n-discs operad. Our equivariant recognition principle says that a grouplike space acted on by fD_n is equivalent to the n-fold loop space on a SO(n)-space. Examples of fD_2-spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of fD_n. Koszul duality for semidirect product operads of chain complexes is defined and applied to compute the double loop space homology as BV-algebra.

2001-06-28math.ATmath.QA

Equivariant K-theory, wreath products, and Heisenberg algebra

Given a finite group G and a G-space X, we show that a direct sum $F_G (X) = \bigoplus_{n \geq 0}K_{G_n} (X^n) \bigotimes \C$ admits a natural graded Hopf algebra and $λ$-ring structure, where $G_n$ denotes the wreath product $G \sim S_n$. $F_G (X)$ is shown to be isomorphic to a certain supersymmetric product in terms of $K_G(X)\bigotimes \C$ as a graded algebra. We further prove that $F_G (X)$ is isomorphic to the Fock space of an infinite dimensional Heisenberg (super)algebra. As one of several applications, we compute the orbifold Euler characteristic $e(X^n, G_n)$.

1999-12-07hep-thmath-phmath.KT

Real Spectral Triples on Crossed Products

Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed product $C^{*}$-algebra $A\rtimes_r G$ was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262-291], extending the construction by Belissard, Marcolli and Reihani in [Dynamical systems on spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the Kasparov product to make an ansatz for the Dirac operator. Supposing …

2022-07-24math-phmath.MPmath.OA

Noncommutativity and theta-locality

In this paper, we introduce the condition of theta-locality which can be used as a substitute for microcausality in quantum field theory on noncommutative spacetime. This condition is closely related to the asymptotic commutativity which was previously used in nonlocal QFT. Heuristically, it means that the commutator of observables behaves at large spacelike separation like $\exp(-|x-y|^2/θ)$, where $θ$ is the noncommutativity parameter. The rigorous formulation given in the paper implies averaging fields with suitable test functions. We define a test function space which most closely correspo…

2008-02-07hep-thmath-phmath.MP

紧与离散量子群在算子系统上的作用

Actions of compact and discrete quantum groups on operator systems

本文引入离散或紧量子群在算子系统上的作用概念，证明了等变内射性与交叉积对偶内射性的对偶结果，并描述了约化交叉积的等变内射包络。

2023-04-27math.GRmath.OAmath.QA