Reflection equation, twist, and equivariant quantization

Donin, J.; Mudrov, Andrey

doi:10.1007/bf02807191

Cited by 34 publications

(66 citation statements)

References 8 publications

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“…In this case, the braided dual goes by many names: it has appeared in the literature as the 'equivariantized quantum coordinate algebra' of Majid [80], 'the reflection equation algebra' (see Remark 6.5) [33,35], and the 'quantum loop algebra' of Alekseev and Schomerus [2,3,6]. A comprehensive reference is the text [64].…”

Section: Braided Dual Of a Quasi-triangular Hopf Algebramentioning

confidence: 99%

Integrating quantum groups over surfaces

Ben‐Zvi

Brochier

Jordan

2018

Journal of Topology

161

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We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the (0,1,2)-dimensional part of Crane-Yetter-Kauffman four-dimensional TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group Uq(g) we obtain in this way an aspect of topologically twisted four-dimensional N = 4 super Yang-Mills theory, the setting introduced by Kapustin-Witten for the geometric Langlands program.For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of G-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to Uq(g), and from the punctured torus we recover the algebra of quantum differential operators associated to Uq(g). From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum D-modules. Contents INTEGRATING QUANTUM GROUPS OVER SURFACES 875[21]. These connections (which are discussed further in Sections 1.4.2 and 1.4.4) suggest many rich structures for quantum character varieties, some of which we discuss in this paper, and many which we plan to explore in future papers. Factorization homology of surfacesFactorization homology was originally introduced by Beilinson and Drinfeld [17] in the setting of chiral conformal field theory, as an abstraction (and geometric interpretation) of the functor of conformal blocks of a vertex algebra. Factorization homology in the topological, rather than conformal, setting is developed in [71] and further in [11,12]. In this paper we use the terminology and formalism of [11,12] (see [54] for a survey and [29] for more general applications to quantum field theory).The algebraic input to factorization homology of surfaces (in the terminology of [11] and subsequent papers) is a '2-disk algebra' in an appropriate symmetric monoidal higher category C . Informally speaking, a 2-disk algebra is an object A ∈ C equipped with operations A k → A parametrized in a locally constant fashion by embeddings of disjoint unions of k disks into a large disk, and satisfying a composition law governed by composition of disk embeddings. There are several variants of 2-disk algebras, named according to the kind of tangential structure carried by the disks and embeddings: framed 2-disk algebras are better known as E 2 algebras (algebras over the little 2-disk operad), while oriented 2-disk algebras are (confusingly) known as framed E 2 -algebras (algebras over the framed little 2-disk operad). We adopt the terminology of [11] as it reflects the type of surfaces over which the corresponding algebras may be int...

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Section: Braided Dual Of a Quasi-triangular Hopf Algebramentioning

confidence: 99%

Integrating quantum groups over surfaces

Ben‐Zvi

Brochier

Jordan

2018

Journal of Topology

161

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“…This quantization in the form of star product was constructed in [DM3]. Below we give a description of C [G] in terms of generators and relations, using the so called reflection equation algebra.…”

Section: Quantization Of the Sts Bracket On The Groupmentioning

confidence: 99%

Quantum Conjugacy Classes of Simple Matrix Groups

Mudrov

2007

Commun. Math. Phys.

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“…It is easy to see that F(T ) is a left module algebra over the Hopf ⊗ denotes the twisted tensor product of two Hopf algebras by a bicharacter F , see [RS] and also [DM3]. We also regard F(T ) as an H-bimodule by the Hopf algebra embedding…”

Section: Dfrt Algebra Associated With a Dynamical Matrixsmentioning

confidence: 99%

“…The FRT algebra is a module over U op ⊗ U. Recall from [DM3] that the RE algebra a module over U R ⊗ U, the twisted tensor square of U, [RS]. Those two Hopf algebras are related by the composition of twists:…”

Section: Twist and Twisted Tensor Squarementioning

confidence: 99%

Dynamical reflection equation

Kulish¹,

Mudrov

2007

Quantum Groups

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We construct a dynamical reflection equation algebra,K, via a dynamical twist of the ordinary reflection equation algebra. A dynamical version of the reflection equation is deduced as a corollary. We show thatK is a right comodule algebra over a dynamical analog of the Faddeev-Reshetikhin-Takhtajan algebra equipped with a structure of right bialgebroid. We introduce dynamical trace and use it for constructing central elements ofK.

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Reflection equation, twist, and equivariant quantization

Cited by 34 publications

References 8 publications

Integrating quantum groups over surfaces

Integrating quantum groups over surfaces

Quantum Conjugacy Classes of Simple Matrix Groups

Dynamical reflection equation

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