Invariant integrals on coideals and their Drinfeld doubles

Kenny De Commer; Joel Right Dzokou Talla

doi:10.1093/imrn/rnae094

Invariant integrals on coideals and their Drinfeld doubles

Kenny De Commer, Joel Right Dzokou Talla

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let A be a CQG Hopf ∗-algebra, i.e. a Hopf ∗-algebra with a positive invariant state. Given a unital right coideal ∗-subalgebra B of A, we provide conditions for the existence of a quasi-invariant integral on the stabilizer coideal B⊥ inside the dual discrete multiplier Hopf ∗-algebra of A. Given such a quasi-invariant integral, we show how it can be extended to a quasi-invariant integral on the Drinfeld double coideal. We moreover show that the representation theory of the Drinfeld double coideal has a monoidal structure. As an application, we determine the quasi-invariant integral for the coideal ∗-algebra Uq(sl(2,R)) constructed from the Podleś spheres.

Original language	English
Article number	rnae094
Pages (from-to)	10650-10677
Number of pages <span style="color:red"p> <font size="1.5"> ✽ </span> </font>	28
Journal	International Mathematics Research Notices
Volume	2024
Issue number	14
DOIs	https://doi.org/10.1093/imrn/rnae094
Publication status	Published - 1 Jul 2024

Bibliographical note

Funding Information:
This work was supported by the FWO grant [G032919N to K.D.C. and J.R.D.T.]. Acknowledgments

Publisher Copyright:
© The Author(s) 2024. Published by Oxford University Press.

Keywords

Quantum groups
coideals
Drinfeld double

Access to Document

10.1093/imrn/rnae094

https://arxiv.org/abs/2112.07476Licence: Unspecified

Cite this

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title = "Invariant integrals on coideals and their Drinfeld doubles",

abstract = "Let A be a CQG Hopf ∗-algebra, i.e. a Hopf ∗-algebra with a positive invariant state. Given a unital right coideal ∗-subalgebra B of A, we provide conditions for the existence of a quasi-invariant integral on the stabilizer coideal B⊥ inside the dual discrete multiplier Hopf ∗-algebra of A. Given such a quasi-invariant integral, we show how it can be extended to a quasi-invariant integral on the Drinfeld double coideal. We moreover show that the representation theory of the Drinfeld double coideal has a monoidal structure. As an application, we determine the quasi-invariant integral for the coideal ∗-algebra Uq(sl(2,R)) constructed from the Podle{\'s} spheres.",

keywords = "Quantum groups, coideals, Drinfeld double",

author = "{De Commer}, Kenny and {Dzokou Talla}, {Joel Right}",

note = "Funding Information: This work was supported by the FWO grant [G032919N to K.D.C. and J.R.D.T.]. Acknowledgments Publisher Copyright: {\textcopyright} The Author(s) 2024. Published by Oxford University Press.",

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AU - De Commer, Kenny

AU - Dzokou Talla, Joel Right

N1 - Funding Information: This work was supported by the FWO grant [G032919N to K.D.C. and J.R.D.T.]. Acknowledgments Publisher Copyright: © The Author(s) 2024. Published by Oxford University Press.

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N2 - Let A be a CQG Hopf ∗-algebra, i.e. a Hopf ∗-algebra with a positive invariant state. Given a unital right coideal ∗-subalgebra B of A, we provide conditions for the existence of a quasi-invariant integral on the stabilizer coideal B⊥ inside the dual discrete multiplier Hopf ∗-algebra of A. Given such a quasi-invariant integral, we show how it can be extended to a quasi-invariant integral on the Drinfeld double coideal. We moreover show that the representation theory of the Drinfeld double coideal has a monoidal structure. As an application, we determine the quasi-invariant integral for the coideal ∗-algebra Uq(sl(2,R)) constructed from the Podleś spheres.

AB - Let A be a CQG Hopf ∗-algebra, i.e. a Hopf ∗-algebra with a positive invariant state. Given a unital right coideal ∗-subalgebra B of A, we provide conditions for the existence of a quasi-invariant integral on the stabilizer coideal B⊥ inside the dual discrete multiplier Hopf ∗-algebra of A. Given such a quasi-invariant integral, we show how it can be extended to a quasi-invariant integral on the Drinfeld double coideal. We moreover show that the representation theory of the Drinfeld double coideal has a monoidal structure. As an application, we determine the quasi-invariant integral for the coideal ∗-algebra Uq(sl(2,R)) constructed from the Podleś spheres.

KW - Quantum groups

KW - coideals

KW - Drinfeld double

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U2 - 10.1093/imrn/rnae094

DO - 10.1093/imrn/rnae094

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VL - 2024

SP - 10650

EP - 10677

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 14

M1 - rnae094

ER -

Invariant integrals on coideals and their Drinfeld doubles

Abstract

Bibliographical note

Keywords

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FWOAL900: Quantum symmetric spaces, operator algebras and quantum cluster algebras

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Invariant integrals on coideals and their Drinfeld doubles

Abstract

Bibliographical note

Keywords

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Other files and links

Embargoed Document

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FWOAL900: Quantum symmetric spaces, operator algebras and quantum cluster algebras

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