TY - JOUR

T1 - Finite dimensional quantum Teichmüller space from the quantum torus at root of unity

AU - Kim, Hyun Kyu

N1 - Funding Information:
This work was supported by the Ewha Womans University Research Grant of 2017. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number 2017R1D1A1B03030230 ). H.K. thanks anonymous referees and the editors for their works.
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2019/3

Y1 - 2019/3

N2 - Representation theory of the quantum torus Hopf algebra, when the parameter q is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a ‘multiplicity module’ tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map T between the multiplicity modules, called the 6j-symbols. We study the left and right dual representations, and correspondingly, the left and right representations on the Hom spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map A on a multiplicity module, encoding the permutation of the roles of the irreducible representations in the identification of the multiplicity module as the space of intertwiners between representations. We show that T and A satisfy certain consistency relations, forming a Kashaev-type quantization of the Teichmüller spaces of bordered Riemann surfaces. All constructions and proofs in the present work use only plain representation theoretic language with the help of the notions of the left and the right dual and Hom representations, and therefore can be applied easily to other Hopf algebras for future works.

AB - Representation theory of the quantum torus Hopf algebra, when the parameter q is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a ‘multiplicity module’ tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map T between the multiplicity modules, called the 6j-symbols. We study the left and right dual representations, and correspondingly, the left and right representations on the Hom spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map A on a multiplicity module, encoding the permutation of the roles of the irreducible representations in the identification of the multiplicity module as the space of intertwiners between representations. We show that T and A satisfy certain consistency relations, forming a Kashaev-type quantization of the Teichmüller spaces of bordered Riemann surfaces. All constructions and proofs in the present work use only plain representation theoretic language with the help of the notions of the left and the right dual and Hom representations, and therefore can be applied easily to other Hopf algebras for future works.

UR - http://www.scopus.com/inward/record.url?scp=85052054897&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2018.08.011

DO - 10.1016/j.jpaa.2018.08.011

M3 - Article

AN - SCOPUS:85052054897

VL - 223

SP - 1337

EP - 1381

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 3

ER -