TY - JOUR

T1 - Algebraic Structures in Group-theoretical Fusion Categories

AU - Morales, Yiby

AU - Müller, Monique

AU - Plavnik, Julia

AU - Ros Camacho, Ana

AU - Tabiri, Angela

AU - Walton, Chelsea

N1 - Funding Information:
Y. Morales was partially supported by the London Mathematical Society, workshop grant #WS-1718-03. M. Müller was partially supported by London Mathematical Society, workshop grant #WS-1718-03 and by Universidade Federal de Viçosa - Campus Florestal. J. Plavnik gratefully acknowledges the support of Indiana University, Bloomington, through a Provost’s Travel Award for Women in Science. A. Ros Camacho was supported by the NWO Veni grant 639.031.758, Utrecht University and Cardiff University. A. Tabiri was supported by the Schlumberger Foundation Faculty for the Future Fellowship, AIMS-Google AI Postdoctoral Fellowship and AIMS-Ghana. C. Walton was supported by a research fellowship from the Alfred P. Sloan foundation. J. Plavnik and C. Walton were also supported by the U.S. NSF with research grants DMS-1802503/1917319, and DMS-1903192/2100756, respectively.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2022

Y1 - 2022

N2 - It was shown by Ostrik (Int. Math. Res. Not. 2003(27), 1507–1520 2003) and Natale (SIGMA Symmetry Integrability Geom. Methods Appl. 13, 042 2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor Φ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties.

AB - It was shown by Ostrik (Int. Math. Res. Not. 2003(27), 1507–1520 2003) and Natale (SIGMA Symmetry Integrability Geom. Methods Appl. 13, 042 2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor Φ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties.

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U2 - 10.1007/s10468-022-10186-7

DO - 10.1007/s10468-022-10186-7

M3 - Article

AN - SCOPUS:85142616523

SN - 1386-923X

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

ER -