Abstract
We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.
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References
Khovanov, M.: Nilcoxeter algebras categorify the Weyl algebra. Comm. Algebra 29(11), 5033–5052 (2001). doi:10.1081/AGB-100106800
Morita, K.: Adjoint pairs of functors and Frobenius extensions. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 9, 40–71 (1965)
Nakayama, T., Tsuzuku, T.: On Frobenius extensions. I. Nagoya Math. J. 17, 89–110 (1960)
Rosso, D., Savage, A.: Towers of graded superalgebras categorify the twisted Heisenberg double. J. Pure Appl. Algebra 219(11), 5040–5067 (2015). doi:10.1016/j.jpaa.2015.03.016
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Presented by Iain Gordon.
The second author was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. The first author was supported by the Discovery Grant of the second author.
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Pike, J., Savage, A. Twisted Frobenius Extensions of Graded Superrings. Algebr Represent Theor 19, 113–133 (2016). https://doi.org/10.1007/s10468-015-9565-4
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DOI: https://doi.org/10.1007/s10468-015-9565-4
Keywords
- Frobenius extension
- Frobenius algebra
- Graded superring
- Graded superalgebra
- Induction
- Restriction
- Adjuction
- Adjoint functors