Characterisation and applications of $\Bbbk$-split bimodules
Abstract
We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk$-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk$-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk$-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\Bbbk[x,y]/(x^2,y^2,xy)$.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2017
- DOI:
- 10.48550/arXiv.1701.03025
- arXiv:
- arXiv:1701.03025
- Bibcode:
- 2017arXiv170103025M
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Rings and Algebras
- E-Print:
- MATH. SCAND. 124 (2019), 161--178