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:

arXiv eprints
 Pub Date:
 January 2017
 DOI:
 10.48550/arXiv.1701.03025
 arXiv:
 arXiv:1701.03025
 Bibcode:
 2017arXiv170103025M
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Rings and Algebras
 EPrint:
 MATH. SCAND. 124 (2019), 161178