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Tetramodules over a bialgebra form a 2-fold monoidal category
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| Let B be an associative bialgebra over any field. A module over B in the sense of deformation theory is a tetramodule over B. All tetramodules form an abelian category. This category was studied by R. Taillefer (Algebr Represent Theory 7(5):471-490, 2004) and R. Taillefer (J Algebra 276(1):259-279, 2004). In particular, she proved that for any bialgebra B, the abelian category has enough injectives, and that Ext (a (TM) aEuro parts per thousand)(B,B) in this category coincides with the Gerstenhaber-Schack cohomology of B. We prove that the category of tetramodules over any bialgebra B is a 2-fold-monoidal category, with B a unit object in it. Roughly, this means that the category admits two monoidal structures, with common unit B, which are compatible in some rather non-trivial way (the concept of an n-fold monoidal category is introduced in Baltenu et al. (Adv Math 176:277-349, 2003)). Within (yet unproven) 2-fold monoidal analogue of the Deligne conjecture, our result would imply that RHom (a (TM) aEuro parts per thousand)(B,B) in the category of tetramodules is naturally a homotopy 3-algebra. | | |
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English
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Applied categorical structures. - Dordrecht, 1993, currens | |
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Dordrecht : 2013
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0927-2852
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21:3(2013), p. 291-309
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000318286100004
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