Invariants of tensor categories and categorification (R-7568)

In the recent ten years, one of the main research streams in Algebra is the categorification of quantum groups or rings. In Mathematics, categorification is the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions by functors, and equations by natural isomorphisms of functors satisfying additional properties. A tensor (or monoidal) category is the categorification of a ring. In such a category one can carry out the "addition" and the "multiplication" on objects like one does on elements in a ring. Decategorification is the reverse process of categorification. It is a systematic process by which isomorphic objects in a category are identified as equal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward, and requires insight into individual situations. The decategorification of a tensor category yields the invariant: the Green ring. If one begins with a tensor category C, after decategorification and then categorification, one may not able to get the category C back. This is because the decategorification process of C losses the information over morphisms and the associativity. In this proposal, we will find back the lost information, two invariants: the Auslander algebra and the associator of C, and use these three invariants to classify and to deform tensor categories.

Keywords:ENVELOPING ALGEBRAS OF LIE ALGEBRAS, INVARIANT THEORY, NON-COMMUTATIVE ALGEBRAIC GEOMETRY, NON-COMMUTATIVE RINGS

Disciplines:Algebra