Diagram categories and invariants of Lie algebras and quantum groups

       Ruibin Zhang         

University of Sydney

Graphical methods have been widely applied in recent years to analyse certain tensor products of modules for Lie (super)algebras and quantum (super)groups. One creates diagram categories, such as the tangle category and Brauer category, and seeks functors from them to categories of modules for these algebraic structures. The graphical description of morphisms of a diagram category usually leads to simpler descriptions of morphisms in the target category of modules, and hence a better understanding of the latter.

We discuss the Brauer category and various incarnations of it, and explain how they provide the natural context for a wide class of algebras, such as the Hecke algebra, BMW algebra, (affine) Brauer algebra, the algebra of chord diagrams, and etc., which play important roles in representation theory. We apply these categories to investigate invariants in modules of the type M ⊗ V ⊗ . . . ⊗ V  for classical Lie (super)algebras and related quantum (super)groups, where V is the natural module and M is an arbitrary module. In particular, if M=V, we obtain generalisations of Schur-Weyl dualities; and if M is taken to be the universal enveloping algebra itself, we obtain explicit generators for the centre, and derive a categorical interpretation of certain ``characteristic identities'' of the orthogonal and symplectic Lie algebras.

Topics to be covered include

1.      Brauer category

-          Brauer category and its oriented analogue

-          Categorical Schur-Weyl- Brauer dualities for classical Lie supergroups

-          Triangular decomposition of Brauer category

-          Deligne category

-          Indecomposable objects in the category of tensor modules for OSp

-          Enhanced Brauer category and invariants of orthogonal Lie algebra

2.      Polar Brauer category

-          polar Brauer category

-          Centre of universal enveloping superalgebra

-          Characteristic identities for Lie algebras

-          Idempotent completion of polar Brauer category

3.      Tangle categories

-          Braid group representations and quantum R-matrices

-          Tangle categories – oriented and un-oriented

-          Categorical Schur-Weyl- Brauer dualities for quantum (super)groups

-          Invariants of quantum G₂-module algebras

4.      Polar tangle categories

-          Polar tangle categories

-          Affine Templerley-Lieb category and category O of U_q(sl₂).