• Prof. Shun-Jen Cheng (Academia Sinica)
  

  Title: Kostant's formula for Lie superalgebras (3-4 lectures)

  Abstract：

       We describe how to use Howe duality and the homology theory of generalized Kac-Moody Lie algebras to give a unified fashion to compute Kostant type (co)homology groups for certain unitarizable representations of Lie superalgebras. Our method covers the oscillator modules of finite-dimensional Lie superalgebras and also the infinite-rank Lie superalgebras of classical types. Specializing our results to Lie algebras we obtain, in an independent way, Enright's Kostant homology formula for the Hermitian symmetric pairs. These talks are based on joint works with Jae-Hoon Kwon and Weiqiang Wang.

• Prof. Meng-Kiat Chuah ( National Tsing Hua University )
  
  Title: Automorphisms on Lie algebras (2-3 lectures)
  
  

  Abstract：

       We discuss the diagrammatic expression of finite order automorphisms on complex simple Lie algebras. This is a result of Kac. It helps to describe finite order automorphisms on real simple Lie algebras. In particular, when the order is 2, they characterize the non-Riemannian locally symmetric pairs.

  
  
• Prof. Kyo Nishiyama ( Kyoto University )
  
  Title: Symmetric pair and orbits on the flag variety (6 lectures)

  Abstract：

       First, I will give a brief survey on spherical varieties, on which a reductive group K acts with multiplicity free. Also I will recall some of the basic properties of the structure of reductive groups, including maximal tori, Borel subgroups, Weyl groups, etc. In the second part, I will give an abstract classification of K-orbits on the flag variety G/B, where K is a symmetric subgroup of a reductive group G. An explicit example will be discussed by Professor Ochiai. Finally, I will discuss on some geometric properties of those orbits, and the relation to the nilpotent orbits in the Lie algebra of G.
  

• Prof. Hiroyuki Ochiai ( Nagoya University )
  
  Title: Double coset decomposition on the flag variety for indefinite unitary group (6 lectures)
  
  Abstract：

       We start with the description of the Grassmannian and the flag variety for general linear groups. We give the combinatorial description for the orbit decomposition by the involutive subgroup on it in two ways; Matsuki-Oshima, and Richardson-Springer. Additional data such as closure relations will be read off. The general case other than U(p,q) will be treated by the lecture of Professor Nishiyama. We will use the terminology mostly from linear algebra instead of root systems andWeyl groups.
  
  
  

• Prof. Ruibin Zhang ( University of Sydney )
  

  Title: Classical invariant theory for Lie algebras and quantum groups (4 lectures)
  
  Abstract：

       These lectures give an introduction to classical invariant theory. They consist of two parts. One part treats the invariant theory for the classical Lie groups, and the other develops the invariant theory for quantum groups. Below is a brief description of the content of the lectures.

       In classical invariant theory, one typically considers the algebra of polynomial functions on some finite dimensional module over a Lie group. The group acts naturally on the polynomial algebra, and one wishes to describe the subalgebra of G -invariants. If the polynomial algebra is completely reducible as a G -module, the subalgebra of G -invariants is finitely generated. When G is one of the classical groups over the complex number field, and the G -module under consideration is a direct sum of multiple copies of the natural module and also of the dual if G is GL n or SL n , the standard polynomial form of the first fundamental theorem of classical invariant theory yields a finite set of generators for the subalgebra of invariants.

       Classical invariant theory can be extended to quantum groups. The quantum group U q ( g ) of a Lie algebra g is a deformation of the universal enveloping algebra of g and has the structure of a noncommutative and noncocommutative Hopf algebra. The general setting for developing an invariant theory for quantum groups is the theory of module algebras for Hopf algebras. A module algebra A over U q ( g ) is an associative algebra such that its underlying vector space is a U q ( g ) -module, and the algebraic structure is preserved by the quantum group action. Because of the noncocommutativity of the quantum group, the module algebra A is noncommutative in general. Nevertheless one can show that the subspace A U q ( g ) of U q ( g ) -invariants forms a (usually noncommutative) subalgebra. The aim is to describe the algebraic structures of the subalgebras of invariants.

       For each quantum group associated with a classical Lie algebra, we construct a noncommutative module algebra which reduces, in the limit q ! 1, to the polynomial algebra of the direct sum of multiple copies of the natural module (and also its dual in type A) of the corresponding classical group. We construct the generators of the subalgebra of quantum group invariants and determine the commutation relations obeyed by the generators. It turns out that in all the cases considered, the subalgebra of invariants is finitely generated. These results may be regarded as a noncommutative analogue of the first fundamental theorem of classical invariant theory.

       A different manifestation of invariant theory for the classical groups G and associated quantum groups U q ( g ) is the study of the structure and representations of the endomorphism algebras End G ( V r ) of tensor powers of the natural G -module V , and the endomorphism algebras End U q ( g ) ( V r q ) of tensor powers of the natural U q ( g ) -module V q . The Schur-Weyl-Brauer dualities and their quantum analogues respectively describe End G ( V r ) as quotients of the symmetric group algebra or the Brauer algebra, and End U q ( g ) ( V r q ) as quotients of the Hecke algebra of type- A or the Birman-Wenzl-Murakami (BMW) algebra. The Hecke algebra, BMW algebra and their classical analogues all have cellular structures in the sense of Graham and Lehrer. Exploring the cellular structures enables one to obtain presentations in terms of generators and relations for the endomorphism algebras in some cases.