 |
Professor Chanyoung Lee Shader
Education
Courses |
| Professor Lee Shader's research
concerns representation theory of Lie (super) algebras and their quantized
algebras. The current research focuses on construction of representations and analyzing
their structures by means of characters.
One of the ways that one can explicitly construct representations is by taking tensor products of well-known simple representations and then decompose them. The celebrated result, nowadays called Schur-Weyl duality, which considers the decomposition of the tensor product of the natural representations of the general linear group (and the general liear Lie algebra) is the origin of this research. Schur-Weyl duality provides a beautiful connection between the representations of general linear groups (and general linear Lie algebras) and the representations of symmetric groups (or their group algebras), and thus provides a crucial link beween representation theory and the combinatorics of partitions, tableaux, and symmetric functions. R. Brauer extended this duality to the other families of the classical groups, symplectic and orthogonal groups (and their Lie algebras) by introducing a new class of algebras, nowadays called Brauer algebras, which play the role of synmmetric groups for the symplectic and orthogonal groups. The quantied universal enveloping algebras, often called quantum groups, have been studied extensively lately, and Iwahori-Hecke algebras and BMW (Berman-Murakami-Wenzl) algebras and their representations are related with the quantum groups and their representations via analogous dualities. Professor Lee Shader's research seeks to obtain similar dualities for the representations of Lie superalgebras and their quantized algebras, and thus to construct representations and to describe their characters explicitly in terms of combinatorial tools and algorithms. Unlike for the representations of the classical Lie algebras and their guantum groups, the representations of Lie superalgebras and their quantum groups are not completely reducible in general. The lack of the analogue of H. Weyl's classical theorem in the super-setting makes the study of representation theory of Lie superalgebras and their quantum groups interesting and challenging. |
G. Benkart, R. King and C. Lee Shader, Spinor representations of orthosymplectic Lie superalgebras and of their quantized enveloping algebras, in preparation.
C. Lee Shader, Representations of Lie superalgebras of type C, to appear in J. Algebra.
C. Lee Shader and D. Moon, Mixed tensor representations and rational representations for the general linear Lie superalgebras. Comm. Algebra 30 (2002), no. 2, 839--857.
C. Lee Shader, Representations for Lie superalgebra spo(2m,1). J. Korean Math. Soc. 36 (1999), no. 3, 593--607.
C. Lee Shader, Typical representations for orthosymplectic Lie superalgebras. Comm. Algebra 28 (2000), no. 1, 387--400.
G. Benkart, C. Lee Shader and A. Ram, Tensor product representations for orthosymplectic Lie superalgebras. J. Pure Appl. Algebra 130 (1998), no. 1, 1--48.
C. Lee Shader, Tensor product module V(\lambda)\otimes V(µ) for general linear Lie superalgebras. Kyungpook Math. J. 36 (1996), no. 2, 337--347.
C. Lee, Construction of modules for Lie superalgebras of type C. J. Algebra 176 (1995), no. 1, 249--264.
G. Benkart and C. Lee, Stability in modules for general linear Lie superalgebras. Nova J. Algebra Geom. 2 (1993), no. 4, 383--409.
C. Lee, Stability of dominant weights of the tensor product module for orthosymplectic Lie superalgebras. Linear and Multilinear Algebra 37 (1994), no. 4, 283--295.
G. Benkart, M. Chakrabarti,
T. Halverson, R. Leduc, C. Lee and J. Stroomer, Tensor product representations
of general linear groups and their connections with
Brauer algebras. J. Algebra 166 (1994),
no. 3, 529--567.
*Receipient of a NSF POWRE grant, 2000-2002.
*Senior personnel in the Middle School Mathematics Initiative Project for Wyoming, NSF 2002-2004.