Xiaoping XU

Professor

                                    lattices and codes

Research Summary:

• Find seven classes of nongraded simple Lie algebras

• Characterize lattice vertex operator algebras and classify their twisted irreducible modules

• Generalize the WZW model in conformal field theory to the ones over high order differential operators

• Classify simple Novikov algebras and their irreducible modules

• Use partial differential equations to determine the singular vectors in highest weight representations of Lie algebras

• Use Lie algebra and its thoughts to solve partial differential equations

• Determine the differential invariants over the basic representations of classical Lie groups

• Find new multi-variable hypergeometric functions in connection with integrable systems, representations of Lie algebras and integer paths.

• Construct self-dual linear lattices and codes

Representative Research Works:

• X. Xu, Representations of centrally-extended Lie algebras over differential operators and vertex algebras, Journal of Pure and Applied Algebra 212 (2008), 1253-1309.

• X. Xu, Flag partial differential equations and representations of Lie algebras, Acta Applicanda Mathematicae 102 (2008), 249-280

• X. Xu, Path hypergeometric functions, Journal of Algebra and Its Applications 6 (2007), no. 4 , 595-653.

• X. Xu, Stable-range approach to the equation of nonstationary transonic gas flows, Quarterly of Applied Mathematics 65 (2007), 529-547.

• X. Xu, Tree diagram Lie algebras of differential operators and evolution partial differential equations, Journal of Lie Theory 16 (2006）, no.4, 691-718.

• X. Xu, Classification of simple Novikov algebras and their irreducible modules of characteristic 0, Journal of Algebra 246 (2001), 673-707.

• X. Xu, New generalized simple Lie algebras of Cartan type, Journal of Algebra 224 (2000), 23-58.

• X. Xu, Differential invariants of classical groups, Duke Mathematical Journal 94 (1998), 543-572.

• X. Xu, Twisted modules of colored lattice vertex operator superalgebras, Quarterly Journal of Mathematics 47 (1996), 233-259.

• X. Xu, Self-dual lattices of type A, Acta Mathematica 175 (1995), 123-150.