The Principal Investigator (PI) will investigate problems involving the connections between representations of algebraic objects and their underlying geometric structures. The basic algebraic structures that the PI proposes to study are Lie superalgebras, algebraic/finite groups, quantum groups, and Frobenius kernels. The algebraic objects have concrete (discrete) realizations, and often times the underlying rich geometric structures arise at the derived level. Cohomological methods are useful for unveiling this geometry. The PI proposes to use new methods involving the Balmer spectrum to describe homological properties of Lie superalgebras. He also plans to make calculations of support varieties for algebraic and quantum groups as a way to connect geometric objects and representation theory. The PI plans to use geometric structures to understand the behavior of the cohomology of finite and algebraic groups.
It is well known that algebraic structures such as groups, rings, Lie algebras, and Lie superalgebras manifest themselves naturally in science. The basic understanding of these objects have been used in many different applications in physics and chemistry. These structures are often complicated. Both algebraic and geometric methods are often necessary to extract the important encoded information within these algebraic objects. In terms of broader impacts, the PI has been active nationally in the promotion of integrating research and education. He will continue to direct the NSF funded VIGRE (Vertical Integration of Research and Education) Program at the University of Georgia (UGA). He is also a co-organizer of the VIGRE Algebra Group at UGA which provides practical training in contemporary mathematics to postdoctoral fellows and graduate students. The PI will continue to organize conferences in algebra with an emphasis toward the development of junior mathematicians, and will promote the working knowledge of cohomology and representation theory as an invited speaker at seminars, workshops, and summer schools in the U.S. and abroad.
Intellectual Merit The Principal Investigator (PI) studied connections between representations of algebraic objects and their underlying geometric structures. Geometric structures often arise at the derived level and can be extracted using tools like cohomology. The basic algebraic structures investigated by the PI were Lie superalgebras, algebraic/finite groups, quantum groups, and Frobenius kernels. In the PI’s work, new methods for calculating cohomology and support varieties for algebraic and quantum groups were developed via these connections. The PI with his collaborators settled open questions and discovered important results in the areas of representation theory and cohomology for these various algebraic structures. The PI’s results for the grant period include the following. i) Lie Superalgebra Representations: The PI with Boe and Kujawa introduced the theory of support varieties for classical Lie superalgebras. Strong connections between the dimensions of support varieties for simple modules and the combinatorial notions of atypicality and defect (due to Kac and Wakimoto) were verified. A theory for the complexity for modules over Lie superalgebras was developed, and sophisticated techniques (involving Kazhdan-Lusztig theory) were employed to compute this invariant for simple and Kac modules for the general linear Lie superalgebra. ii) Structure of the Endotrivial Group: The PI with Carlson investigated the behavior of endotrivial modules for arbitrary finite group schemes. In a series of papers, the group of endotrivial modules for a wide class of infinitesimal group schemes were computed. It was demonstrated that for any fixed dimension (for a given group scheme) there are finitely many endotrivial modules. iii) Cohomological Calculations: The PI with Drupieski and Ngo computed the ring structure of the cohomology of the first Frobenius kernel of the unipotent radical of a Borel subgroup for a simple algebraic group. For quantum groups at roots of unity when the order is less than the Coxeter number, the PI with Bendel, Parshall, and Pillen calculated the cohomology ring for the small quantum group. This result extended prior work of Ginzburg and Kumar. iv) Support Variety Calculations: The PI with Drupieski and Parshall calculated the support varieties for all simple modules for the small quantum group when order of the root of unity is larger than the Coxeter number. This computation utilized the validity of the Lusztig Character Formula (for the small quantum group) and the positivity of parabolic Kazhdan-Lusztig polynomials. Moreover, in joint work with Bendel, Parshall and Pillen, the support varieties of quantum induced modules were computed when the root of unity is arbitrary. v) Finite Groups of Lie Type: Bendel, Pillen, and the PI located the first non-trivial cohomology class for the cohomology of finite groups of Lie type. This answered a question posed by Friedlander. The PI with the University of Georgia (UGA) VIGRE Algebra Group developed new theoretical methods to compute cohomology for finite groups of Lie type with coefficients in a simple module. These results presented a complete picture for the first and second cohomology for simple modules with fundamental dominant weights. vi) Symmetric and the General Linear Group: The PI with Cohen and Hemmer employed classical algebraic topological constructions (with their natural general linear group actions) to determine the structure of the symmetric group cohomology of the tensor space as a general linear group module. These results provided an algorithm (involving decomposition matrices) to completely describe the extensions in all degrees between Young modules. Broader Impacts The PI has been heavily involved in the professional development of graduate students and postdocs in the area of representation theory. For this grant period, the PI hosted two international Ph.D graduate exchange students, graduated two Ph.D students, and mentored three postdocs at UGA. The PI also co-led the UGA VIGRE Algebra Research Group. This large group setting (often involving 12-15 mathematicians) provided a natural place for mentoring relationships to develop between faculty, postdocs and students by bringing individuals with varying backgrounds and expertise to work collaboratively on a common research project. From 2003-2012, the UGA VIGRE Research Group published seven papers. During the last six years (2008-14) the PI served as the VIGRE Director at UGA. The PI coorganized a two week VIGRE Summer School on Geometry and Representation Theory (for postdocs and graduate students) at the UGA (2010), and the Summer School at the University of Washington on Cohomology and Supports in 2012. In addition to numerous formal research talks, the PI presented lecture series aimed at junior mathematicians on symmetric group representations (Oberwolfach Mini-Workshop, 2011) and tensor triangular geometry (Morningside Institute, Beijing, China, 2013). The PI has been engaged in the national mathematical community. From 2011-13 the PI served on the American Mathematical Society Committee on Committees, and the Southeastern Section Program Committee (Chair in 2012-13). Since 2010 the PI has been an editor for the Journal of Pure and Applied Algebra.
