Christopher Drupieski

Overview of my research

My research interests are in the branch of algebra called representation theory. Roughly speaking, representation theory is the study of how algebraic structures like groups or rings can act on vector spaces by linear transformations. A vector space equipped with a group or ring action is called a module. Given two modules V and W, one can try to classify all modules U that have V as a submodule and W as the quotient module U/V. This is the extension problem, and its solutions are classified by objects called cohomology groups. Cohomology groups are some of the central objects of study in my research.

My particular research focus is the modular representation theory of algebraic groups and related algebraic structures, including Lie algebras, finite group schemes, finite groups of Lie type, quantized enveloping algebras, Lie superalgebras, and Hopf algebras. I am especially interested in the representation-theoretic and cohomological connections between these different algebraic structures, and in how these connections can be exploited to make various computations.

Click here to find my papers on the arXiv.

Papers published or accepted

C. M. Drupieski, Representations and cohomology for higher Frobenius-Lusztig kernels, J. Pure Appl. Algebra 215 (2011), no. 6, 1473-1491.
C. M. Drupieski, On injective modules and support varieties for the small quantum group, Int. Math. Res. Not. 2011 (2011), no. 10, 2263-2294. (corrigendum)
C. M. Drupieski, D. K. Nakano, and B. J. Parshall, Differentiating the Weyl generic dimension formula with applications to support varieties, Adv. Math. 229 (2012), no. 5, 2656-2668.
C. M. Drupieski, D. K. Nakano, and N. V. Ngo, Cohomology for infinitesimal unipotent algebraic and quantum groups, Transform. Groups. 17 (2012), no. 2, 393-416.
University of Georiga VIGRE Algebra Group, Second cohomology for finite groups of Lie type, J. Algebra 360 (2012), 21-52.
University of Georgia VIGRE Algebra Group, First cohomology for finite groups of Lie type: Simple modules with small dominant weights, Trans. Amer. Math. Soc. 365 (2013), no. 2., 1025-1050.
C. M. Drupieski, On projective modules for Frobenius kernels and finite Chevalley groups, Bull. London Math. Soc. 45 (2013), no. 4, 715-720.
C. M. Drupieski, Cohomology rings for quantized enveloping algebras, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3739-3753.
C. M. Drupuieski, Cohomological finite generation for restricted Lie superalgebras and finite supergroup schemes, Represent. Theory 17 (2013), 469-507.
C. M. Drupieski, Universal extension classes for GL2, Algebr. Represent. Theor. 17 (2014), no. 6, 1853-1860.
C. P. Bendel, B. D. Boe, C. M. Drupieski, D. K. Nakano, B. J. Parshall, C. Pillen, and C. B. Wright, Bounding the dimensions of rational cohomology groups, Developments and Retrospectives in Lie Theory, Develop. Math., vol. 38, Springer, 2014, pp. 51-69.
C. M. Drupieski, Cohomological finite-generation for finite supergroup schemes, Adv. Math. 288 (2016), 1360-1432. Corrigendum: Adv. Math. 311 (2017), 935-937.
C. M. Drupieski and J. R. Kujawa, On support varieties for Lie superalgebras and finite supergroup schemes, J. Algebra 525 (2019), 64-110.
C. M. Drupieski and J. R. Kujawa, On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes, Advances in Algebra, Springer Proc. Math. Stat., vol. 277, Springer, 2019, pp. 121-167.
C. M. Drupieski and J. R. Kujawa, Graded analogues of one-parameter subgroups and applications to the cohomology of GL_{m|n(r)}, Adv. Math. 348 (2019), 277-352.
C. M. Drupieski and J. R. Kujawa, Support schemes for infinitesimal unipotent supergroups, Adv. Math. 384 (2021), Paper No. 107754, 61pp.
C. M. Drupieski and J. R. Kujawa, Support varieties and modules of finite projective dimension for modular Lie superalgebras. With an appendix by Luchezar L. Avramov and Srikanth B. Iyengar., Algebra Number Theory 15 (2021), no. 5, 1157-1180.
C. M. Drupieski and J. R. Kujawa, Superized Troesch complexes and cohomology for strict polynomial superfunctors, J. Pure Appl. Algebra 226 (2022), no. 12, Paper No. 107136, 44pp.
C. M. Drupieski and J. R. Kujawa, A survey of support theories for Lie superalgebras and finite supergroup schemes, to appear in Contemporary Mathematics.
C. M. Drupieski and J. R. Kujawa, Support varieties for Lie superalgebras in characteristic 2, to appear in Proceedings of Symposia in Pure Mathematics.

Papers under submission

C. M. Drupieski and J. R. Kujawa, The Lie superalgebra of transpositions

Other materials

GAP Code from the 2009-2010 and 2010-2011 UGA VIGRE Algebra Group: This file contains GAP code to perform calculations necessary for the papers Cohomology for finite groups of Lie type: Simple modules with small dominant weights and Second cohomology for finite groups of Lie type.
Cohomology products: Some notes I wrote about the cup product in cohomology.
Irreducible modular representations of finite and algebraic groups: Notes by Christopher Drupieski and Terrell Hodge, written for the 2007 AIM workshop on Cohomology and Representation Theory for Finite Groups of Lie Type.
Cohomology of Frobenius-Lusztig kernels of quantized enveloping algebras: My Ph.D. thsis, a dissertation presented to the graduate faculty of the University of Virginia in candidacy for the degree of Doctor of Philosophy.

Conference talks

Polynomial superfunctors, with applications to and from finite supergroup schemes. IPAM workshop on Symmetric Tensor Categories & Representation Theory, January 2024. Video of talk.
The Lie superalgebra of transpositions. AMS Fall Southeastern Sectional Meeting, Special Session on Representation Theory of Finite and Algebraic Groups, October 2023
Superized Troesch complexes and cohomology for strict polynomial superfunctors. AMS Spring Western Sectional Meeting, Special Session on Methods in Non-Semisimple Representation Categories, May 2023
The sulfuric stink of super: An homage to results of Brian Parshall by way of Lie superalgebras. AMS Southeastern Sectional Meeting; Special Session on Representation Theory of Algebraic Groups and Quantum Groups: A Tribute to the Work of Cline, Parshall and Scott; March 2023
Support varieties and modules of finite projective dimension for modular Lie superalgebras. AMS Spring Eastern Virtual Sectional Meeting; Special Session on Linear Algebraic Groups: their Structure, Representations, and Geometry; March 2022
Support varieties for modular Lie superalgebras. AMS Fall Central Sectional Meeting, Special Session on Lie Representation Theory, September 2019
Support schemes for infinitesimal unipotent supergroups. AMS Spring Central and Western Joint Sectional Meeting, Special Session on Recent Advances in Lie and Related Algebras and their Representations, March 2019
Cohomology and support varieties for Lie superalgebras and finite supergroup schemes. Oberwolfach Mini-Workshop: Cohomology of Hopf Algebras and Tensor Categories, March 2019
Support schemes for infinitesimal unipotent supergroups. Sixth Conference on Geometric Methods in Representation Theory, University of Iowa, November 2018
Support schemes for infinitesimal unipotent supergroups. AMS Western Section Meeting, Special Session on Homological Aspects in Commutative Algebra and Representation Theory, October 2018
Rank varieties for infinitesimal supergroup schemes. Southeastern Lie Theory Workshop X, June 2018
Cohomology and support varieties for unipotent supergroup schemes. AMS Eastern Sectional Meeting; Special Session on Hopf Algebras, Tensor Categories, and Homological Algebra; April 2018
Some Z/2Z-graded analogues of one-parameter subgroups and applications to the cohomology of GL{m|n(r)}. AMS Southeastern Sectional Meeting, Special Session on Categorial Methods in Rrepresentation Theory, 09/2017
Some Z/2Z-graded analogues of one-parameter subgroups and applications to the cohomology of GL{m|n(r)}. Mathematical Congress of the Americas, Special Session on Cohomology of Groups, July 2017
Some Z/2Z-graded analogues of one-parameter subgroups and applications to the cohomology of GL_{m|n(r)}. Southern Regional Algebra Conference, March 2017
Support varieties for Lie superalgebras. AMS Central Sectional Meeting, Special Session on Noncommutative Algebras and their Representations, October 2016
Support varieties for Lie superalgebras and finite graded group schemes. Southeastern Lie Theory Workshop on Algebraic Groups, Quantum Groups, and Geometry; May 2016
Support varieties for Lie superalgebras and graded group schemes. AMS Southeastern Sectional Meeting; Session on Lie Theory, Representation Theory, and Geometry; March 2016
Support varieties for Lie superalgebras and graded group schemes. AMS Joint Meetings, Special Session on Geometric and Categorical Methods in Representation Theory, January 2016
Support varieties for Lie superalgebras and graded group schemes. Workshop on Representations, Support, and Cohomology (in honor of Rolf Farnsteiner's 60th birthday), December 2015
Universal extension classes for algebraic supergroups. Groups, Representations, and Cohomology (Conference in honor of Dave Benson's 60th birthday), June 2015
Finite-generation for cohomology rings of finite supergroup schemes. Southwestern Group Theory Day, March 2015
Finite-generation for cohomology rings of finite supergroup schemes. AMS Central Sectional Meeting, Special Session on Groups and Representations, March 2015
Polynomial superfunctors and universal extension classes for algebraic supergroups. Southeastern Lie Theory Workshop 2014, May 2014
Cohomology for restricted Lie superalgebras and finite supergroup schemes. Cohomology in Lie Theory, University of Oxford, June 2013
Second cohomology for finite groups of Lie type. Seattle Summer Pi-School and Workshop on Cohomology and Support in Representation Theory, August 2012
1- and 2-cohomology for algebraic groups and finite groups of Lie type. AIM Workshop on Cohomology Bounds and Growth Rates, June 2012
Support varieties for irreducible modules of small quantum groups. Southeastern Lie Theory Workshop on Categorification of Quantum Groups and Representation Theory, April 2012
Projective modules for Frobenius kernels and finite groups of Lie type. AMS Southeastern Sectional Meeting, Special Session on Representations of Algebraic Groups, March 2012
Some quantum analogues of results from Lie algebra cohomology. AMS Central Sectional Meeting, Special Session on Quantum Groups and Representation Theory, October 2011
Comparing low degree cohomology for algebraic groups and finite groups of Lie type. Southeastern Lie Theory Workshop on Algebraic and Finite Groups, June 2011
Cohomology rings for quantized enveloping algebras. AMS Joint Meetings, Special Session on Lie Algebra, Algebraic Groups, and Related Topics, January 2011
On injective modules and support varieties for the small quantum group. Southeastern Lie Theory Conference on Homological Methods in Representation Theory, May 2010
Cohomology rings of infinitesimal algebraic and quantum groups. AMS Central Sectional Meeting, Special Session on Representation Theory of Algebraic Groups, April 2010
Cohomology of finite-dimensional quantized enveloping algebras, the mixed case. AMS Joint Meetings, Special Session on Representation Theory of Lie Algebras and Algebraic Groups, January 2009
Adjoint actions on integral forms in quantized enveloping algebras. AMS Central Sectional Meeting, Special Session on Computation in Modular Representation Theory, October 2008

Seminar and colloquium talks

The Lie superalgebra of transpositions. Topology, Algebra, Combinatorics, and Operators Seminar; Loyola University Chicago; September 2023
Support varieties for Lie superalgebras and finite supergroup schemes. Superalgebra Theory and Representations Seminar, Ben Gurion University/Bar Ilan University/Weizmann Institute of Science (virtual), May 2022
Superized Troesch complexes and resolutions for the Frobenius twist functor. Algebra and Representation Theory Seminar, University of Oklahoma, April 2022
Superized Troesch complexes and resolutions for the Frobenius twist functor. Topology, Algebra, Combinatorics, and Operators Seminar; Loyola University Chicago; March 2022
Cohomological finite generation for finite supergroup schemes. Learning seminar on the work of Friedlander-Suslin (2 lectures), MIT (virtual), September 2021
DePaul Math Club, DePaul University, April 2020
Some very simple, very explicit examples related to detecting projectivity of modules. Algebra and Combinatorics Seminar; Loyola University Chicago, March 2019
Some geometric invariants associated to representations, and how to describe them. Algebra, Combinatorics, and Number Theory Seminar; DePaul University, February 2019
DePaul Math Club, DePaul University, October 2018
DePaul Math Club, DePaul University, May 2018
Support varieties for infinitesimal (super)group schemes. GRTA Young Researchers Seminar, Mathematical Sciences Research Institute (MSRI), February 2018
DePaul Math Club, DePaul University, April 2017
Some geometric objects associated to representations of Lie (super)algebras. Algebra and Combinatorics Seminar, Loyola University Chicago, March 2016
Support varieties for Lie superalgebras and finite graded group schemes. Séminaire de topologie algébrique, Université Lille 1 - Sciences et Technologies, November 2015
Three games, and a mathematical connection to Matthew Broderick and Jeff Bridges. DePaul Math Club, DePaul University, May 2015
Finite-generation for cohomology rings of finite supergroup schemes. Algebra Seminar, University of Virginia, April 2015
Cohomology and Geometry for Groups and Related Structures. Mathematics Colloquium, Northern Illinois University, March 2015
Examples and results on strict polynomial superfunctors. Algebra and Combinatorics Seminar, Loyola University Chicago, November 2014
Symmetry and Groups. North Central Math Circle, North Central College, February 2014
Polynomial (super)functors and cohomology. Algebra and Combinatorics Seminar, DePaul University, January 2014
Polynomial Functors and Cohomology, II. Algebra Seminar, University of South Alabama, November 2013
Polynomial Functors and Cohomology. Mathematics Colloquium, University of South Alabama, October 2013
The Infinitude of the Primes. DePaul Math Club, DePaul University, October 2013
What is a cohomology ring, and why should I care if it is finitely-generated? Mathematics Colloquium, University of Oklahoma, September 2013
Cohomological finite generation for restricted Lie superalgebras and finite supergroup schemes. Algebra Seminar, University of Virginia, March 2013
The finite simple groups. Graduate Seminar, University of Virginia, March 2013
The Greatest Mathematical Achievement of the 20th Century. DePaul Math Club, DePaul University, February 2013
Finite-generation problems for cohomology rings, Parts I & II. Algebra and Combinatorics Seminar, DePaul University, October-November, 2012
Finite-generation problems for cohomology rings. Algebra and Combinatorics Seminar, Loyola University Chicago, September 2012
Support varieties for restricted Lie algebras (or, an application of Calculus to Abstract Algebra). Algebra Seminar, University of Georgia, January 2012
Comparing low degree cohomology for algebraic groups and finite groups of Lie type. Algebra Seminar, Christian-Albrechts Universitat zu Kiel (Kiel, Germany), June 2011
Some quantum analogues of results from Lie algebra cohomology. Algebra Seminar, Christian-Albrechts Universitat zu Kiel (Kiel, Germany), June 2011
Comparing low degree cohomology for algebraic groups and finite groups of Lie type. BIREP Seminar, Universitat Bielefeld (Bielefeld, Germany), June 2011
Calculus, group theory, and the infinitude of primes. Undergraduate Math Club, University of Georgia, January 2011
Cohomology rings for quantized enveloping algebras. Algebra Seminar, University of Georgia, August 2010
Cohomology of finite groups: an introduction to the Algebra VRG. VIGRE Algebra Seminar, University of Georgia, August 2010
Cohomology and support varieties. VIGRE Graduate Seminar, University of Georgia, November 2009
Support varieties for small quantum groups. Algebra Seminar, University of South Alabama, October 2009
Cohomology and support varieties. Mathematics Colloquium, University of South Alabama, October 2009
Cohomology of Frobenius-Lusztig kernels of quantized enveloping algebras. Algebra Seminar, University of Georgia, August 2009
Fun with Hopf algebras. Graduate Seminar, University of Virginia, September 2008
Calculus and other things I didn't learn until graduate school. Undergraduate Mathematics Seminar, McDaniel College, September 2008
Cohomology of infinitesimal algebraic groups and quantized enveloping algebras. Algebra Seminar, University of New South Wales (New South Wales, Australia), August 2008
The Steinberg module and Steinberg's tensor product theorem. Algebra Seminar, University of Virginia, 09/2007
Dynkin diagram automorphisms and realizations of twisted affine Lie algebras. Algebra Seminar, University of Virginia, November 2006
Divsion algebra theorems of Frobenius and Wedderburn. Algebra Seminar, University of Virginia, November 2005