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Algebra, combinatorics, category theory
My research interests include topics in noncommutative algebra, category theory and algebraic combinatorics, with Hopf algebras and their generalizations appearing prominently. A goal of my past work has been to build a conceptual framework for the study of Hopf algebraic structures in combinatorics and to clarify its significance to concrete applications. Part of my current work is devoted to enlarging the scope of classical Hopf-Lie theory. The new theory includes hyperplane arrangements and idempotent semigroups as part of the fundamental data.
The Steinberg torus of a Weyl group as a module over the Coxeter complex (with Kyle Petersen) Journal of Algebraic Combinatorics (2015), Volume 42, Issue 4, 1135--1175.
The characteristic polynomial of the Adams operators on graded connected Hopf algebras (with Aaron Lauve) Algebra & Number Theory 9-3 (2015), 547-583.
Hopf monoids in the category of species (with Swapneel Mahajan), Contemporary Mathematics 585 (2013), 17-124.
Generalized Hopf Modules for bimonads (with Steve Chase). Theory and Applications of Categories, Vol. 27 (2013), No. 13, 263-326.
Monoidal functors, species and Hopf algebras (with Swapneel Mahajan), CRM Monograph Series Vol. 29 (2010), AMS, Providence, RI. lii+784 pp.
Combinatorial Hopf algebras and generalized Dehn-Sommerville equations (with Nantel Bergeron and Frank Sottile), Compositio Mathematica 142 (2006), 1-30.
On the associative analog of Lie bialgebras, Journal of Algebra 244 (2001), 492-532.