Research

RodinMuseum2-JSmith-1169-587-587x0

 

 

 

 

 

My research interest is Noncommutative Algebra, which has motivating contexts in noncommutative geometry, number theory, combinatorics, and mathematical physics. My recent projects are focusing on the following four areas.

 

Hopf algebra was born in earlier 1950s with two honorable parents-algebraic topology and algebraic geometry. Later until 1980s, it had enjoyed a blooming period where a deluge of interactions with knot theory and topology, conformal field theory, ring theory, category theory, combinatorics, etc. appeared. Nowadays, Hopf algebra and its categorification is essential in the design of modern quantum computers. I am interested in Frobenius-Schur indicators and cohomology rings of Hopf algebras.

Poisson Algebra has lately been playing an important role in algebra, geometry, mathematical physics and other subjects. For example, Poisson structures can be obtained through the semiclassical limits of quantized coordinate rings studied by Goodearl and can be used in the study of noncommutative discriminants in the work of Nguyen, Trampel and Yakimov. My interest of Poisson algebra involves applying Poisson geometry to study the representation theory and cohomological behaviors of noncommutative algebras through the idea of symplectic foliation.

Noncommutative Invariant Theory has been among the most fast growing and influential topics concerning Hopf algebras and their (co)actions on rings. It has deep relations with Calabi-Yau manifold, Chevalley-Shephard-Todd theorem, Auslander theorem, McKay quiver, Dynkin diagram and etc. The main mission is to broaden the classical picture of actions of finite groups on polynomial rings by its quantum counterpart of (co)actions of (semisimple) Hopf algebras on regular graded algebras. My work is to study various properties of generalized quantum symmetry groups associated to Artin-Schelter regular algebras.

Noncommutative Projective Algebraic Geometry was initiated by Artin and Schelter in a program of classifying some graded algebras of dimension 3, among which the family of 3 dimensional Sklyanin algebras is the most difficult one to deal with. I study the representation theory of PI Sklyanin algebras of dimension 3 and 4 through Poisson geometry using the symplectic foliation on the maximal spectrum of the center of these PI algebras, which has application in String Theory, namely to the understanding of marginal supersymmetric deformations of the N = 4 super-Yang-Mills theory in 4 dimensions.

Here are my publication list and research statement.

Publications and preprints

  1. Poisson geometry and representations of PI 4-dimensional Sklyanin algebras, (with C. Walton and M. Yakimov), submitted.
  2. The cohomology ring of some Hopf algebras, (with K. Erdmann and O. Solberg), submitted.
  3. Finite generation of some cohomology rings via twisted tensor product and Anick resolutions, (with V. Nguyen and S. Witherspoon), to appear Pure Appl. Algebra.
  4. On q-commutative power and Laurent series rings at roots of unity, (with E. S. Letzter and L.-H. Wang), submitted.
  5. A note on the bijectivity of antipode of a Hopf algebra and its applications, (with J.-F. Lv, S.-Q. Oh and X.-L. Yu), to appear Proc. Amer. Math. Soc.
  6. Indicators of Hopf algebras in positive characteristic, (with L.-H. Wang), submitted.
  7. Computing indicators of Radford algebras (with H. Hu, X.-Y. Hu and L.-H. Wang), Involve, a journal of mathematics 11, 325-334 (2018).
  8. The Poisson geometry of the 3-dimensional Sklyanin algebras (with C. Walton and M. Yakimov), submitted.
  9. The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras (with M.-T. Guo, X.-G. Hu and J.-F. Lv), submitted.
  10. PBW-basis for universal enveloping algebras of differential graded Poisson algebras (with X.-G. Hu and J.-F. Lv), submitted.
  11. Enveloping algebras of double Poison-Ore extension (with J.-F. Lv, S.-Q. Oh and X.-L. Yu), to appear Comm. Algebra.
  12. Calabi-Yau property under monoidal Morita-Takeuchi equivalence (with X.-L. Yu and Y.-H. Zhang), Pacific J. Math. 290 , 481-510 (2017).
  13. Pointed p3-dimensional Hopf algebras in positive characteristic (with V. Nguyen), submitted.
  14. Homological unimodularity and Calabi-Yau condition for Poisson algebras (with J.-F. Lv and G.-B. Zhuang), Lett. Math. Phys. 107, 1715-1740 (2017).
  15. On quantum groups associated to a pair of preregular forms (with A. Chirvasitu and C. Walton), to appear in J. Noncommut. Geom.
  16. Primitive deformations of quantum p-groups (with V. Nguyen and L.-H. Wang), submitted.
  17. On quantum groups associated to non-noetherian regular algebras of dimension 2 (with C. Walton), Math. Z. 284, 543-574 (2016).
  18. DG Poisson algebra and its universal enveloping algebra (with J.-F. Lv and G.-B. Zhuang), Sci. China Math. 59,  849-860 (2016).
  19. Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic, Adv. Math. 281, 594-623 (2015).
  20. A note on the duality between Poisson homology and cohomology (with J.-F. Lv and G. Zhuang), preprint.
  21. Universal enveloping algebras of differential graded Poisson algebras (with J.-F. Lv and G.-B. Zhuang), preprint.
  22. Universal enveloping algebras of Poisson Ore-extensions (with J.-F. Lv and G.-B. Zhuang), Proc. Amer. Math. Soc. 143, 4633-4645 (2015).
  23. Universal enveloping algebras of Poisson Hopf algebras (with J. F. Lv and G.-B. Zhuang), J. Algebra 426, 92-136 (2015).
  24. Classification of connected Hopf algebras of dimension p3 I (with V. Nguyen and L.-H. Wang), J. Algebra 424, 473–505 (2015).
  25. Classification of pointed Hopf algebras of dimension p3 over any algebraically closed field (with L.-H. Wang), Algebr. Represent. Theory 17, 1267-1276 (2014).
  26. Local criteria for cocommutative Hopf algebras, Comm. Algebra 42, 5180-5191 (2014).
  27. Another proof of Masuoka’s Theorem for semisimple irreducible Hopf algebras, preprint.
  28. Connected Hopf algebras of dimension p2, J. Algebra 391, 93-113 (2013).