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University of Illinois – Urbana-Champaign

1. Tian, Hongfei. On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic.

Degree: PhD, Mathematics, 2017, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/99314

In this thesis we prove the existence of Jordan Decomposition in D_{G/k}, the ring of invariant differential operators on a semisimple algebraic group over a field of positive characteristic, and its corollaries. In particular, we define the semisimple center of D_{G/k}, denoted by Z_{s}(D_{G/k}), as the set of semisimple elements of its center. Then we show that if G is connected, the semisimple center Z_{s}(D_{G/k}) contains Z_{s}(D_{G/k}^{(ν)}) for any positive interger ν, where Z_{s}(D_{G/k}^{(ν)}) is the ring of invariant differential operators on a Frobenius kernel derived from G.
*Advisors/Committee Members: Haboush, William J (advisor), Bergvelt, Maarten J (Committee Chair), Yong, Alexander (committee member), Nevins, Thomas A (committee member).*

Subjects/Keywords: Representation theory; Positive characteristic; Invariant differential operators; Semisimple center

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Tian, H. (2017). On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/99314

Chicago Manual of Style (16^{th} Edition):

Tian, Hongfei. “On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic.” 2017. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 29, 2020. http://hdl.handle.net/2142/99314.

MLA Handbook (7^{th} Edition):

Tian, Hongfei. “On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic.” 2017. Web. 29 Sep 2020.

Vancouver:

Tian H. On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2017. [cited 2020 Sep 29]. Available from: http://hdl.handle.net/2142/99314.

Council of Science Editors:

Tian H. On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2017. Available from: http://hdl.handle.net/2142/99314

2. Im, Mee Seong. On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution.

Degree: PhD, 0439, 2014, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/49392

We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of n × n matrices to (block) upper triangular matrices up to conjugation by invertible (block) upper triangular matrices. With this notion in mind, we describe the ring of invariant polynomials for interesting families of quivers, namely, finite ADE-Dynkin quivers and affine type \widetilde{A}-Dynkin quivers. We then study their relation to an important and fundamental object in representation theory called the Grothendieck-Springer resolution, and we conclude by stating several conjectures, suggesting further research.
*Advisors/Committee Members: Nevins, Thomas A. (advisor), Kedem, Rinat (Committee Chair), Nevins, Thomas A. (committee member), Bergvelt, Maarten J. (committee member), Schenck, Henry K. (committee member).*

Subjects/Keywords: Algebraic geometry; representation theory; quiver varieties; filtered quiver variety; quiver flag variety; semi-invariant polynomials; invariant subring; Derksen-Weyman; Domokos-Zubkov; Schofield-van den Bergh; ADE-Dynkin quivers; affine Dynkin quivers; quivers with at most two pathways between any two vertices; filtration of vector spaces; classical invariant theory; the Hamiltonian reduction of the cotangent bundle of the enhanced Grothendieck-Springer resolution; almost-commuting varieties; affine quotient

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Im, M. S. (2014). On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/49392

Chicago Manual of Style (16^{th} Edition):

Im, Mee Seong. “On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution.” 2014. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 29, 2020. http://hdl.handle.net/2142/49392.

MLA Handbook (7^{th} Edition):

Im, Mee Seong. “On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution.” 2014. Web. 29 Sep 2020.

Vancouver:

Im MS. On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2014. [cited 2020 Sep 29]. Available from: http://hdl.handle.net/2142/49392.

Council of Science Editors:

Im MS. On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2014. Available from: http://hdl.handle.net/2142/49392