#### Advisor(s)

Valerio Toledano Laredo, Andrey V. Zelevinsky (1953-)

#### Contributor(s)

Venkatraman V. Lakshmibai, P. I. (Pavel I.) Etingof (1969-)

#### Date of Award

6-2011

#### Date Accepted

6-2011

#### Degree Grantor

Northeastern University

#### Degree Level

Ph.D.

#### Degree Name

Doctor of Philosophy

#### Department or Academic Unit

College of Science, Department of Mathematics

#### Keywords

mathematics, representation theory

#### Disciplines

Mathematics

#### Abstract

The aim of the current dissertation is to address certain problems in the representation theory of simple Lie algebras and associated quantum algebras.

In Part I, we study the simple Lie group of type $G_2$ from the cluster point of view. We prove a conjecture of Geiss, Leclerc and Schröer, relating the geometry of the partial flag varieties to cluster algebras in the case of $G_2$.

In Part II, we establish a concrete relationship between certain infinite dimensional quantum groups, namely the quantum loop algebra $U_{ \hbar}(L \mathfrak{g})$ and the Yangian $Y_{ \hbar}(\mathfrak{g})$, associated with a simple Lie algebra $\mathfrak{g}$. The main result of this part gives a construction of an explicit isomorphism between completions of these algebras, thus strengthening the well known Drinfeld’s degeneration homomorphism.

In Part III, we give a proof of the monodromy conjecture of Toledano Laredo for the case of $\mathfrak{g} = \mathfrak{sl}_2$. The monodromy conjecture relates two classes of representations of the affine braid group, the first arising from the quantum Weyl group operators of the quantum loop algebra and the second coming from the monodromy of the trigonometric Casimir connection, a flat connection introduced by Toledano Laredo.

#### Document Type

Dissertation

#### Rights Information

copyright 2011

#### Rights Holder

Sachin Gautam

#### Permanent URL

#### Recommended Citation

Gautam, Sachin, "Three contributions in representation theory: (1) cluster algebras and Grassmannians of type $G_2$ (2) Yangians and quantum loop algebras (3) monodromy of the trigonometric Casimir connection of $\mathfrak{sl}_2$" (2011). *Mathematics Dissertations.* Paper 17. http://hdl.handle.net/2047/d20001058

Click button above to open, or right-click to save.

## Notes

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Mathematics in the Graduate School of Arts and Sciences of Northeastern University.