Exceptional Lie algebra g2 and its representations

Download

thesis.pdf

Date

2022-9-01

Author

Kayakökü, Mehmet Mustafa

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

517
views

249
downloads

In the classification of complex simple Lie algebras, there are five of them whose Dynkin diagrams are of exceptional type. The Lie algebra g_2 has the smallest dimension among these exceptional Lie algebras and together with its corresponding Lie group G_2, it plays an important role in differential geometry, mathematical physics, and modern string theory. In this thesis after a general introduction to Lie algebras, we show the classification of complex simple ones. Afterward, we give several constructions of the exceptional Lie algebra g_2 and investigate its fundamental representations.

Subject Keywords

Lie algebra, Lie group, exceptional Lie algebra, exceptional Lie group

URI

https://hdl.handle.net/11511/99442

Collections

Graduate School of Natural and Applied Sciences, Thesis

Suggestions

OpenMETU
Core

Bivariate polynomial mappings associated with simple complex Lie algebras Küçüksakallı, Ömer (2016-11-01) There are three families of bivariate polynomial maps associated with the rank-2 simple complex Lie algebras A(2), B-2 congruent to C-2 and G(2). It is known that the bivariate polynomial map associated with A(2) induces a permutation of F-q(2) if and only if gcd(k, q(3) - 1) = I. for s = 1, 2, 3. In this paper, we give similar criteria for the other two families. As an application, a counterexample is given to a conjecture posed by Lidl and Wells about the generalized Schur's problem.
Value sets of bivariate folding polynomials over finite fields Küçüksakallı, Ömer (2018-11-01) We find the cardinality of the value sets of polynomial maps associated with simple complex Lie algebras B-2 and G(2) over finite fields. We achieve this by using a characterization of their fixed points in terms of sums of roots of unity.
Value sets of folding polynomials over finite fields Küçüksakallı, Ömer (2019-01-01) Let k be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra g. We find the cardinality of the value sets of the folding polynomials P-g(k)(x) is an element of Z[x] of arbitrary rank n >= 1, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.
Algebraic curves hermitian lattices and hypergeometric functions Zeytin, Ayberk; Önsiper, Mustafa Hurşit; Department of Mathematics (2011) The aim of this work is to study the interaction between two classical objects of mathematics: the modular group, and the absolute Galois group. The latter acts on the category of finite index subgroups of the modular group. However, it is a task out of reach do understand this action in this generality. We propose a lattice which parametrizes a certain system of ”geometric” elements in this category. This system is setwise invariant under the Galois action, and there is a hope that one can explicitly under...
Mutation classes of finite type cluster algebras with principal coefficients Seven, Ahmet İrfan (Elsevier BV, 2013-06-15) Cluster algebras of finite type is a fundamental class of algebras whose classification is identical to the famous Cartan Killing classification. More recently, Fomin and Zelevinslcy introduced another central notion of cluster algebras with principal coefficients. These algebras are determined combinatorially by mutation classes of certain rectangular matrices. It was conjectured, by Fomin and Zelevinsky, that finite type cluster algebras with principal coefficients are characterized by the mutation classe...

Citation Formats

M. M. Kayakökü, “Exceptional Lie algebra g2 and its representations,” M.S. - Master of Science, Middle East Technical University, 2022.