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University of Saskatchewan

1. Azam, Saeid. Extended affine lie algebras and extended affine weyl groups.

Degree: 1997, University of Saskatchewan

URL: http://hdl.handle.net/10388/etd-10212004-001324

This thesis is about extended affine Lie algebras and extended affine Weyl groups. In Chapter I, we provide the basic knowledge necessary for the study of extended affine Lie algebras and related objects. In Chapter II, we show that the well-known twisting phenomena which appears in the realization of the twisted affine Lie algebras can be extended to the class of extended affine Lie algebras, in the sense that some extended affine Lie algebras (in particular nonsimply laced extended affine Lie algebras) can be realized as fixed point subalgebras of some other extended affine Lie algebras (in particular simply laced extended affine Lie algebras) relative to some finite order automorphism. We show that extended affine Lie algebras of type A1, B, C and BC can be realized as twisted subalgebras of types A§¤(l ¡Ã 2) and D algebras. Also we show that extended affine Lie algebras of type BC can be realized as twisted subalgebras of type C algebras. In Chapter III, the last chapter, we study the Weyl groups of reduced extended affine root systems. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements which in turn leads to a presentation of the Weyl group, called a presentation by conjugation. Using a new notion, called the index, which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems. We also show that extended affine Weyl groups (of index zero) are homomorphic images of some indefinite Weyl groups where the homomorphism and its kernel are given explicitly.
*Advisors/Committee Members: Berman, Stephen.*

Subjects/Keywords: mathematics; Lie algebra; extended affine Lie algebras; extended affine Weyl groups; automorphism

Record Details Similar Records

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

APA (6^{th} Edition):

Azam, S. (1997). Extended affine lie algebras and extended affine weyl groups. (Thesis). University of Saskatchewan. Retrieved from http://hdl.handle.net/10388/etd-10212004-001324

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

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

Azam, Saeid. “Extended affine lie algebras and extended affine weyl groups.” 1997. Thesis, University of Saskatchewan. Accessed September 21, 2020. http://hdl.handle.net/10388/etd-10212004-001324.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Azam, Saeid. “Extended affine lie algebras and extended affine weyl groups.” 1997. Web. 21 Sep 2020.

Vancouver:

Azam S. Extended affine lie algebras and extended affine weyl groups. [Internet] [Thesis]. University of Saskatchewan; 1997. [cited 2020 Sep 21]. Available from: http://hdl.handle.net/10388/etd-10212004-001324.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Azam S. Extended affine lie algebras and extended affine weyl groups. [Thesis]. University of Saskatchewan; 1997. Available from: http://hdl.handle.net/10388/etd-10212004-001324

Not specified: Masters Thesis or Doctoral Dissertation

University of Saskatchewan

2. Tan, Shaobin. A study of vertex operator constructions for some infinite dimensional lie algebras.

Degree: 1998, University of Saskatchewan

URL: http://hdl.handle.net/10388/etd-10212004-001516

In Chapter one of the thesis we construct a module for the toroidal Lie algebra and the extended toroidal Lie algebra of type A1. The Fock module representation obtained here is faithful and completely reducible over the extended toroidal Lie algebra. We also study the level two vertex operator representation of the toroidal Lie algebra of type A1. This generalizes the Lepowsky-Wilson study of the principal level two standard module for A(1)a. In Chapter two and three we study the core of the smallest extended affine Lie algebra which is not of finite or affine type. Let T(S) be the Jordan algebra constructed from a semilattice S of Rν (ν ≥ 1). Let K(T(S)) be the Lie algebra obtained from the Jordan algebra T(S) by the Tits-Kantor-Koecher construction. The TKK algebra K(T(S)) is the universal central extension of the Lie algebra K(T(S)). When one specializes this construction to the (non-lattice) semilatticeS of R2, one obtains the core of the smallest extended affine Lie algebra which is not of finite or affine type. We present a complete description of this TKK algebra K(T(S)), which then allows us to give a faithful representation to this Lie algebra by vertex operators. It is interesting that in the construction of this TKK algebra the Clifford algebra enters the picture. The situation here is similar to, but more complicated than, that for the level 2 standard A(1)1-module and the level 1 standard B(1)1 module, where the Lie algebras of operators act on a vector space of mixed boson-fermion states. In the last chapter of the thesis we give vertex operator constructions for the toroidal Lie algebra of type Bl (l ≥ 3). The constructions are related to the folding of Dynkin diagram of D(1)l+1 and a two-cocycle necessary for the vertex operator constructions. From the construction it follows that the Fock space also affords a representation of the Clifford algebra W, which is spanned by the operators ωj(j in 2Z+1) with the relation ωiωj+ωj
ωi=-2 δi+j,0 (i,j in 2Z+1). Moreover, the construction also suggests a direct construction of the toroidal Lie algebra T(Bl) by vertex operators. In fact, the second construction generalizes the Lepowsky-Primc construction of the level one standard module of B(1)l to the toroidal case.
*Advisors/Committee Members: Berman, Stephen.*

Record Details Similar Records

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

APA (6^{th} Edition):

Tan, S. (1998). A study of vertex operator constructions for some infinite dimensional lie algebras. (Thesis). University of Saskatchewan. Retrieved from http://hdl.handle.net/10388/etd-10212004-001516

Not specified: Masters Thesis or Doctoral Dissertation

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

Tan, Shaobin. “A study of vertex operator constructions for some infinite dimensional lie algebras.” 1998. Thesis, University of Saskatchewan. Accessed September 21, 2020. http://hdl.handle.net/10388/etd-10212004-001516.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Tan, Shaobin. “A study of vertex operator constructions for some infinite dimensional lie algebras.” 1998. Web. 21 Sep 2020.

Vancouver:

Tan S. A study of vertex operator constructions for some infinite dimensional lie algebras. [Internet] [Thesis]. University of Saskatchewan; 1998. [cited 2020 Sep 21]. Available from: http://hdl.handle.net/10388/etd-10212004-001516.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Tan S. A study of vertex operator constructions for some infinite dimensional lie algebras. [Thesis]. University of Saskatchewan; 1998. Available from: http://hdl.handle.net/10388/etd-10212004-001516

Not specified: Masters Thesis or Doctoral Dissertation