On a class of extended toroidal Lie algebras coming from untwisted affine Yangians
Date
2024-07-24
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
ORCID
0009-0007-0981-9986
Type
Thesis
Degree Level
Masters
Abstract
The purpose of this thesis is to construct so-called \say{$\gamma$-extended toroidal Lie algebras}. Originally, these $\gamma$-extended toroidal Lie algebras were created to aid with endowing \say{toroidal Lie algebras}, i.e. universal central extensions:
$$\toroidal$$
of the Lie algebras of the form:
$$\g[v^{\pm 1}, t^{\pm 1}] := \g \tensor_{\bbC} \bbC[v^{\pm 1}, t^{\pm 1}]$$
(so-called \say{double-loop algebras}), where $\g$ is a finite-dimensional simple Lie algebra over $\bbC$, with Lie bialgebra structures, and the point of doing this is so that toroidal Lie bialgebras can be recognised as classical limits of certain quantum groups known as affine Yangians. Per the general theory of quantisations, such a Lie bialgebra structure on $\toroidal$ can by constructed by means of Manin triples of the form:
$$(\toroidal, \toroidal^{\positive}, \toroidal^{\negative})$$
In doing so, we must endow $\toroidal$ with an invariant bilinear form satisfying some conditions, but an issue that we will encounter in attempting this is that, any invariant bilinear form on a universal central extension is necessarily \textit{degenerate}. As such, we are motivated to enlarge toroidal Lie algebras into $\gamma$-extended Lie algebras:
$$\extendedtoroidal$$
and this is done in such a way that the resulting larger Lie algebras can then be endowed with \textit{invariant} symmetric bilinear forms that are also \textit{non-degenerate}. Importantly, the construction of these bilinear forms depends entirely on a certain linear map:
$$\gamma: \bbC[v^{\pm 1}, t^{\pm 1}] \to \bbC$$
(and hence the name of our Lie algebras).
We shall see that the Lie algebras $\extendedtoroidal$ all arise as \say{twists} of the semi-direct product $\toroidal \rtimes \der_{\gamma}(\bbC[v^{\pm 1}, t^{\pm 1}])$ by Lie $2$-cocycles $\sigma \in Z^2_{\Lie}(\der_{\gamma}(\bbC[v^{\pm 1}, t^{\pm 1}]), \z(\toroidal))$, with $\der_{\gamma}(\bbC[v^{\pm 1}, t^{\pm 1}])$ being a certain ($\Z^2$-graded) Lie subalgebra of the Lie algebra $\der(\bbC[v^{\pm 1}, t^{\pm 1}])$ of all derivations on $\bbC[v^{\pm 1}, t^{\pm 1}]$. Moreover, we will see that there is a readily available example of such a $2$-cocycle giving rise to a $\gamma$-extended toroidal Lie algebra that is \textit{not} isomorphic to the aforementioned semi-direct product.
Description
Keywords
Algebra, representation theory, Lie algebras, infinite-dimensional Lie algebras, quantum groups, affine Yangians
Citation
Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Mathematics