Deformation Theory For Some Infinite-dimensional Graded Lie Algebras In Mathematical Physics

In this thesis, we explore deformation theory of some infinite-dimensional gradedLie algebras, including Schr(o|¨)dinger-Virasoro Lie algebra, extended Schr(o|¨)dinger-Viras-oro Lie algebra and W(2, 2) Lie algebra, in mathematical physics problems. Moreover,we introduce a new class of algebra, called Hom-Lie color algebra, which can also beviewed as a certain deformation of Lie algebra.Quantum groups originated in theoretical physics. The term first appeared in theinverse scattering method which was used to construct and solve quantum integrablesystems, and was then gradually developed by Drinfel'd and Jimbo, et al. Up to now,there is no single and precise definition for it. In general, a quantum group is somekind of Hopf algebras, which, in many cases, become universal enveloping algebras ofa certain Lie algebra, i.e., quantum universal enveloping algebras. Roughly speaking,a quantum group is a Hopf algebra neither commutative nor cocommutative. Drinfel'dgave a general method, based on the concepts of classical limit and quantization, forconstructing such kind of Hopf algebras. But the Drinfel'd twist, which is essentialfor quantization, is determined by the Lie bialgebra structures. This makes it verynecessary to investigate Lie bialgebra structures. There is no uniformed method toquantize all the Lie bialgebras so far. Actually, investigating Lie bialgebra structuresand quantizations of a specific Lie algebra is a complicated problem.We study Lie bialgebra structures and quantizations of the Schr(o|¨)dinger-VirasoroLie algebra and the extended Schr(o|¨)dinger-Virasoro Lie algebra. These two Lie alge-bras are both 2/1Z-graded and infinite-dimensional. The Schr(o|¨)dinger-Virasoro Lie alge-bra was introduced during the process of investigating the free Schr(o|¨)dinger equations.It is closely related to Schr(o|¨)dinger algebra and Virasoro algebra, both of which playimportant roles in many areas of mathematics and physics. While in the process ofprobing vertex representations of the Schr(o|¨)dinger-Virasoro Lie algebra, J. Unterbergerintroduced the extended Schr(o|¨)dinger-Virasoro Lie algebra, which can be viewed as anextension of the original Schr(o|¨)dinger-Virasoro algebra. But they differ greatly both in structure and representation theory. Such difference also appear in their Lie bial-gebra structures. Han-Li-Su proved that not all the Lie bialgebra structures on theSchr(o|¨)dinger-Virasoro Lie algebra are triangular coboundary. But this is not the casefor the extended Schr(o|¨)dinger-Virasoro Lie algebra, namely, all the Lie bialgebra struc-tures on it are triangular coboundary.In chapter 3, we quantize both the Schr(o|¨)dinger-Virasoro Lie algebra and the ex-tended Schr(o|¨)dinger-Virasoro Lie algebra. With the suitable choice of two pairs ofelements contained in the Schr(o|¨)dinger-Virasoro Lie algebra, say h,e with [h,e] = e,we construct two different Drinfel'd twists which are definitely determined by the Liebialgebra structures. Then by using the Drinfel'd twists, we quantize the Schr(o|¨)dinger-Virasoro Lie algebra and obtain two noncommutative and noncocommutative Hopf al-gebra structures. Similarly, we accomplish the quantization of the extended Schr(o|¨)dinger-Virasoro Lie algebra and get another new Hopf algebra neither commutative nor co-commutative.Exploring quantum deformations of Lie algebras is an important means of pro-ducing new algebras, and also one of the highlights of quantum group research. Weinvestigate the quantum deformation of the centerless W(2, 2) Lie algebra, denoted byW, which is an infinite-dimensional Z-graded Lie algebra. Firstly, we give an explicitrealization of this Lie algebra using the famous bosonic and fermionic oscillators intheoretical physics. Based on this, we define the quantum deformation W_q and themore general one W_qc of the centerless W(2, 2) Lie algebra, which can recover fromboth W_q and W_q^c in the q→1 limit. Furthermore, we construct the quantum groupstructures on W_q and obtain a noncommutative and noncocommutative Hopf algebra.Finally, we study the 1-dimensional central extension W_q = W_q Cc of W_q, whichturns out that W_q is also coincided with the conventional W(2, 2) Lie algebra definedby Zhang-Dong in the q→1 limit. In other words, the 1-dimensional central extensionW_q is exactly the quantum deformation of the conventional W(2,2) Lie algebra.The motivations to study Hom-Lie algebras are related to both physics and quan-tum deformations of Lie algebras, in particular Lie algebras of vector fields. Theparadigmatic examples are quantum deformations of Witt and Virasoro algebras. But,it is noteworthy that Hom-Lie algebras are indeed certain deformations of Lie algebras. We introduce a new class of algebras, called Hom-Lie color algebra. It can be regardedas a natural generalization of Hom-Lie algebras as well as Lie color algebras, and alsoa new deformation of Lie algebras. We study homomorphisms between Hom-Lie coloralgebras. In order to show the existence of such kind of algebras, especially the uni-versality, we present three options from which one can obtain a new Hom-Lie coloralgebra: a Hom-color algebra, a Lie color algebra along with an even algebra endo-morphism and a Hom-Lie color algebra by aσ-twist. We also offer some interestingand useful examples. Finally, we extend Hom-Lie admissible algebras to Hom-Lieadmissible color algebras and describe all these classes via G-Hom-associative coloralgebras, where G is a subgroup of the symmetric group S₃.