| Representation theory has been widely used in the study of abstract algebraic structures byrepresenting their elements as linear transformations of vector spaces, and studies modules overthese abstract algebraic structures. The algebraic objects include groups, associative algebrasand Lie algebras, etc. The vertex representation theory of affine Lie algebras is particularly im-portant in mathematics and physics. In the representation of affine Kac-Moody Lie algebras,the basic module can be defined with a Fock space while the affine Lie algebra is certain vertexoperators on the Fock space. One construction for the basic module is called principal, whichwas found by Lepowsky and Wilson[36]for affine Lie algebra(1)1, and Kac et al.[33]generalizedall affine Lie algebra of type ADE. Another construction is called homogeneous, which wasdeveloped by Frenkel and Kac[14]. In1988, Drinfeld[6]introduced a new realization of quantumaffine algebras g, which is usually called Drinfeld quantum affinization of g. The newrealization has various applications, such as vertex representation[9,23,44], finite dimensional rep-resentation[2,3]etc. Including Frenkel and Jing[9]established untwisted quantized affine algebrasby homogeneous vertex operator representations. Later, Jing[23]showed the twisted cases.In the theory of the extended affine Lie algebras, toroidal algebras play an especially im-portant role in the study. The first realization of a toroidal algebra appeared as a vertex represen-tation of the affinized Kac-Moody algebra[8]. The loop algebraic representation of toroidal Liealgebras was given by Moody et al.[39], which shows the similarity with the affine Kac-Moodyalgebras. Quantum toroidal algebras appeared in the study of the Langlands reciprocity for al-gebraic surfaces[18]. More general quantum toroidal Kac-Moody algebras were constructed[23]in using Drinfeld presentation and vertex representations, and the quantum Serre relations werefoundtobecloselyconnectedwithsomenontrivialrelationsofHall-Littlewoodsymmetricfunc-tions. Several other interesting examples appeared in various contexts, such as toroidal Schur-Weyl duality[46], vertex representations[42], McKay correspondence[13]and toroidal actions onlevel one representations[43,47]and higher level analogs for quantum affine algebras[45].Like the theory of affine algebras, quantum toroidal algebras also have twisted analogs.Based on the quantum general linear algebra action on the quantum toroidal algebra[16], a quan-tum TKK algebra was constructed[17]using homogeneous-deformed vertex operators in con- nection with special unitary Lie algebras, where the Serre relation was found to be equivalent tosome non-trivial combinatorial identity of Hall-Littlewood polynomials. In chapter2, we con-struct a new twisted quantum toroidal algebra of type A1[27]as an analog of the quantum TKKalgebra. Explicit realization of the new quantum TKK algebra is constructed with the help oftwisted quantum vertex operators over a Fock space.On one hand, a principal quantum affine algebra was constructed[24]by deforming theKac-Moody algebras with an involution. However that algebra is larger than its classical analogas the Serre relations were not known. On the other hand, based on quantum homogeneousconstructions[9,23], the quantum principal realization was also excepted. In charpter3, as anelementary study of quantization of principally affine Lie algebra, we introduce a principallygraded quantum Kac-Moody algebras and get the associated Serre relations[28]. It is hopeful thatit can be improved the representation theory. |
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