Representations Of The Klein-bottle Lie Algebras And Their Q-analog

Infinite-dimensional Lie algebras and their representations play an increasinglyimportant role in several branches of mathematics and physics. Representations ofseveral infinite-dimensional Lie algebras are studied in this paper.In chapter1, we study the representations of a class of infinite-dimensionalLie algebrasBrelated to the Klein-bottle. We give the necessary and sufcientconditions for all weight spaces of the highest weight irreducible module V (φ)of the Klein-bottle Lie algebraBto be finite dimensional. We prove that theirreducible highest weight module V (φ) equals to the VermaB-moduleVˉ(φ) if andonly if V (φ) has at least one infinite-dimensional weight space. Furthermore, wegive the classification of the quasi-finiteB-modules with nontrivial center, and weprove that the quasi-finiteB-module with nontrivial center is the highest weightB-module or the lowest weightB-module.In chapter2, we study the representations of the q-analog Klein-bottle LiealgebrasBq. We give the necessary and sufcient conditions for all weight spacesof the highest weight irreducible module V (φ) of the q-analog Klein-bottle LiealgebraBqto be finite dimensional. We prove that the irreducible highest weightBq-module V (φ) equals to the VermaBq-moduleVˉ(φ) if and only if V (φ) hasat least one infinite-dimensional weight space. We notice that the Lie algebraB′qCc can be embedded into the Lie algebra b∞. Next we give the maximalproper submodule of the Verma b∞-module when the highest weight is dominantintegral. Using the given maximal proper submodule, we finally characterize themaximal proper submodule of the VermaBq-module and give the e-character of theirreducible highest weightBq-module V (φ) when the highest weight φ satisfies somenatural conditions. Furthermore, we give the classification of the quasi-finiteBq-modules with nontrivial center, and we prove that the quasi-finiteBq-module with nontrivial center is the highest weightBq-module or the lowest weightBq-module.In chapter3, we construct a class of irreducible highest weight representationsof the q-analog Virasoro-like Lie algebraA qin terms of vertex operator. We noticethat the Lie algebraA qcan be embedded into a∞, and give the embedding formula,corresponding b∞, c∞, d∞-series of a∞, we give theB q,C q,D q-series of the Lie alge-brasA qand the embedding formulas. We construct the irreducible highest weightrepresentations ofB q,C q-series ofA qin terms of vertex operator. Furthermore, wegive the completely reducible representations of a∞on the space C[xj; j∈Z], andgive the polynomial representations ofA qandB q,C q,D q-series ofA q.