Structures And Representations Of The Twisted N=1Schr(o|¨)dinger-Neveu-Schwarz Algebra And Some Related Algebras

It is well known that infinite-dimensional Lie algebras and Lie super-algebras aitract widespread attention of mathematicians and physicists because of their deep physical background, whose struc-ture theory and representation theory have the vital significance and profound influence to many branches of mathematical physics. The structure theory and representation theory of the infinite-dimensional Lie algebras and Lie super-algebras related to the Virasoro Lie algebra and the N=1Neveu-Schwarz algebra have become one of the hot topics in Lie algebra research during the recent years. In this thesis, we mainly investigate some problems concerning some infinite-dimensional Lie algebras and Lie super-algebras such as the deformativc Schrodinger-Virasoro algebras, the twisted N=2super-conformal algebra and the twisted N=1Schrodinger-Neveu-Schwarz alge-bra, all of which are closely related to the Virasoro Lie algebra or the N=1Neveu-Schwarz algebra and possess their own physical and mathematical background.In the second chapter, we prove that all derivations of the twisted N=2super-conformal algebra (?) are inner ones and all Lie super-bialgebra structures over (?) are triangular coboundary according to the known related results of the corresponding centerless algebra (?). Meanwhile, we express the automorphism group of (?) as the semidirect product of its inner automorphism group and Z/4Z and in particular, we provide the generator of its outer automorphism group.In the third chapter, we determine all symmetric invariant bilinear forms of the deformative Schrodinger-Virasoro algebras in the context of different parameters. Then combining the known results of their second cohomology groups, we obtain the corresponding second Leibniz cohomol-ogy groups.In the forth chapter, we deduce that both of the second cohomology group and the Leibniz case over the twisted N=1Schrodinger-Neveu-Schwarz algebra without universal central extensions are one-dimensional and then obtain its universal central extension, which is denoted by tsns. We prove that tsns possesses two outer derivations and its automorphism group is isomorphism to that of its even part tsns0Meanwhile, we find non-triangular coboundary super-bialgebra structures on tsns and give the necessary and sufficient conditions making them to become triangular cobound-ary. Besides, we also prove that there does not exist irreducible mixed modules over tsns, and that any indecomposable module of the intermediate series over the twisted N=1Schrodinger-Neveu-Schwarz algebra is simply an indecomposable module of the intermediate series over the N=1Neveu-Schwarz algebra (in other words, the maximal ideal acts trivially on such kind of modules).