The Lie bialgebra structures on the Witt and Virasoro algebras

A b s tr a c t ...................................................................................................................................... ii
A cknow ledgem ents .............................................................................................................. iii
D edication ............................................................................................................................... iv
1 . Introduction ...................................................................................................................... 1
2 . Q uantization of Lie b ia lg e b ra s ................................................................................. 4
2.1. Hopf algebras ........................................................................................................... 4
2.2. Quantization of Lie algebras ................................................................................ 8
2.3. Lie bialgebras ........................................................................................................... 10
2.4. Quasitriangular Hopf algebras ............................................................................ 12
3. Cohomology of graded Lie a lg e b ra s ................................................................... 16
3.1. Graded Lie alg eb ras ............................................................................................. 16
3.2. p-modules ................................................................................................................. 18
3.3. Duality of homology and cohomology of Lie algebras .................................... 22
3.4. Universal central extension of Lie subalgebras of V containing W . . . . 26
4. Lie bialgebra structures on the W itt and V irasoro a lg e b ra s ............... 33
4.1. Classical Yang Baxter equation ......................................................................... 34
4.2. Classification of finite dimensional subalgebras of the Witt and Virasoro
algebras ..................................................................................................................... 38
4.3. The Lie bialgebra structures on W \\ .................................................................. 40
4.4. Some cohomology results .................................................................................. 41
4.5. Proof of Main R e su lts ......................................................................................... 47
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5. Lie bialgebra structures of the saturated Lie subalgebras of the W itt
a lg eb ra ....................................................................................................................................... 50
5.1. Saturated subalgebras of the Witt alg eb ra ...................................................... 50
5.2. Calculations of the cohomology group L(I) A L{I)) .................... 52
5.3. Lie bialgebra structures on the saturated Lie subalgebras of W ............. 55
6. Uniqueness of the Taft Lie Bialgebras In Characteristic p .................... 57
6.1. Uniqueness for the case i = 0 (m odp) ................................................................ 57
6.2. Uniqueness for the case i « 0 (m odp) ................................................................ 63
R eferences ................................................................................................................................ 72
V ita .............................................................................................................................................. 75
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