臺灣博碩士論文加值系統

　　一個代數群如果和一個對角線值均為一的上三角矩陣群的子群同構，則該代數群被稱做冪一代數群。了解冪一代數群的分類結構是一個重要的課題，因為如果了解冪一代數群與約化群的分類足以使我們研究任意的線性代數群。然而冪一代數群的分類是至今仍未解決的。

　　在本論文中，我們將探究目前所知的冪一代數群分類理論。我們將廣泛地使用各種子群間的中心群擴張與非中心群擴張理論，並且在代數群是定義在特徵值為0的體上時，探討其相應的李代數。藉由這些工具，我們可以給出關於冪一代數群在等胚變換之內的分類結果，而我們的結論主要可應用於二維與三維的情形。

An algebraic group is said to be unipotent if it is isomorphic a subgroup of triangular matrix having only 1’s on the principal diagonal. Understanding the classification of unipotent groups has been an interesting topic, since such understanding as well as the understanding of the structure of reductive groups would be sufficient to study an arbitrary linear algebraic group. However the problem is so far unsolved.

In this thesis, we will give a survey on the classification on unipotent algebraic groups. We extensively analyze central and non-central extensions on various subgroups, and Lie algebras of groups on fields of characteristic 0 are also considered. Using these tools, we will give results on the classification of unipotent algebraic groups up to isogeny, mainly on dimension 2 and 3.

口試委員會審定書 i
誌謝 ii
中文摘要 iii
英文摘要 iv
第一章　Introduction 1
第二章　Unipotent Algebraic Groups of Dimension 2 1
2.1　Group Extensions 1
2.2　Unipotent Groups of Dimension 2 4
第三章　Unipotent Algebraic Groups of Dimension 3 7
3.1　The Case when k is of characteristic 0 7
3.2　The Case when G is commutative 8
3.3　The Case when G is non-commutative 9
3.3.1　dim G(1) = 1 15
3.3.3　dim G(1) = 2 17
參考文獻 20

[1] Cartan, H., Eilenberg, S., Homological Algebra, Princeton Math. Ser., no. 19.
[2] Demazure, M., Gabriel, P., Groupes Alg ebriques, Masson & Cie, 1991.
[3] Di Bartolo, A., Falcone, G., Plaumann, P. and Strambach K., Algebraic Groups and Lie
Groups with Few Factors, Lecture Notes in Mathematics, Vol. 1944, Springer, 2008.
[4] Fauntleroy, A., De ning Normal Subgroups of Unipotent Algebraic Groups, Proceedings
of the American Mathematical Society, Vol. 50, No. 1, pp. 14-20, 1976.
[5] Gong, M. P., Classi cation of Nilpotent Lie Algebras of Dimension 7 (over Algebraically
Closed Field and R), PhD thesis, University of Waterloo, Waterloo, Canada, 1998.
[6] Jacobson, N., Basic Algebra II, second edition, Dover Publications, 2009.
[7] Jacobson, N., Lie Algebras, Interscience Publishers, 1962.
[8] Kambayashi, T., Miyanishi, M. and Takeuchi, M., Unipotent Algebraic Groups, Lecture
Notes in Mathematics, Vol. 414, Springer-Verlag, 1974.
[9] Rosenlicht, M., Some Basic Theorems on Algebraic Group, Amer. J. Math. LXXVIII,
2 pp. 401-443, 1956.
[10] Serre, J-P., Algebraic Groups and Class Fields, Springer-Verlag, 1988.
[11] Springer, T.A., Linear Algebraic Groups, Birkh�auser, 2008.