李代数是一类重要的非结合代数,随着时间的推移,李代数在数学以及古典力学和量子力学中的地位不断上升,其理论也在不断完善和发展,很多理论与方法已经渗透到了数学和理论物理的许多领域。
《李代数的表示:通过gln进行介绍(英文)》采用大胆而新颖的方法对李代数及其表示进行了论述。
《李代数的表示:通过gln进行介绍(英文)》共分八章,从对李代数概念的介绍入手,阐述了李代数及其表示的相关性质及理论,重点介绍了李代数在表示论中取得的一个重要成果——一般线性李代数不可约模的高权分类。
《李代数的表示:通过gln进行介绍(英文)》适合大学师生、研究生及数学爱好者参考阅读。
Why another introduction to Lie algebras? The subject of this book is one of the areas of algebra that has been most written about. The basic theory was unearthed more than a century ago and has been polished in a long chain of textbooks to a sheen of classical perfection. Experts' shelves are graced by the three volumes of Bourbaki; for students with the right background and motivation to learn from them, the expositions in the books by Humphreys, Fulton and Harris, and Carter could hardly be bettered; and there is a recent undergraduate-level introduction by Erdmann and Wildon. So where is the need for this book?
The answer comes from my own experience in teaching courses on Lie algebras to Australian honours-level undergraduates (see the Acknowledgements section). Such courses typically consist of 24 one-hour lectures. At my own university the algebraic background knowledge of the students would be: linear algebra up to the Jordan canonical form, the basic theory of groups and rings, the rudiments of group representation theory, and a little multilinear algebra in the context of differential forms. From that starting point, I have found it difficult to reach any peak of the theory by following the conventional route. My definition of a peak includes the classification of simple Lie algebras, the highest-weight classification of their modules, and the combinatorics of characters, tensor products, and crystal bases; by 'the conventional route' I mean the path signposted by the theorems of Engel and Lie (about solvability), Cartan (about the Killing form), Weyl (about complete reducibility), and Serre, as in the book by Humphreys. Following that path without skipping proofs always seemed to require more than 24 lectures.