Du kanske gillar. Spara som favorit. Skickas inom vardagar. These lectures focus on the idea of a highest weight representation, which goes through four different incarnations. It is a Kac—Moody algebra for which the generalized Cartan matrix is positive semi-definite and has corank 1. From purely mathematical point of view, affine Lie algebras are interesting because their representation theory , like representation theory of finite dimensional, semisimple Lie algebras is much better understood than that of general Kac—Moody algebras.
As observed by Victor Kac , the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.
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Their central extensions are precisely the twisted affine Lie algebras. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations transform amongst themselves under the modular group. The Lie bracket is defined by the formula. The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands. There is a distinguished derivation of the affine Lie algebra defined by.
The Dynkin diagram of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root.
Of course, such a node cannot be attached to the Dynkin diagram in just any location, but for each simple Lie algebra there exists a number of possible attachments equal to the cardinality of the group of outer automorphisms of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an untwisted affine Lie algebra.
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When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to twisted affine Lie algebras. The attachment of an extra node to the Dynkin diagram of the corresponding simple Lie algebra corresponds to the following construction. An affine Lie algebra can always be constructed as a central extension of the loop algebra of the corresponding simple Lie algebra.
If one wishes to begin instead with a semisimple Lie algebra, then one needs to centrally extend by a number of elements equal to the number of simple components of the semisimple algebra.
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The second one is a very readable survey of Lie group and Lie algebra cohomology. This questions seems to be closely related to the study of classical integrable systems and mathematical quantization.
enter Am I right? I think lot of information you are interested in contained there, albeit maybe not in the most explicit form. Algebra in mid s, book by Strade - start with a pretty much general context arbitrary field, arbitrary dimension and may be also relevant for your purpose.
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XIV , Academic Press Differential Geom. I do not have the chance to do it at present, but I remember some other refs by Lichnerowicz at around the same years. Being in French they're often forgot in the literature.
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Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. References: Infinite dimensional Lie algebras Ask Question. Asked 7 years, 7 months ago. Active 7 years, 5 months ago.
Viewed times. Weinstein has many papers along these lines. If you can browse the titles using MathSciNet or other sources you will probably find some relevant ones.