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Math. colloquium/MP2 lecture

2010-03-17, 15:30

Euler, Skeppsgränd 3

Alexander Stolin: Classification of double Lie algebras

Abstract :
Let me first remind you of the definition of dual numbers over the complex numbers: They are numbers of the form a+eb, where a,b are complex numbers and e is a formal number such that the square of e is zero. Clearly, the set fo dual numbers forms a vector space of dimension 2. More generally, we call any algebra of dimension 2 a "double algebra". It is not difficult to prove that, along with the dual numbers, there exists only one other double algebra, namely the algebra of pairs (a,b) with componentwise operations.

It turns out that double Lie algebras can be defined for any Lie algebra in a similar (but, of course, much more complicated) way.

In my talk I will present a classification of double Lie algebras over Lie algebras of the form g[u] and g[[u]], where g is a simple finite dimensional Lie algebra and u is a formal variable (usually referred to in physics as the "spectral parameter"). This is joint work with F. Montaner (Spain) and E. Zelmanov (USA).