Triangle Seminars
Tuesday, 1 Apr 2008
Making modules into algebras
Jan Grabowski
(Oxford University)
Abstract:
Modules for associative or Lie algebras are just vector spaces acted on by the algebra. This means that we can take the product of an algebra element with a module element in a sensible way. Typically, one cannot take the product of two module elements, though.
However, in certain circumstances, it is possible to make a module into an associative or Lie algebra, in a way that is compatible with the action. I will start with some examples for finite-dimensional Lie algebras, then infinite-dimensional Kac-Moody Lie algebras and finally quantized enveloping algebras. The products we get will turn out to involve braidings, giving us braided Lie algebras and braided enveloping algebras. I will also demonstrate one use for this extra structure, namely gluing together the module and the original algebra to get bigger algebras.
Modules for associative or Lie algebras are just vector spaces acted on by the algebra. This means that we can take the product of an algebra element with a module element in a sensible way. Typically, one cannot take the product of two module elements, though.
However, in certain circumstances, it is possible to make a module into an associative or Lie algebra, in a way that is compatible with the action. I will start with some examples for finite-dimensional Lie algebras, then infinite-dimensional Kac-Moody Lie algebras and finally quantized enveloping algebras. The products we get will turn out to involve braidings, giving us braided Lie algebras and braided enveloping algebras. I will also demonstrate one use for this extra structure, namely gluing together the module and the original algebra to get bigger algebras.
Posted by: KCL