Thanks, but they basically deal with Lie groups. The chapter about Lie algebras contains only the standard results as far as I can judge from the list of contents, and which are closely related to semisimple Lie algebras, as they are - with a few exceptions (Heisenberg, Poincaré) - the only ones with physical relevance. Then they have a little bit of the BRST calculus which again is physics.dextercioby said:If you cannot find here: Azcarraga, J., Izquierdo, J. - Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics (CUP, 1995), then I wonder which better source you can get.
The Chevalley Eilenberg Complex is a mathematical tool used in algebraic topology to study the cohomology of topological spaces. It is a cochain complex, which is a sequence of abelian groups connected by homomorphisms, that encodes information about the topology of a space.
The complex is constructed by taking the space of multilinear maps from a vector space to itself, and then taking the direct sum of these spaces over all possible numbers of inputs. The differential is then defined as the alternating sum of the partial derivatives.
The complex is significant because it allows for the computation of the cohomology of a space, which is an algebraic invariant that captures topological information. It also has applications in other areas of mathematics, such as algebraic geometry and representation theory.
In practice, the complex is often used to compute the cohomology of specific spaces, such as Lie groups or Lie algebras. It can also be used to prove theorems about topological spaces, such as the Brouwer fixed point theorem.
Yes, there are several variations and extensions of the complex, such as the Hochschild-Kostant-Rosenberg complex and the de Rham complex. These variations have different constructions and may be used to study different types of spaces or in different areas of mathematics.