I'm given to understand that a lot of result on representation theory of Lie Algebras can be obtained by applying known result of representation theory of associative algebras to the enveloping algebras of lie algebras.

I understand why it works and that's not the problem. I'd just like to have a reference of some book that follow this kind of approach.

I mean a book that:

  • first develops a general theory for representation theory of associative algebras;
  • introduces universal enveloping algebras for a Lie algebra;
  • deduce most of we have to deduce from representation of Lie algebras.

Does anybody know the standard reference for this kind of program?

  • 1
    $\begingroup$ While a lot of things can be shown from associative algebra representation theory, anything that depends on the tensor product over the base field also being a representation won't be, and will essentially require the Hopf algebra structure to be known. You might be able to get some fruitful information from an intro to Hopf algebras? $\endgroup$ Commented Jan 27, 2017 at 23:22
  • $\begingroup$ Yes indeed, I won't have any problem with an Hopf algebra intro that manifestely treat this topic. $\endgroup$
    – Dac0
    Commented Jan 27, 2017 at 23:35

1 Answer 1


The universal enveloping algebra $U(L)$ of a finite-dimensional Lie algebra $L$ of characteristic zero is infinite-dimensional. So it seems much easier to develop representation theory of finite-dimensional Lie algebras on its own right. Still, there are very good lecture notes developing representation theory in general (of finite groups, of associative algebras, of Lie algebras, of Jordan algebras etc.), for example the following:

Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina: Introduction to representation theory.

  • $\begingroup$ Hi Dietrich, indeed I know these notes and I like them. Indeed was part of my motivation of asking this question. I was wondering if there was a more developed book on the argument. I also know that this way of proceeding would not be pedagogical. $\endgroup$
    – Dac0
    Commented Nov 23, 2016 at 11:09

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