# Working with quadratic Lie algebras

A quadratic Lie algebra is a Lie algebra with an invariant inner product and the main examples are semisimple Lie algebras. This definition then makes sense in any linear symmetric monoidal category. I have a series of questions about generalising constructions from semisimple Lie algebras to this abstract setting. Instead of one post with a list of questions (which is frowned on) I have in mind a series of posts. This will also allow me to "edit" posts to take into account any responses before posting.

The first construction is the universal enveloping algebra. This is not usually regarded as mysterious but there are some points to ponder. There are actually two constructions one is as a deformation of the symmetric algebra of the adjoint representation. This was Poincare's construction. This is an early example of a universal quantisation. This makes sense in the abstract setting of symmetric monoidal categories. The other approach was taken by Birkoff-Witt (independently, and decades later) and constructs the universal enveloping algebra as a quotient of the tensor algebra of the adjoint representation. This does not make sense in a symmetric monoidal category which is not abelian.

If we assume/impose the condition that the Casimir is non-zero on every non-trivial irreducible representation then I think the category of representations is semisimple. In this case the Birkoff-Witt construction is defined (and agrees with Poincare's approach).

Now we come to the Yangian. This is defined by Drinfeld by a presentation. More specifically as a quotient of the semi-direct product of the universal enveloping algebra and the tensor algebra. I have not seen a proof that this has the properties it should have. My first question is then whether there is anywhere I can find a proof?

The real question then is what is the simplest construction of the Yangian for a general quadratic Lie algebra? I say simplest because I believe the high-powered machinery of universal quantisations does give one construction. My hope would be constructions analogous to one or both of the constructions of the universal enveloping algebra.

• Bruce, have you looked at Molev's relatively recent book on Yangians for your foundational question? (He is mostly interested in the classical Lie algebra cases, which admit alternative presentations related to twisted Yangians. I am not convinced that there is only one construction of Yangian: Drinfeld famously gave two very different presentations.) For universal enveloping algebra in symmetric monoidal categories, see mathoverflow.net/questions/25020/… Jul 2, 2010 at 3:35
• Thanks Victor. I have added an answer to question 25020. I will see if I can find Molev's book. My concern is that $SL(n)$ and $GL(n)$ are exceptional in the sense that the adjoint representation appears in the symmetric square of the adjoint representation. Most of the work on Yangians relies on this which is not what I want. Jul 2, 2010 at 8:07
• This book relies on the evaluation homomorphism. The Yangians for orthogonal and symplectic groups are "twisted Yangians" which are not the usual ones. Jul 2, 2010 at 9:43