# Reference on Highest Weight Module of Kac-Moody Algebra

I am trying to understand this paper. The construction requires the understanding of the following concepts in the representation theory of simple and affine Lie algebras:

• The construction of Verma module for a general (not necessarily integral) highest-weight state;
• The character for these modules (the Weyl-Kac character formula cannot be applied for a generic non-integral highest-weight module);
• BRST reduction of affine Lie algebras;
• Quantum Drinfeld-Sokolov reduction;
• ...

I am looking for some references that explain these concepts or some detailed examples of the construction for simplest cases. I appreciate any comment.

• It would help to clarify here what you mean by the symbol $\mathfrak{u}(n)$. Aside from that, your screen name seems to imply that you are looking at Kac-Moody theory from the physics viewpoint. I'm mainly acquainted with some texts taking a more mathematical viewpoint, which might or might not be useful to you: Kac Infinite Dimensional Lie Algebras (3rd ed., Cambridge, 1990); Moody & Patera Lie Algebras with Triangular Decompositions (Wiley-Interscience, 1995); Carter Lie Algebras of Finite and Affine Type (Cambridge, 2005). All treat highest weight modules. – Jim Humphreys Oct 30 '17 at 13:56
• There is also Kac & Raina Highest Weight Representations of Infinite-Dimensional Lie Algebras (1987). – M.G. Oct 30 '17 at 16:06
• @JimHumphreys, Thank you for the references. Are they considering infinite-dimensional highest weight modules as well? I mean modules labeled by a set of complex parameters? – QGravity Oct 31 '17 at 0:29
• @July, Thank you for the reference, Does it explore infinite-dimensional highest weight modules as well? – QGravity Oct 31 '17 at 0:29
• @QGravity: if by module you mean a vector space acted upon by something like $\mathrm{GL}_{\infty}$ (i.e. in the sense that a $k[G]$-module is the same thing as a rep.), then yes. – M.G. Oct 31 '17 at 1:18