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.

  • 1
    $\begingroup$ 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. $\endgroup$ – Jim Humphreys Oct 30 '17 at 13:56
  • $\begingroup$ There is also Kac & Raina Highest Weight Representations of Infinite-Dimensional Lie Algebras (1987). $\endgroup$ – M.G. Oct 30 '17 at 16:06
  • $\begingroup$ @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? $\endgroup$ – QGravity Oct 31 '17 at 0:29
  • $\begingroup$ @July, Thank you for the reference, Does it explore infinite-dimensional highest weight modules as well? $\endgroup$ – QGravity Oct 31 '17 at 0:29
  • $\begingroup$ @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. $\endgroup$ – M.G. Oct 31 '17 at 1:18

I highly recommend the book

Affine Lie Algebras, Weight Multiplicities, and Branching Rules
by Kass, Moody, Patera, and Slansky.


  • $\begingroup$ This is an unusual book, with a second volume I don't have consisting of tables (now rather outmoded by the internet). The first volume has rather old-fashioned typesetting, in an oversized hardback format, and by now is hard to acquire but may be found in some libraries. It's #9 in a series Los Alamos Series in Basic and Applied Sciences, published in 1990 by U. California Press. But as my comment above indicates, all books listed involve overkill unless one really wants to study an affine Lie algebra (of infinite dimension) and its representations. $\endgroup$ – Jim Humphreys Nov 1 '17 at 16:02
  • $\begingroup$ Yes, it's an unusual book. I loved that book! $\endgroup$ – André Henriques Nov 1 '17 at 21:02
  • $\begingroup$ @AndréHenriques Thank you very much for the recommendation. $\endgroup$ – QGravity Nov 7 '17 at 5:20

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