Many more occurrences at Victor Kac…

]]>Thanks for alerting me. I thought I was following references I saw, but checking now it seems I was hallucinating. Have fixed it now.

]]>Urs, I am puzzled, why do you use this strange spelling “Kač” ? I know it is Jewish family name in Slavic environment. In this family name it is pronounced ts (both in East and West) which is normally spelled c in Slavic languages, no diacritics in any dialect. I know he lived in Moldova in part of his youth so maybe there is indeed some reason there (I have seen this occasionally somewhere, e.g. https://vdocuments.mx/contents-laksovnotesinvariantpdf-invariant-theory-victor-kac-abstract-notes.html but I see no parallel in any other word that ts would be spelled as č). Wikipedia uses c both in Latin and Cyrillic. https://en.wikipedia.org/wiki/Victor_Kac

]]>and added pointer to:

- Minoru Wakimoto,
*Lectures on Infinite-Dimensional Lie Algebra*, World Scientific (2001) [doi:10.1142/4269]

[almost one year later…]

finally added these references (from #1-#3 above) to the entry

]]>i am worried that this is a boundary case in which the formula is not actually meant to apply.

Just to add that I checked with an expert, and the formula on that p. 9 indeed has a typo: The range of the variable $n$ there should exclude 0, hence the correct condition is instead $n \,\in\, \{1, \cdots, p -1\}$.

This way, the admissible weights at integer level $k$ are still just the usual “integrable” values, namely the elements of $\{0, 1, \cdots, k\}$.

But…

…apparently the extra weight $-1 \,mod\, k + 2$ – that I was hoping to see –, does appear as the weight of one extra highest weight module which is not integrable, but is an “affine Kac module”, according to Rasmussen arXiv:1812.08384.

Or so it seems (e.g. p. 19), I am still in the process of absorbing this.

]]>I am trying to get a better feeling for the non-integrable but “admissible” irreps of an affine Lie algebra $\widehat{\mathfrak{g}}_k$, due to

Victor G. Kač, Minoru Wakimoto,

*Modular invariant representations of infinite-dimensional Lie algebras and superalgebras*, PNAS**85**14 (1988) 4956-4960 (doi:10.1073/pnas.85.14.4956)Victor G. Kač, Minuro Wakimoto,

*Classification of modular invariant representations of affine algebras*, p. 138-177 in V. G. Kač (ed.):*Infinite dimensional lie algebras and groups*Advanced series in Mathematical physics**7**, World Scientific 1989 (pdf, cds:268092)

For the special case $\mathfrak{g} = \mathfrak{sl}_2$ the formula for the “admissible” weights is made explicit in

- B. Feigin, F. Malikov,
*Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level*, p. 357-405 in: Loday, Stasheff, Voronov (eds.)*Operads: Proceedings of Renaissance Conferences*, Contemporary Mathematics**202**, AMS 1997 (arXiv:q-alg/9511011, ams:conm-202)

(on p. 9).

Am I reading this right that even for integral level $k \,\in\, \mathbb{N}$, the weight $\Lambda = k + 1$ (which violates the integrability bound $\Lambda \leq k$) is “admissible”?

That would indeed be what their formula on p. 9 says (taking $q = 1$, $m = 1$, $n = 0$ and $p = k$ the level), but i am worried that this is a boundary case in which the formula is not actually meant to apply. But if it does, what is the role of that non-integrable but admissible module in the WZW theory (which would seem to be the usual one at integral level)?

]]>And, of course, the entry needs to be given full pointer to:

- Victor G. Kač,
*Infinite Dimensional Lie Algebras*, Progress in Mathematics**44**Springer 1983 (doi:10.1007/978-1-4757-1382-4), Cambridge University Press 1990 (doi:10.1017/CBO9780511626234)

For when the editing functionality is back, to add some of these references on relation of *affine Lie algebra*s to modular forms and $SL(2,\mathbb{Z})$-representations:

Victor G. Kač, Dale H. Peterson,

*Affine Lie algebras and Hecke modular forms*, Bull. Amer. Math. Soc. (N.S.)**3**3 (1980) 1057-1061 (bams:1183547694)Victor G. Kač, Dale H. Peterson,

*Infinite-dimensional Lie algebras, theta functions and modular forms*, Advances in Mathematics**53**2 (1984) 125-264 (doi:10.1016/0001-8708(84)90032-X)

review:

I. G. MacDonald,

*Affine Lie algebras and modular forms*, Séminaire Bourbaki : vol. 1980/81, exposés 561-578, Séminaire Bourbaki, no. 23 (1981), Exposé no. 577 (numdam:SB_1980-1981__23__258_0)Victor Kac, Minoru Wakimoto,

*Modular and conformal invariance constraints in representation theory of affine algebras*, Advances in Mathematics**70**2 (1988) 156-236 (doi:10.1016/0001-8708(88)90055-2, spire:275458)

(I am giving this comment the category “latest changes” not because I made a change to the entry *affine Lie algebra* – which is impossible at the moment – but just so that once it’s possible again, this here will be, I believe, in the same thread as whatever edit logs will come. )