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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 10th 2022
• (edited Feb 10th 2022)

For when the editing functionality is back, to add some of these references on relation of affine Lie algebras 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. )

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 11th 2022

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 14th 2022
• (edited Feb 14th 2022)

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)?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 16th 2022
• (edited Feb 16th 2022)

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.