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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,)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 𝔤^ k\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 𝔤=𝔰𝔩 2\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 𝔰𝔩 2^\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 kk \,\in\, \mathbb{N}, the weight Λ=k+1\Lambda = k + 1 (which violates the integrability bound Λk\Lambda \leq k) is “admissible”?

    That would indeed be what their formula on p. 9 says (taking q=1q = 1, m=1m = 1, n=0n = 0 and p=kp = 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 nn there should exclude 0, hence the correct condition is instead n{1,,p1}n \,\in\, \{1, \cdots, p -1\}.

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

    But…

    …apparently the extra weight 1modk+2-1 \,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.

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