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Towards functor exponentiation. (English) Zbl 1391.18006

In Lie theory, the exponential map connects a Lie algebra and its Lie group. Idempotented version of quantized universal enveloping algebras of simple Lie algebras have been categorified by M. Khovanov and A. D. Lauda [Represent. Theory 13, 309–347 (2009; Zbl 1188.81117); Quantum Topol. 1, No. 1, 1–92 (2010; Zbl 1206.17015)] and R. Rouquier [“\(2\)-Kac-Moody algebras”, Preprint, arXiv:0812.5023].
This paper can be viewed as a small step towards lifting the exponential map to the categorical level. The authors focus on the case of \(\mathfrak{sl}_2\) and consider a possible framework to categorify the exponential map \(\exp(-f)\) given the categorification of a generator \(f\) of \(\mathfrak{sl}_2\) by A. D. Lauda [Adv. Math. 225, No. 6, 3327–3424 (2010; Zbl 1219.17012)]. In this setup the Taylor expansions of \(\exp(-f)\) and \(\exp(f)\) turn into complexes built out of categorified divided powers of \(f\).
The proposed framework is only an approximation to categorification of exponentiation, because the functors categorifying \(\exp(f)\) and \(\exp(-f)\) are not invertible. One possible application of categorified exponentiation would be the categorification of integral forms of Lie groups and the exponential map between a Lie algebra and its Lie group.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
16E20 Grothendieck groups, \(K\)-theory, etc.
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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References:

[1] Church, T.; Ellenberg, J.; Farb, B., FI-modules and stability for representations of symmetric groups, Duke Math. J., 164, 9, 1833-1910, (2015) · Zbl 1339.55004
[2] Khovanov, M., Heisenberg algebra and a graphical calculus, Fund. Math., 225, 169-210, (2014) · Zbl 1304.18019
[3] Khovanov, M.; Lauda, A. D., A diagrammatic approach to categorification of quantum groups I, Represent. Theory, 13, 309-347, (2009) · Zbl 1188.81117
[4] Khovanov, M.; Lauda, A. D., A categorification of quantum sl(n), Quantum Topol., 1, 1, 1-92, (2010) · Zbl 1206.17015
[5] Khovanov, M.; Sazdanovic, R., Categorification of the polynomial ring, Fund. Math., 230, 3, 251-280, (2015) · Zbl 1335.16016
[6] Khovanov, M.; Tian, Y., How to categorify the ring of integers localized at two · Zbl 1454.16005
[7] Lauda, A. D., A categorification of quantum sl(2), Adv. Math., 225, 3327-3424, (2010) · Zbl 1219.17012
[8] Rouquier, R., 2-Kac-Moody algebras
[9] Tian, Y., Towards a categorical boson-fermion correspondence · Zbl 1453.18015
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