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Braiding via geometric Lie algebra actions. (English) Zbl 1249.14005
Let $$\mathfrak g$$ be a simply-laced Kac-Moody algebra. The notion of strong categorical $$\mathfrak{sl}_2$$-action (as functors between triangulated categories) was introduced in [J. Chuang and R. Rouquier, Ann. Math. (2) 167, No. 1, 245–298 (2008; Zbl 1144.20001)] and extended to general $$\mathfrak g$$ as above in [R. Rouquier, 2-Kac-Moody algebras, preprint (2008; arXiv:0812.5023)], and also in [M. Khovanov and A. D. Lauda, Represent. Theory 13, 309–347 (2009; Zbl 1188.81117); Trans. Am. Math. Soc. 363, No. 5, 2685–2700 (2011; Zbl 1214.81113)]. In turn, the notion of geometric categorical actions of $$\mathfrak{sl}_2$$ was introduced in [S. Cautis, J. Kamnitzer and A. Licata, Duke Math. J. 154, No. 1, 135–179 (2010; Zbl 1228.14011)] (as functors between derived categories of algebraic varieties); in this same paper it was shown that a geometric categorical $$\mathfrak{sl}_2$$-action gives rise to a strong categorical $$\mathfrak{sl}_2$$-action. In the present paper the definition of geometric categorical action is extended to general $$\mathfrak g$$ as above. Although the authors do not prove in this general context that a geometric categorical action is necessarily strong categorical (as they do expect), they show that, whether such an action is strong or geometrical, it gives rise to an action of the braid group of $$\mathfrak g$$. Two examples of geometrical categorical actions are discussed: one from minimal resolutions of Kleinian singularities (where $$\mathfrak g$$ arises from the MacKay correspondence) and another from coherent sheaves on cotangent bundles to $$n$$-step partial flag varieties (where $$\mathfrak g = \mathfrak{sl}_n$$).

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14M15 Grassmannians, Schubert varieties, flag manifolds 18F99 Categories in geometry and topology
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##### References:
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