4,584 research outputs found
Some results on homology of Leibniz and Lie n-algebras
From the viewpoint of semi-abelian homology, some recent results on homology
of Leibniz n-algebras can be explained categorically. In parallel with these
results, we develop an analogous theory for Lie n-algebras. We also consider
the relative case: homology of Leibniz n-algebras relative to the subvariety of
Lie n-algebras.Comment: 14 page
Two-dimensional categorified Hall algebras
In the present paper, we introduce two-dimensional categorified Hall algebras
of smooth curves and smooth surfaces. A categorified Hall algebra is an
associative monoidal structure on the stable -category
of complexes of sheaves with
bounded coherent cohomology on a derived moduli stack .
In the surface case, is a suitable derived enhancement
of the moduli stack of coherent sheaves on the surface. This
construction categorifies the K-theoretical and cohomological Hall algebras of
coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case,
we define three categorified Hall algebras associated with suitable derived
enhancements of the moduli stack of Higgs sheaves on a curve , the moduli
stack of vector bundles with flat connections on , and the moduli stack of
finite-dimensional local systems on , respectively. In the Higgs sheaves
case we obtain a categorification of the K-theoretical and cohomological Hall
algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in
the other two cases our construction yields, by passing to , new
K-theoretical Hall algebras, and by passing to ,
new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and
the non-abelian Hodge correspondences can be lifted to the level of our
categorified Hall algebras of a curve.Comment: 54 page
The centroid of extended affine and root graded Lie algebras
We develop general results on centroids of Lie algebras and apply them to
determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody
Lie algebras, and Lie algebras graded by finite root systems.Comment: 35 page
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