Derived representation theory of Lie algebras and stable homotopy categorification of \(sl_{k}\). (English) Zbl 1453.57013
Khovanov homology [M. Khovanov, Duke Math. J. 101, No. 3, 359–426 (2000; Zbl 0960.57005)] is now a well established categorification of the Jones polynomial which can be defined with fairly elementary means. But it can also be viewed as a categorification of representations of \(sl_2\). This less elementary point of view has led to the generalization of \(sl_k\)-Khovanov homology.
A different kind of generalization was the introduction of a stable homotopy type by R. Lipshitz and S. Sarkar [J. Am. Math. Soc. 27, No. 4, 983–1042 (2014; Zbl 1345.57014)] whose homology reduces to Khovanov homology. The paper here is now a step in obtaining a stable homotopy type for \(sl_k\)-Khovanov homology for \(k>2\). The approach is given by a generalization of a construction of J. Sussan [“Category \(\mathcal O\) and \(\mathfrak{sl}_k\) link invariants”, Preprint, arXiv:math/0701045] for \(sl_k\)-homology which requires a representation theory over the sphere spectrum \(S\). This creates various technical difficulties that the authors circumvent by working over a large prime (linearly larger than \(k\)). While this is a notable restriction, one can still expect extra information for complicated links relative to \(sl_k\)-homology.
A different kind of generalization was the introduction of a stable homotopy type by R. Lipshitz and S. Sarkar [J. Am. Math. Soc. 27, No. 4, 983–1042 (2014; Zbl 1345.57014)] whose homology reduces to Khovanov homology. The paper here is now a step in obtaining a stable homotopy type for \(sl_k\)-Khovanov homology for \(k>2\). The approach is given by a generalization of a construction of J. Sussan [“Category \(\mathcal O\) and \(\mathfrak{sl}_k\) link invariants”, Preprint, arXiv:math/0701045] for \(sl_k\)-homology which requires a representation theory over the sphere spectrum \(S\). This creates various technical difficulties that the authors circumvent by working over a large prime (linearly larger than \(k\)). While this is a notable restriction, one can still expect extra information for complicated links relative to \(sl_k\)-homology.
Reviewer: Dirk Schütz (Durham)
MSC:
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
| 55P42 | Stable homotopy theory, spectra |
| 55P48 | Loop space machines and operads in algebraic topology |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Keywords:
homotopy categorification; Khovanov homology; spectral algebra; spectral representation theoryReferences:
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