Looijenga–Lunts–Verbitsky (LLV) decomposition
The Looijenga–Lunts–Verbitsky (LLV) Lie algebra acts on cohomology of every compact hyperkähler manifold. For a very general member of a deformation type, this gives a decomposition of $\operatorname{H}^\ast(X,\mathbb{Q})$ into irreducible representations of an orthogonal Lie algebra determined by $\mathrm{b}_2(X)$.
For ${\mathrm K3}^{[n]}$-type this is controlled by $\mathfrak{so}(25)$; for $\mathrm{Kum}_n$-type by $\mathfrak{so}(9)$; and for the sporadic types by their corresponding LLV algebras. Each LLV summand is compatible with the Hodge decomposition: it breaks into irreducible Hodge structures, and the shifts in degree are visible on the per-type panels on the main page.
The notation in the interface is: $V_\lambda$ for the irreducible LLV representation of highest weight $\lambda$, and $\bar V$ for the induced Hodge structure on $\operatorname{H}^2(X,\mathbb{Q})$. We display \[ \operatorname{H}^\ast(X,\mathbb{Q})=\bigoplus_{\lambda}V_\lambda^{\oplus m_\lambda} \] and refine each $V_\lambda$ into irreducible Hodge structures by degree shifts.
Lie algebra structure used in the display.
- For ${\mathrm K3}^{[n]}$-type: LLV algebra $\mathfrak{g}_{\mathrm{LLV}}\cong\mathfrak{so}(4,\mathrm{b}_2-2)\cong\mathfrak{so}(4,21)$, complexified to type $\mathrm{B}_{12}$.
- For $\mathrm{Kum}_n$-type: $\mathfrak{g}_{\mathrm{LLV}}\cong\mathfrak{so}(4,5)$, complexified to type $\mathrm{B}_{4}$.
- For $\mathrm{OG}_6$: the decomposition table used on this site is modeled with the same rank-$4$ orthogonal type as the $\mathrm{Kum}_n$ panel, namely type $\mathrm{B}_{4}$ (complex algebra $\mathfrak{so}(9,\mathbb{C})$).
- For $\mathrm{OG}_{10}$: $\mathfrak{g}_{\mathrm{LLV}}\cong\mathfrak{so}(4,20)$, complexified to type $\mathrm{D}_{12}$.
Background and expository references: [MR1465328, MR1406664, MR4516501].
References
- MR1465328
- Looijenga, E., & Lunts, V. A. (1997). A Lie algebra attached to a projective variety. Invent. Math., 129(2), 361–412. doi:10.1007/s002220050168 MR1465328
- MR1406664
- Verbitsky, M. (1996). Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal., 6(4), 601–611. doi:10.1007/BF02246797 MR1406664
- MR4516501
- Oberdieck, G., & Song, J. (2022). The LLV decomposition of hyperkähler cohomology and applications to the Nagai conjecture (after Green-Kim-Laza-Robles). Milan J. Math., 90(2), 485–501. doi:10.1007/s00032-022-00362-1 MR4516501