For a complex graded semisimple Lie algebra, Vinberg's theory of carrier algebras reduces the classification of nilpotent orbits in a homogeneous component to the classification of certain regular subalgebras, yielding a bijection and effective algorithms. Over the real numbers this bijection fails: the carrier map need be neither injective nor surjective, and the step from the (computable) list of real carrier algebras to the nilpotent orbits has only been completed case by case. The open problem is a general, provably complete classification: for an arbitrary real graded semisimple Lie algebra, a complete and irredundant list of representatives of the nilpotent orbits in the graded component, together with a proof of completeness and irredundancy. Such a classification must reduce the continuous families of elements in general position to finitely many conjugacy representatives, and must certify non-conjugacy of the surviving representatives.

Background

A semisimple real Lie algebra g\mathfrak{g} is Zm\mathbb{Z}_m-graded if it decomposes as g=g0⊕g1⊕⋯⊕gm−1\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\cdots\oplus\mathfrak{g}_{m-1} with [gi,gj]⊆gi+j mod m[\mathfrak{g}_i,\mathfrak{g}_j]\subseteq\mathfrak{g}_{i+j\bmod m}; the gradings of interest here admit a Cartan involution (an involutive automorphism with compact fixed-point algebra) that reverses the grading. Such gradings were classified by Djoković in 1982, and the graded components g1\mathfrak{g}_1 carry actions of the group G0G_0 with Lie algebra g0\mathfrak{g}_0 — in the convention of Dietrich–Faccin–de Graaf (2015), G0G_0 is the connected Lie subgroup of the adjoint group GG of g\mathfrak{g} (the connected Lie subgroup of Aut(g)\mathrm{Aut}(\mathfrak{g}) with Lie algebra ad g\mathrm{ad}\,\mathfrak{g}) having Lie algebra g0\mathfrak{g}_0; in Vinberg's terminology these actions are studied through his theory of θ\theta-groups (the class of graded actions obtained from a periodic automorphism θ\theta of a semisimple Lie algebra). An element e∈g1e\in\mathfrak{g}_1 is nilpotent if the operator adg(e)\mathrm{ad}_{\mathfrak{g}}(e) is nilpotent; by a graded version of the Jacobson–Morozov theorem, nilpotent elements lie in homogeneous sl2\mathfrak{sl}_2-triples, i.e. triples (h,e,f)(h,e,f) with [h,e]=2e[h,e]=2e, [h,f]=−2f[h,f]=-2f, [e,f]=h[e,f]=h and h∈g0h\in\mathfrak{g}_0, e∈g1e\in\mathfrak{g}_1, f∈g−1f\in\mathfrak{g}_{-1}. Classifying the nilpotent G0G_0-orbits in g1\mathfrak{g}_1 is the real analogue of classifying nilpotent orbits in a representation, and it underlies concrete problems such as the classification of alternating kk-forms on Rn\mathbb{R}^n up to GL(n,R)GL(n,\mathbb{R}) (important in differential geometry, e.g. trivectors in dimensions 8 and 9) and the classification of stationary black-hole solutions of extended supergravities, where duality groups act on charge vectors via symmetric pairs.

Over C\mathbb{C}, Vinberg (1979) showed that this classification reduces to a question about subalgebras: to each nilpotent element one associates its carrier algebra, a regular (meaning normalized by a Cartan subalgebra of the ambient algebra) Z\mathbb{Z}-graded semisimple subalgebra c=c−1⊕c0⊕c1\mathfrak{c}=\mathfrak{c}_{-1}\oplus\mathfrak{c}_0\oplus\mathfrak{c}_1 containing the element in general position (meaning [c0,e]=c1[\mathfrak{c}_0,e]=\mathfrak{c}_1), and the map sending a nilpotent orbit to the conjugacy class of its carrier algebra is a bijection. Since carrier algebras correspond to certain regular semisimple subalgebras, classifiable through root-system data, this yields an effective classification over C\mathbb{C}.

Dietrich, Faccin, and de Graaf (2015) extended the machinery to real graded semisimple Lie algebras: they gave an algorithm for the regular semisimple subalgebras of a real semisimple Lie algebra (implemented in GAP via the package CoReLG, building on the real Weyl group), and hence an algorithm listing the real carrier algebras up to G0G_0-conjugacy. However, over R\mathbb{R} the carrier map ψ\psi, from nilpotent G0G_0-orbits in g1\mathfrak{g}_1 to G0G_0-classes of carrier algebras, need be neither injective (one carrier algebra can contain representatives of several orbits) nor surjective (some carrier algebras are the carrier of no nilpotent element). They provided computable necessary and sufficient tests (their Proposition 35, via real ranks of centralizers) deciding whether a given carrier algebra is the carrier of a given element, and demonstrated the whole approach on three benchmark gradings of E8E_8-type and G2G_2-type, recovering known classifications. But going from the list of carrier algebras to the orbit list was, in their words, 'not an algorithm, but more a structured programme of work': it is not known in general how to reduce the continuous family of general-position elements in a carrier algebra to finitely many G0G_0-conjugacy representatives (this requires knowledge of the image ρ(G0)\rho(G_0) in GL(g1)GL(\mathfrak{g}_1)), nor how to certify that surviving candidate orbits are pairwise non-conjugate using G0G_0-invariants. Subsequent work has developed an alternative route through Galois cohomology: real orbits inside a complex nilpotent orbit correspond to the first Galois cohomology set H1(R,Z)H^1(\mathbb{R},Z) of the stabilizer ZZ of a homogeneous sl2\mathfrak{sl}_2-triple (a standard nonabelian cohomology set that classifies the real forms of a complex homogeneous space), and computing such cohomology sets has recently been algorithmized — powerful building blocks, but still assembled only case by case. No structural theory of the real carrier map, and no provably terminating complete classification algorithm for arbitrary real graded semisimple Lie algebras, is known.

Problem Statement

Develop a general classification theory for nilpotent orbits in real graded semisimple Lie algebras. Precisely: given an arbitrary real Zm\mathbb{Z}_m-graded semisimple Lie algebra g=⨁igi\mathfrak{g}=\bigoplus_i\mathfrak{g}_i with a grading-reversing Cartan involution and the connected real Lie group G0G_0 with Lie algebra g0\mathfrak{g}_0 (in the convention of Dietrich–Faccin–de Graaf (2015), the connected Lie subgroup of the adjoint group of g\mathfrak{g}), produce a complete and irredundant set of representatives of the nilpotent G0G_0-orbits in g1\mathfrak{g}_1 together with a proof of completeness and irredundancy. Such a classification must reduce the continuous family of elements in general position in each carrier algebra to finitely many G0G_0-conjugacy representatives, and must certify pairwise non-conjugacy of the resulting candidate orbits.

References

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