Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in $${\mathfrak {sl}}_2$$ sl 2 corresponds to the KdV hierarchy, and the triple $$(f,0,e_\theta )$$ ( f , 0 , e θ ) , where f is the sum of negative simple root vectors and $$e_\theta $$ e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy.

Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.

Airy Structures for Semisimple Lie Algebras

We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $${\mathbb {C}}$$ C , and to some extent also over $${\mathbb {R}}$$ R , up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $$\mathfrak {sl}_2$$ sl 2 , $$\mathfrak {sp}_4$$ sp 4 and $$\mathfrak {sp}_{10}$$ sp 10 . Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a number of interesting properties of these Airy structures and constructed many examples. Techniques used to derive our results may be described, broadly speaking, as an application of representation theory in semiclassical analysis.

Split Casimir Operator and Universal Formulation of the Simple Lie Algebras

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T⊗2 in the case when T is the adjoint representation. These projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.