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.

Simply transitive NIL-affine actions of solvable Lie groups

Every simply connected and connected solvable Lie group 𝐺 admits a simply transitive action on a nilpotent Lie group 𝐻 via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups 𝐺 can act simply transitively on which Lie groups 𝐻. So far, the focus was mainly on the case where 𝐺 is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action ρ : G → Aff ⁡ ( H ) \rho\colon G\to\operatorname{Aff}(H) is simply transitive by looking only at the induced morphism φ : g → aff ⁡ ( h ) \varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h}) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group 𝐺 acts simply transitively on a given nilpotent Lie group 𝐻, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.

Jordan–Chevalley Decomposition in Lie Algebras

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and $A\in \mathfrak{s}$ , then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in \mathfrak{s}$ , $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$ ) such that $A=S+N$ .

ITERATED POWER INTERSECTIONS OF IDEALS IN RINGS OF ITERATED DIFFERENTIAL POLYNOMIALS

Let R be an n-iterated ring of differential polynomials over a commutative noetherian domain which is a ℚ-algebra. We will prove that for every proper ideal I of R, the (n + 1)-iterated intersection I(n + 1) of powers of I equals zero. A standard application includes the freeness of non-finitely generated projective modules over such rings. If I is a proper ideal of the universal enveloping algebra of a finite-dimensional solvable Lie algebra over a field of characteristic zero, then we will improve the above estimate by showing that I(2) = 0.

Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

We present a systematic procedure for the determination of a complete set ofkth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of twokth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case ofk= 2 and 31 classes for the case ofk≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of twokth-order (k≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

GROUND RING OF α-SYMMETRIES AND SEQUENCE OF RNS STRING THEORIES

We construct a sequence of nilpotent BRST charges in RNS superstring theory, based on new gauge symmetries on the worldsheet, found in this paper. These new local gauge symmetries originate from the global nonlinear space–time α-symmetries, shown to form a noncommutative ground ring in this work. The important subalgebra of these symmetries is U (3) × X6, where X6 is solvable Lie algebra consisting of six elements with commutators reminiscent of the Virasoro type. We argue that the new BRST charges found in this work describe the kinetic terms in string field theories around curved backgrounds of the AdS × CP n-type, determined by the geometries of hidden extra dimensions induced by the global α-generators. The identification of these backgrounds is however left for the work in progress.