On the Amount of Nondegenerate Tubular Orbits of 7-Dimensional Lie Algebras in $$\mathbb C^4$$

For any complex 6-dimensional nilpotent Lie algebra \(\mathfrak{g},\) we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain \(\mathfrak{g}\) by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras.

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in $ \Bbb C^4 $ we study in this article the 7-dimensional orbits of real Lie algebras in this space. By the well-known Morozov theorem, any nilpotent 7-dimensional Lie algebra has at least a 4-dimensional Abelian ideal. The article considers nilpotent indecomposable 7-dimensional Lie algebras containing a 5-dimensional Abelian ideal. It is proved that in the space $ \Bbb C^4 $ all the orbits of such algebras are Levi degenerate. This statement covers 73 algebras from the complete list of 149 indecomposable 7-dimensional nilpotent Lie algebras.

Special Hermitian metrics and Lie groups

An algorithm to determine the isomorphism classes of 4-dimensional complex Lie algebras

Moduli spaces of low dimensional complex Lie algebras have a unique stratification, determined by deformation theory, in which each stratum has the structure of a projective orbifold, in fact, it is given by an action of a finite group on complex projective space. For low dimensional real Lie algebras, the picture is similar, except that the strata are orbifolds given by the action of finite groups on spheres. We give a complete decomposition of the moduli spaces of real three and four dimensional Lie algebras and compare the structure of these moduli spaces to that of the corresponding complex Lie algebras.

Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2

Finite dimensional Lie algebras are semi-direct product of the semi-simple Lie algebras and solvable Lie algebras. 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. It is extremely important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this article, a new natural connection between the set of complex analytic isolated hypersurface singularities and the set of finite dimensional solvable (nilpotent) Lie algebras has been constructed. We construct finite dimensional solvable (nilpotent) Lie algebras naturally from isolated hypersurface singularities. These constructions help us to understand the solvable (nilpotent) Lie algebras from the geometric point of view. Moreover, it is known that the classification of nilpotent Lie algebras in higher dimensions ($\gt 7$) remains to be a vast open area. There are one-parameter families of non-isomorphic nilpotent Lie algebras (but no two-parameter families) in dimension seven. Dimension seven is the watershed of the existence of such families. It is well-known that no such family exists in dimension less than seven, while it is hard to construct one-parameter family in dimension greater than seven. In this article, we construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $11$ (resp. $10$) from $\tilde{E}_7$ singularities and show that the weak Torelli-type theorem holds. We shall also construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $12$ (resp. $11$) from $\tilde{E}_8$ singularities and show that the Torelli-type theorem holds. Moreover, we investigate the numerical relation between the dimensions of the new Lie algebras and Yau algebras. Finally, the new Lie algebras arising from fewnomial isolated singularities are also computed.

We classify kinematical Lie algebras in dimension D + 1 for D > 3 up to isomorphism. This is part of a series of papers applying deformation theory to the classification of kinematical Lie algebras in arbitrary dimension. This is approached via the classification of deformations of the relevant static kinematical Lie algebra. We also classify the deformations of the universal central extension of the static kinematical Lie algebra in dimension D + 1 for D > 3. In addition, we determine which of these Lie algebras admit an invariant inner product.

Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial) are, if the depth of their grading is greater than 1, results of generalized Cartan–Tanaka–Shchepochkina (CTS) prolongs. Here we are looking for new examples of simple finite dimensional modular Lie (super)algebras in characteristic 2 obtained as Cartan prolongs. We consider pairs (an (ortho-)orthogonal Lie (super)algebra or its derived algebra, its irreducible module) and compute the Cartan prolongs of such pairs. The derived algebras of these prolongs are simple Lie (super)algebras. We point out several amazing phenomena in characteristic 2: a supersymmetry of representations of certain Lie algebras, latent or hidden over complex numbers, becomes manifest; the adjoint representation of some simple Lie superalgebras is not irreducible.

Constants of Lie algebra actions

Classification of Orbit Closures of 4-Dimensional Complex Lie Algebras

A Lie algebra L over a field $$\mathbb {F}$$ is said to be zero product determined (zpd) if every bilinear map with the property that $$f(x,y)=0$$ , whenever x and y commute, is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra , where V is a simple $$\mathfrak {sl}_2$$ -module, is zpd if and only if $$\dim V =2$$ or $$\dim V$$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $$\mathscr {L}_q$$ and $$\mathscr {L}^+_q$$ , the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most 3, while the class of non-zpd Lie algebras includes the (4-dimensional) aging Lie algebra and all Lie algebras of dimension more than 3 in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of zpd Lie algebra by using it in the study of commutativity preserving linear maps.

In this paper we develop a method of construction of complex rigid solvable Lie algebras which is independent of coho-mological techniques or classification of Lie algebras. As an application, we classify all rigid solvable Lie algebras in dimension less or equal than eight, and we obtain partial results in dimension nine. Moreover, we give several examples of families of rigid Lie algebras in arbitrary dimension, some of them having its second cohomology group, in the Chevalley cohomology, non trivial.

We study one-parameter families of 7-dimensional nilpotent indecomposable Lie algebras and the orbits of holomorphic realizations of such algebras in a 4-dimensional complex space. It is shown, in contrast to the orbits of 5-dimensional nilpotent Lie algebras in 3-dimensional case, that two such families (out of the existing nine ones) admit orbits that are Levi non-degenerate (homogeneous) real non-spherical hypersurfaces. Up to holomorphic equivalence, all obtained non-degenerate nonspherical orbits are graphs of polynomials of degree 4

In this note we re-derive and generalize a dimension formula for block ideals of reduced universal enveloping algebras of a finite dimensional nilpotent restricted Lie algebra due to V. M. Petrogradsky and provide related results on the corresponding projective indecomposable modules. In particular, the dimension and the composition length of every projective indecomposable module is always a p-power. The example at the end of the paper shows that even for algebraically closed fields our results are more complicated than the analoguous results for finite p-nilpotent groups (cf. [8, Proposition 2.4 (b)]). In the following F will always be an algebraically closed field of prime characteristic p. For other undefined notation and some fundamental facts from restricted Lie algebra theory we refer to [11] or [5]. According to a classical result of H. Zassenhaus [11, Corollary 1.4.4 and Proposition 1.4.6], for every simple module S of a finite dimensional nilpotent Lie algebra L, there exists a function co : L -~ F such that