Related papers: V-systems, holonomy Lie algebras and logarithmic v…

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie…

This paper establishes a categorical framework for phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical s-matrices, and relative Rota-Baxter operators by introducing the concept of coherent homomorphisms. Starting with…

This paper studies the algebraic structure of a new class of hyperplane arrangement $A$ obtained by deleting two hyperplanes from a free arrangement. We provide information on the minimal free resolutions of the logarithmic derivation…

To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close…

An algorithm for embedding finite dimensional Lie algebras into Lie algebras of vector fields (and Lie superalgebras into Lie superalgebras of vector fields) is offered in a way applicable over ground fields of any characteristic. The…

We study the category of modules admitting compatible actions of the Lie algebra $\mathcal{V}$ of vector fields on an affine space and the algebra $\mathcal{A}$ of polynomial functions. We show that modules in this category which are…

We show that an arithmetic lattice $\Gamma$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $\delta(v)$ where $v = \mu (G/\Gamma)$ is the co-volume of the lattice. We prove that $\delta$ is polynomial in general and…

If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if $Ext_A^*(M,A) \neq 0$ for some A-module M of at most polynomial growth. Theorem 1: If f : X \to Y is a continuous map of finite category, and…

We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3).…

For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations $V(t)$, we decompose the tensor powers of $V(t)$ into irreducible factors in a uniform manner, using a tool we call {\it diagram…

Let $L$ be a finite dimensional Lie algebra over a field of characteristic $0$. Then by the original Levi theorem, $L = B \oplus R$ where $R$ is the solvable radical and $B$ is some maximal semisimple subalgebra. We prove that if $L$ is an…

Let $A$ be the path algebra of a Dynkin quiver $Q$ over a finite field, and $\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\mathscr{P})$ the category of 1-cyclic complexes over $\mathscr{P}$, and…

Given a flat connection on a manifold with values in a filtered L-infinity-algebra, we construct a morphism of coalgebras that generalizes the holonomies of flat connections with values in Lie algebras. The construction is based on…

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is…

A diagram obtained from the Carter diagram $\Gamma$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the {\it linkage diagram}. Given a linkage diagram, we associate…

We classify irreducible finite-dimensional modules of a collection of real Lie superalgebras that includes the simple ones, their classical variants, complex Lie superalgebras after restriction of scalars, and all real Lie algebras. Our…

We show how various constructions of $\mathbb{Z}$-graded Lie superalgebras are related to each other. These Lie superalgebras have a Lie algebra $\mathfrak{g}$ as the subalgebra at degree 0, an odd $\mathfrak{g}$-module V as the subspace at…

In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A…

We will prove the freeness of multi-Coxeter arrangements by constructing a basis of the module of vector fields which contact to each reflecting hyperplanes with some multiplicities using K. Saito's theory of primitive derivation.

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology…