Related papers: Free frobenius algebra on the differential forms o…
We discuss natural operations on loops in a quasi-surface and show that these operations define a structure of a quasi-Lie bialgebra in the module generated by the set of free homotopy classes of non-contractible loops.
Consider the real free Lie algebra $\mathfrak{fr}_n$ with generators $\omega_1$, \dots, $\omega_n$. Since it is positively graded, it has a completion $\overline{\mathfrak{fr}}_n$ consisting of formal series. By the Campbell--Hausdorff…
The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times…
We determine the equivariant real structures on nilpotent orbits and the normalizations of their closures for the adjoint action of a complex semisimple algebraic group on its Lie algebra.
We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators…
In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex…
In this paper we propose a general method for computing a minimal free right resolution of a finitely presented graded right module over a finitely presented graded noncommutative algebra. In particular, if such module is the base field of…
The rationality of the parafermion vertex operator algebra associated to any finite dimensional simple Lie algebra and any nonnegative integer is established and the irreducible modules are determined.
In this paper, we construct a pre-Lie structure on the free Lie algebra L(E) generated by a set E, giving an explicit presentation of L(E) as the quotient of the free pre-Lie algebra generated by E, by some ideal I. The main result in this…
We study post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic…
In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some "perfection" property (simpleness, primitiveness, primeness or…
We characterize prelie algebras in words of left ideals of the enveloping algebras and in words of modules, and use this result to prove that a simple complex finite-dimensional Lie algebra is not prelie, with the possible exception of f4.
We examine free orientation-reversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free…
A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type…
This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the…
In this work, we investigate anti-derivations and biderivation of Leibniz algebras. We describe general form of anti-derivations and biderivations on null-filiform and filiform Leibniz algebras. Moreover, we show how to construct Leibniz…
A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for…
The notion of an F-manifold algebra is the underlying algebraic structure of an $F$-manifold. We introduce the notion of pre-Lie formal deformations of commutative associative algebras and show that F-manifold algebras are the corresponding…
This is a survey article on the currently very active research area of free (=non-commutative) real algebra and geometry. We first review some of the important results from the commutative theory, and then explain similarities and…
We discuss free differential algebras (FDA's), a generalization of the Cartan-Maurer equations for the group manifold vielbein, appropriate for theories containing $p$-forms ($p >1$). Their dual formulation is an extension of Lie algebras,…