Related papers: Lie-Butcher series, Geometry, Algebra and Computat…
We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such…
We generalize the Umbral Calculus of G-C. Rota by studying not only sequences of polynomials and inverse power series, or even the logarithms studied in, but instead we study sequences of formal expressions involving the iterated logarithms…
We consider the Lie algebra associated with the descending central series filtration of the pure braid group of a closed surface of arbitrary genus. R. Bezrukavnikov gave a presentation of this Lie algebra over the rational numbers. We show…
A complex $\omega$-Lie algebra is a vector space $L$ over the complex field, equipped with a skew symmetric bracket $[-,-]$ and a bilinear form $\omega$ such that $$[[x,y],z]+[[y,z],x]+ [[z,x],y]=\omega(x,y)z+\omega(y,z)x+\omega(z,x)y$$ for…
This paper describes a formal theory of smooth vector fields, Lie groups and the Lie algebra of a Lie group in the theorem prover Isabelle. Lie groups are abstract structures that are composable, invertible and differentiable. They are…
In quantum groups coproducts of Lie-algebras are twisted in terms of generators of the corresponding universal enveloping algebra. If representations are considered, twists also serve as starproducts that accordingly quantize representation…
Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in…
In order to derive a class of geometric-type deformations of post-Lie algebras, we first extend the geometrical notions of torsion and curvature for a general bilinear operation on a Lie algebra, then we derive compatibility conditions…
Formal Laurent-Puiseux series are important in many branches of mathematics. This paper presents a {\it Mathematica} implementation of algorithms developed by the author for converting between certain classes of functions and their…
We analyze log-algebraic power series identities for formal groups of elliptic curves over $\mathbb{Q}$ which arise from modular parametrizations. We further investigate applications to special values of elliptic curve $L$-functions.
The set of primitive elements of a Hopf algebra in the braided category of group graded vector spaces (with a commutative group) carry the structure of a generalized Lie algebra. In particular the graded derivations of an associative…
Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms.…
The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson-Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic…
The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a canonical projection from the enveloping algebra $U(\mathfrak{g})$ to $\mathfrak{g}$.…
The notion of Lie algebroids over a topological ringed space provides a unified framework to study various geometric structures. This geometric concept is intimately connected with well-known algebraic structures, including Gerstenhaber…
We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) which we call aromatic B-series. We obtain an explicit…
We proof some basic tools about spaces of H"older-continuous functions between (in general infinite dimensional) Banach spaces and use them to construct new examples of infinite dimensional (LB)-Lie groups.
Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant…
In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…
We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space…