Related papers: Integration of the Baker-Campbell-Hausdorff produc…
This short paper presents an efficient implementation of Baker-Campbell-Hausdorff formula for calculating the logarithm of product of two possibly non-commutative Lie group elements using only Lie algebra terms.
After the torch of Anders Kock [Taylor series calculus for ring objects of line type, Journal of Pure and Applied Algebra, 12 (1978), 271-293], we will establish the Baker-Campbell-Hausdorff formula as well as the Zassenhaus formula in the…
We solve the differentiation problem for Lie $\infty$-groups. Our approach builds on a classical version of Cartier duality which canonically identifies the Hopf algebra of point distributions supported at the identity of a Lie group with…
For a Lie groupoid G we prove an analogous of the Baker-Campbell-Hausdorff formula and we calculate the structure functions of the Lie algebroid associated to G.
The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product e X e Y can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction t{\'o}…
This paper presents new research in infinitesimal algebra by introducing the concept of an infinitesimal group and exploring its properties and ramifications. The author investigates first- and second-order subgroups of Lie groups and…
The main goal of this article is to develop integration theory for absolute partition $L_\infty$-algebras, which are point-set models for the (spectral) partition Lie algebras of Brantner-Mathew where infinite sums of operations are…
The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series H=ln(e^X e^Y) for non-commuting X,Y. Formally H lives in the graded completion of the free Lie algebra L generated by X,Y. We present a…
We prove that the spatial realization of a rational complete Lie algebra $L$, concentrated in degree 0, is isomorphic to the simplicial bar construction on the group, obtained from the Baker-Campbell-Hausdorff product on $L$.
The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators ${\bf X}$ and ${\bf Y}$, according to the Campbell-Baker-Hausdorff-Dynkin theorem,…
In this paper, first we revisit the formal integration of Lie algebras, which give rise to braces in some special cases. Then we establish the formal integration theory for complete Rota-Baxter Lie algebras, that is, we show that there is a…
The theory of Poisson-Lie groups and Lie bialgebras plays a major role in the study of one dimensional integrable systems; many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra…
This paper has two parts. The first part is a review and extension of the methods of integration of Leibniz algebras into Lie racks, including as new feature a new way of integrating 2-cocycles (see Lemma 3.9). In the second part, we use…
Using duality and topological theory of well behaved Hopf algebras (as defined in [2]) we construct star-product models of non compact quantum groups from Drinfeld and Reshetikhin standard deformations of enveloping Hopf algebras of simple…
We develop a homotopy theory of $L_\infty$ algebras based on the Lawrence-Sullivan construction, a complete differential graded Lie algebra which, as we show, satisfies the necessary properties to become the right cylinder in this category.…
We introduce the notion of a bicocycle double cross product (resp. sum) Lie group (resp. Lie algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a…
The Baker-Campbell-Hausdorff series computes the quantity \begin{equation*} Z(X,Y)=\ln\left( e^X e^Y \right) = \sum_{n=1}^\infty z_n(X,Y), \end{equation*} where $X$ and $Y$ are not necessarily commuting, in terms of homogeneous multinomials…
We introduce a simplified (coarse) version of pseudo-differential calculus for operators of order zero on complete Riemannian manifolds. This calculus works for the usual Hormander (1,0) class of operators, as well as for…
We reconsider the quantization of symbols defined on the product between a nilpotent Lie algebra and its dual. To keep track of the non-commutative group background, the Lie algebra is endowed with the Baker-Campbell-Hausdorff product,…
In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, 'model' and 'realization', between the categories of simplicial sets and complete differential…