On Injective Homomorphisms for Pure Braid Groups, and Associated Lie Algebras
نویسندگان
چکیده
The purpose of this article is to record the center of the Lie algebra obtained from the descending central series of Artin’s pure braid group, a Lie algebra analyzed in work of Kohno [12, 13, 14], and Falk-Randell [9]. The structure of this center gives a Lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criterion used to test whether certain morphisms out of free groups are faithful [6]. However, it is as unclear whether this criterion for faithfulness can be applied to any open cases concerning representations of Pn such as the Gassner representation.
منابع مشابه
Approximation of a generalized Euler-Lagrange type additive mapping on Lie $C^{ast}$-algebras
Using fixed point method, we prove some new stability results for Lie $(alpha,beta,gamma)$-derivations and Lie $C^{ast}$-algebra homomorphisms on Lie $C^{ast}$-algebras associated with the Euler-Lagrange type additive functional equation begin{align*} sum^{n}_{j=1}f{bigg(-r_{j}x_{j}+sum_{1leq i leq n, ineq j}r_{i}x_{i}bigg)}+2sum^{n}_{i=1}r_{i}f(x_{i})=nf{bigg(sum^{n}_{i=1}r_{i}x_{i}bigg)} end{...
متن کاملFixed point approach to the Hyers-Ulam-Rassias approximation of homomorphisms and derivations on Non-Archimedean random Lie $C^*$-algebras
In this paper, using fixed point method, we prove the generalized Hyers-Ulam stability of random homomorphisms in random $C^*$-algebras and random Lie $C^*$-algebras and of derivations on Non-Archimedean random C$^*$-algebras and Non-Archimedean random Lie C$^*$-algebras for the following $m$-variable additive functional equation: $$sum_{i=1}^m f(x_i)=frac{1}{2m}left[sum_{i=1}^mfle...
متن کاملBraid Groups, Free Groups, and the Loop Space of the 2-sphere
The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin’s braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor [25], and Habegger-Lin [17, 22] on ”homotopy string links”. The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno [19, 20], ...
متن کاملOn braid groups and homotopy groups
The purpose of this article is to give an exposition of certain connections between the braid groups [1, 3] and classical homotopy groups which arises in joint work of Jon Berrick, Yan-Loi Wong and the authors [8, 2, 32]. These connections emerge through several other natural contexts such as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invari...
متن کاملCentralizers of Lie Algebras Associated to the Descending Central Series of Certain Poly-free Groups
Poly-free groups are constructed as iterated semidirect products of free groups. The class of poly-free groups includes the classical pure braid groups, fundamental groups of fiber-type hyperplane arrangements, and certain subgroups of the automorphism groups of free groups. The purpose of this article is to compute centralizers of certain natural Lie subalgebras of the Lie algebra obtained fro...
متن کامل