On the structure of the centralizer of a braid
نویسندگان
چکیده
The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on n strands, which has at most k(k+1) 2 elements if n = 2k , and at most k(k+3) 2 elements if n = 2k+1. These bounds are shown to be sharp, by an example due to S. J. Lee. We conjecture that the set of generators given in this paper has the smallest possible number of elements. Finally, we describe how one can compute this generating set. AMS Classification 20F36; 20E07, 20F65.
منابع مشابه
Examples of centralizers in the Artin braid groups
The normalizer of an element of a group is defined as the subgroup of all elements commuting with it; this subgroup is often also called the centralizer of the given element. In the second version [F-GM] the authors switched to the latter terminology, and we will also use the term “centralizer”. The conjecture was supported by a new algorithm for finding generators of a centralizer, suggested i...
متن کاملDegenerate two-boundary centralizer algebras
Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the action of a complex reductive Lie algebra g on tensor space of the form M ⊗ N ⊗ V ⊗k. We define the degenerate two-boundary braid group Gk and show that centrali...
متن کاملOn centralizers of prime rings with involution
Let $R$ be a ring with involution $*$. An additive mapping $T:Rto R$ is called a left(respectively right) centralizer if $T(xy)=T(x)y$ (respectively $T(xy)=xT(y)$) for all $x,yin R$. The purpose of this paper is to examine the commutativity of prime rings with involution satisfying certain identities involving left centralizers.
متن کاملIrreducibility of the tensor product of Albeverio's representations of the Braid groups $B_3$ and $B_4$
We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$. We specialize the indeterminates used in defining these representations to non zero complex numbers. We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$. We then determine necessary and sufficient conditions that guarantee the irreducibility of th...
متن کاملConjugacy problem for subgroups with applications to Artin groups and braid type groups
Let G be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in G of any element of G. Let H be a subgroup of G. We give some conditions on H , under which we provide a solution to the conjugacy problem in H . We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy probl...
متن کامل