In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids u, v ∈ Bn and an automorphism φ ∈ Aut(Bn), decides whether v = (φ(x))−1ux for some x ∈ Bn. As a corollary, we deduce that each group of the form Bn o H, a semidirect product of the braid group Bn by a torsion-free hyperbolic group H, has solvable conjugacy problem.

We describe three ways of modifying the relativistic Heisenberg algebra first one not linked with quantum symmetries, second and third related with the formalism of quantum groups. The third way is based on the identification of generalized deformed phase space with the semidirect product of two dual Hopf algebras describing quantum group of motions and the corresponding quantum Lie algebra. As...

The Schur-Weyl Duality Theorem motivates the general deenition of dual representations. Such representations arise in a very large class of groups, including all compact, nilpotent and semidirect product groups, as well as the complementary groups in the physics literature and the reductive dual pairs in the mathematics literature. Several applications of the notion of duality are given, includ...

Let R be the pseudovariety of all finite semigroups whose principal right ideals have a unique generator, let L be its dual, and let J = R∩L. Using the word problem for free pro-J semigroups, it is shown that the bilateral semidirect product Sl ∗∗ J is local, where Sl denotes the pseudovariety of all finite semilattices. The global of the pseudovariety R ∨ L is also computed and it is shown tha...

A notion of graph-wreath product of groups is introduced. We obtain sufficient conditions for these products to satisfy the topologically inspired finiteness condition type Fn. Under various additional assumptions we show that these conditions are necessary. Our results generalize results of Bartholdi, Cornulier and Kochloukova about wreath products. Graphwreath products of groups include class...

In a previous paper, the authors introduced and characterized a new kind of matrices called {K,s+1}-potent. In this paper, an associated group to a {K, s+1}-potent matrix is explicitly constructed and its properties are studied. Moreover, it is shown that the group is a semidirect product of Z2 acting on Z(s+1)2−1. For some values of s, more specifications on the group are derived. In addition,...

We develop a theory of twisted actions of categorical groups using a notion of semidirect product of categories. We work through numerous examples to demonstrate the power of these notions. Turning to representations, which are actions that respect vector space structures, we establish an analog of Schur’s lemma in this context. Keeping new terminology to a minimum, we concentrate on examples e...

The Nordstrom-Robinson code NR is a non-linear binary code of length 16, with 2 codewords and minimum distance 6. Its automorphism group is a semidirect product of an elementary abelian group of order 16 and the alternating group A7. This group and the corresponding action of A7 is also at the origin of the sporadic A7-geometry. We construct this geometry and derive the Nordstrom-Robinson code ...

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups Uq(g). They have the same FRT generators l± but a matrix braided-coproduct ∆L = L⊗L where L = l+Sl−, and are se...