Any deformation of a Weyl or Clifford algebra can be realized through some change of generators in the undeformed algebra. Here we briefly describe and motivate our systematic procedure for constructing all such changes of generators for those particular deformations where the original algebra is covariant under some Lie group and the deformed algebra is covariant under the corresponding quantu...

In his initial paper on braids E. Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin’s presentation for various generalizations of braids. The diverse aspects of presentations of braid groups and their generalizations continue to attract attention [12], [5], [15]. The canonical presentation of the braid group Brn was given by E. Artin [1]...

Let Fn be the free group on n generators, and PΣn be the group of automorphisms of Fn which send each generator to a conjugate of itself. Let Kn be the kernel of the homomorphism PΣn → PΣn−1 induced by mapping one of the free group generators to the identity. We show that Kn has cohomological dimension (n − 1), and that Hi(Kn, Z) is infinitely generated for 2 ≤ i ≤ (n − 1).

We present a method of producing normal subgroups of the free group on r generators not containing a given word. Special attention is given to the normal subgroups of the free group on two generators. In considering the exponent sum of the generators in a given word, we produce such subgroups and clearly state for what class of words these subgroups are not easily obtainable. We also provide a ...

We show that Szepietowski’s system of generators [8] for the mapping class group of the non-orientable surface is the minimal generating set by Dehn twists and Y -homemorphisms. Let Ng be a non-orientable surface which is a connected sum of g projective planes. We assume that g ≥ 4. Let M(Ng) be the group of isotopy class of homeomorphisms over Ng, i.e, the mapping class group of Ng. We introdu...

Hidden symmetries of ordinary differential equations (ODEs) are studied with nonlocal group generators. General forms are given for an exponential nonlocal group generator of an ODE that is reduced from a higher-order ODE, which is expressed in canonical variables and which is invariant under a two-parameter Lie group. The nonlocal group generator identifies a type I hidden symmetry. Type II hi...

In this expository article, which is a slightly expanded version of the lecture given at the All Ireland Algebra Days (Belfast, 16–19 May, 2001), we first recall a technique recently developed by F. Dalla Volta and A. Lucchini to study generation properties of finite groups. We then discuss some problems in permutation groups, linear groups and profinite groups where this technique has proved u...

Free Groups are, in a sense, the most generic kind of group. They are defined over a set of generators with no additional relations in between them. They play an important role in the definition of group presentations and in other fields. This theory provides the definition of Free Group as the set of fully canceled words in the generators. The universal property is proven, as well as some isom...

The diameter of a group G with respect to a set S of generators is the maximum over g E G of the length of the shortest word in S U S-’ representing g. This concept arises in the contexts of efficient communication networks and Rubik’s cube type puzzles. “Best” generators (giving minimum diameter while keeping the number of generators limited) are pertinent, to networks, “worst” and “average” g...

We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy the relations of Drinfel ′ d's new realization. Coproduct formulas are given and a PBW type basis is constructed. §0. Introduction. The purpose of this pape...