Abstract The general integer mapping theory of invertible linear transforms is considered in this paper. Two terms of integer factor and elementary triangular matrix are introduced. We prove that there exists an integer implementation if a linear transform is invertible and in finite dimensional space. The integer implementation of an invertible linear transform has several advantages such as (...

We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) of low rank. give lower bounds for the ranks in terms rank adjoint subcategory order group invertible objects. As an application these results, we prove that MNSD 13 15 are pointed. In addition, show 17, 19, 21 23 either pointed or perfect.

A group–category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1–dimensional representations are the invertible simple objects. This paper gives a detailed exploration of “topological quantum field theories” for group–categories, in hopes of finding clues to a better u...

Let G be a group. An endomorphism φ of G is called rational if there exist a1, . . . , ar ∈ G and h1, . . . , hr ∈ Z, such that φ(x) = (xa1)1 . . . (xar)r for all x ∈ G. We denote by Endr(G) the group of invertible rational endomorphisms of G. In this note, we prove that G is nilpotent of class c (c ≥ 3) if and only if Endr(G) is nilpotent of class c − 1. Mathematics Subject Classification: 20E...

Given a categorical crossed module H→ G, where G is a group, we show that the category of derivations, Der(G,H), from G into H has a natural monoidal structure. We introduce the Whitehead categorical group of derivations as the Picard category of Der(G,H) and then we characterize the invertible derivations, with respect to the tensor product, in this monoidal category. 2000 Mathematics Subject ...

We establish commutativity theorems for certain classes of rings in which every invertible element is central, or, more generally, in which all invertible elements commute with one another. We prove that if R is a semiexchange ring (i.e. its factor ring modulo its Jacobson radical is an exchange ring) with all invertible elements central, then R is commutative. We also prove that if R is a semi...

Abstract. Let GL(2,H) be the group of invertible 2 × 2 matrices over the division algebra H of quaternions. GL(2,H) acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classe...

Abstract. Let nn(C) be the algebra of strictly upper-triangular n × n matrices and X2 = {u ∈ nn(C) : u2 = 0} the subset of matrices of nilpotent order 2. Let Bn(C) be the group of invertible upper-triangular matrices acting on nn by conjugation. Let Bu be the orbit of u ∈ X2 with respect to this action. Let S2n be the subset of involutions in the symmetric group Sn. We define a new partial orde...

We give explicit representations of the generalized Drazin inverse of a block matrix having generalized Schur complement generalized Drazin invertible in Banach algebras. Also we give equivalent conditions under which the group inverse of a block matrix exists and a formula for its computation. The provided results extend earlier works given in the literature. 2010 Mathematics Subject Classific...

1: We give an easy proof of Schützenberger’s Theorem stating that non-commutative formal power series are rational if and only if they are recognisable. A byproduct of this proof is a natural metric on a subgroup of invertible rational non-commutative power series. We describe a few features of this metric group.