Hyperbolic Geometry with Clifford Algebra

The Clifford algebra in D. Hestenes formulation is used to study hyperbolic geometry and some interesting theorems are obtained. The computational power of this formulation is fully revealed by the ease of extending old results and discovering new ones. An important new result is the formulas on the area and perimeter of a convex polygon, based on extending Gauss equalities.     

A Numerical Comparison of Some Modified Controlled Random Search Algorithms

In this paper we propose a new version of the Controlled Random Search(CRS) algorithm of Price. The new algorithmhas been tested on thirteen global optimization test problems. Numericalexperiments indicate that the resulting algorithm performs considerablybetter than the earlier versions of the CRS algorithms. The algorithm,therefore, could offer a reasonable alternative to many currently availablestochastic algorithms, especially for problems requiring direct searchtype methods. Also a classification of the CRS algorithms is made based onglobal technique – local technique and the relative performance ofclasses is numerically explored.     

Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category . Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group the corresponding braided category of modules is identified with a full subcategory in . The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.     

Simplicial Homotopical Algebra and Satellites

We study here a notion of simplicial satellites, as a first step towards a characterisation of simplicial derived functors, a problem unsolved since the latter were introduced.The problem comes from the fact that, in contrast with the abelian case, simplicial derived functors do not produce by themselves an exact sequence. Our solution consists in extending them to commutative k-cubes, for all k, forming thus an exact system of functors universal within the connected ones; or, in other words, a system of simplicial satellites. The tool we develop here for this extension is the homotopy kernel of a commutative k-dimensional cubic diagram, generalising the homotopy kernel of a map; its 2-dimensional version has already been proved essential in other homotopical topics.     

Explicit Betti numbers for a family of nilpotent Lie algebras

Betti numbers for the Heisenberg Lie algebras were calculated by Santharoubane in his 1983 paper. However few other examples have appeared in the literature. In this note we give the Betti numbers for a family of -dimensional 2-step nilpotent extensions of by .

     

Weak amenability and the second dual of the Fourier algebra

Let be a locally compact group. We will consider amenability and weak amenability for Banach algebras which are quotients of the second dual of the Fourier algebra. In particular, we will show that if is weakly amenable, then has no infinite abelian subgroup.

     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

Minimal Model Fusion Rules From 2-Groups

The fusion rules for the (p,q)-minimal model representations of the Virasoro algebra are shown to come from the group in the following manner. There is a partition into disjoint subsets and a bijection between and the sectors of the (p,q)-minimal model such that the fusion rules correspond to where .     

Vertex Operators of the q-Virasoro Algebra; Defining Relations,Adjoint Actions and Four Point Functions

Primary fields of the q-deformed Virasoro algebra are constructed. Commutation relations among the primary fields are studied. Adjoint actions of the deformed Virasoro current on the primary fields are represented by the shift operator f(x) = f(x). Four point functions of the primary fields enjoy the connection formula associated with the Boltzmann weights of the fusion Andrews–Baxter–Forrester model.     

Lattice Ordered Polynomial Algebras

In this paper we define a lattice order on a set F of binary functions. We then provide necessary and sufficient conditions for the resulting algebra F to be a distributive lattice or a Boolean algebra. We also prove a Cayley theorem for distributive lattices by showing that for every distributive lattice , there is an algebra F of binary functions, such that is isomorphic to F and we show that F is a distributive lattice iff the operations and are idempotent and cummutative, showing that this result cannot be generalized to non-distributive lattices or quasilattices without changing the definitions of and . We also examine the equational properties of an Algebra for which , now defined on the set of binary -polynomials is a lattice or Boolean algebra.