<Emphasis Type="Italic">N</Emphasis>-homogeneous <Emphasis Type="Italic">C</Emphasis>*-algebras generated by idempotents

In this paper we consider n-homogeneous C*-algebras generated by idempotents. We prove that a finitely generated unital n-homogeneous (when n is greater than or equals 2) C*-algebra A can be generated by a finite set of idempotents if and only if the algebra A contains at least one nontrivial idempotent.     

Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following.Let p ₁,...,p _k be idempotents in a Banach algebra B, and assume p ₁+...+p _k =0.Does it follow that p _j =0,j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four.     

The p-Blocks of the Mackey Algebra

Let p be a prime number. This paper describes the primitive idempotents and prime spectrum of the crossed Burnside algebra of a finite group over a p-local ring. The main application is a formula for the block idempotents of the p-local Mackey algebra of the group, in terms of the corresponding bloks of the group algebra.     

Duadic Z4-Codes

The structure of abelian Z₄-codes (and more generally Z_p^m-codes) is studied. The approach is spectral: discrete Fourier transform and idempotents. A criterion for self-duality is derived. An arithmetic test on the length for the existence of nontrivial abelian self-dual codes is derived. A natural generalization of both the supplemented quadratic residue codes and the binary duadic codes is introduced. Isodual abelian Z₄ codes are considered, constructed, and used to produce 4-modular lattices.     

Additivity of n-Multiplicative Maps on Alternative Rings

Let ? and ?′ be alternative rings. We study the additivity of n-multiplicative isomorphisms from ? onto ?′ and of n-multiplicative derivations of ?. We prove that, if ? contains a family of nontrivial idempotents satisfying Martindale's conditions, then these two maps are additives.     

A torsion-free abelian group of finite rank exists whose quotient group modulo the square subgroup is not a nil-group

The first example of a finite rank torsion-free abelian group A such that the quotient group of A modulo the square subgroup of A is not a nil-group is indicated (in both cases of associative and general rings). In particular, the answer to the question posed by A.E. Stratton and M.C. Webb in [18], Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130, is given for finite rank torsion-free groups. A relationship between nontrivial p-pure subgroups of the additive group of p-adic integers and nontrivial ? [p^?1]-submodules of the field of p-adic numbers is investigated. In particular, a bijective correspondence between these structures is proven using only elementary methods.     

Triangular Matrix Representations

In this paper we develop the theory of generalized triangular matrix representation in an abstract setting. This is accomplished by introducing the concept of a set of left triangulating idempotents. These idempotents determine a generalized triangular matrix representation for an algebra. The existence of a set of left triangulating idempotents does not depend on any specific conditions on the algebras; however, if the algebra satisfies a mild finiteness condition, then such a set can be refined to a “complete” set of left triangulating idempotents in which each “diagonal” subalgebra has no nontrivial generalized triangular matrix representation. We then apply our theory to obtain new results on generalized triangular matrix representations, including extensions of several well known results.     

The Primitive Idempotents of a Cyclic Group Algebra

Explicit expressions are obtained for the 2n + 1 primitive idempotents in FG, the semisimple group algebra of the cyclic group G of order pⁿ (p an odd prime, n ≥ 1) over the finite field F of prime power order q, when q has order φ(pⁿ)/2 modulo pⁿ.AMS Mathematical Subject Classification (2000): 20C05, 94B05, 12E20, 16S34.     

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p¹ -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Z_p? Z_p? Z_pcan be embedded in U(KG).     

A Basis for the Graded Identities of the Matrix Algebra of Order Two over a Finite Field of Characteristic p≠2

Let K be a finite field of characteristic p>2, and let M₂(K) be the matrix algebra of order two over K. We describe up to a graded isomorphism the 2-gradings of M₂(K). It turns out that there are only two nonisomorphic nontrivial such gradings. Furthermore, we exhibit finite bases of the graded polynomial identities for each one of these two gradings. One can distinguish these two gradings by means of the graded polynomial identities they satisfy.     

On Some Geometric Representations of GL N (𝔬)

We study a family of complex representations of the group GL _n (𝔬), where 𝔬 is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL _n (F) to its maximal compact subgroup GL _n (𝔬). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra associated with the representation and an algebraic basis that consists of its minimal idempotents. The transition matrix involves combinatorial invariants of lattices of submodules of finite 𝔬-modules. The idempotents are p-adic analogs of the multivariable Jacobi polynomials.     

The Idempotent Manifold in a Clifford Algebra

The idempotent sets in sufficiently sophisticated algebras form manifolds and Hausdorff spaces. In this paper it is shown how the idempotents in a real Clifford algebra Cl_p,q can be calculated by nilpotents and reflections. Minimal sets of nilpotents are given and generating relations are defined. It is shown that the manifold thus constructed is complete. Every idempotent in the manifold can be calculated in the way proposed here, namely by a nilpotent multinomial form.     

On Maps Determined by Zero Products

The main result describes a bijective additive map h between prime rings with nontrivial idempotents that satisfies h(x)h(y)h(z) = 0 whenever xy = yz = 0. The proof is based on the consideration of a multiadditive map satisfying a related condition.     

Field sensitivity to variations of a scatterer

For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational (Fréchet) derivative of the scattered field measured in the H¹ (root-mean-square-gradient) norm as a function of the material coefficients measured in an L^p (p-power integral) norm, with 2<p<∞, as long as these coefficients are bounded above and below by positive constants and do not admit resonance. The derivative is Lipschitz continuous. We also establish the variational derivative of the transmitted energy with respect to the material coefficients in L^p.     

On Schur's Q-functions and the Primitive Idempotents of a Commutative Hecke Algebra

Let B _n denote the centralizer of a fixed-point free involution in the symmetric group S _2n . Each of the four one-dimensional representations of B _n induces a multiplicity-free representation of S _2n , and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of S _n.     

The Number of Permutation Polynomials of a Given Degree Over a Finite Field

We relate the number of permutation polynomials in F_q[x] of degree d≤q−2 to the solutions (x₁,x₂,…,x_q) of a system of linear equations over F_q, with the added restriction that x_i≠0 and x_i≠x_j whenever i≠j. Using this we find an expression for the number of permutation polynomials of degree p−2 in F_p[x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.     

Biordered sets and fundamental semigroups

Given any biordered set E, a natural construction yields a semigroup T _E that is always fundamental, in the sense that T _E possesses no nontrivial idempotent-separating congruence. In the case that E=E(S) is the biordered set of idempotents of a semigroup S generated by regular elements, there is a natural representation of S by T _E , such that S becomes a biorder-preserving coextension of a fundamental and symmetric subsemigroup of T _E . If further S is regular then this yields the fundamental constructions of Nambooripad, Grillet and Hall, which in turn generalise the construction of Munn of a maximum fundamental inverse semigroup from its semilattice of idempotents.     

Spontaneous Generation of Effective Interaction in a Renormalizable Quantum Field Theory Model

We consider the theory of a massless scalar field with the g³ coupling in a six-dimensional space. We use Bogoliubov's method of quasiaverages to study the possibility of a breaking of the original scaling symmetry and of the corresponding spontaneous generation of the effective G⁴ coupling. We show that the linearized compensation equation for the form factor of this coupling has a nontrivial solution through the third-order approximation. In the same approximation, the Bethe–Salpeter equation for a massless scalar bound state of two fields also has a solution. Matching the values of the form factor and the scalar field mass m at zero leads to a unique solution that gives a relation between the parameters of the g³ coupling and the parameters G and m. We argue in favor of the stability of the nontrivial solution obtained.     

Reflection Theorems and the Tame Kernel of a Number Field

Let L be a number field containing the pth primitive root of unity ζ _p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L.

Let F be a number field and 𝒪 _F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K ₂𝒪 _F and that of some direct summands of the ideal class group of F(ζ _p ).     

Group actions in a unit-regular ring

Let Rbe a unit-regular ring , let Xbe the set of all nonzero, nonunits of Rand let Gbe the group of all units of R. In this paper, some finiteness properties of Rare investigated by considering group actions of Gon Xas follows:First, in case of half-transitive regualr action if 2 is unit in Ror the number of idempotents in Ris finite, then Ris finite. Secondly, if Gis cyclic and 2 is unit in R, then every orbit under regualr action is a finite set, and so in this case, if Rhas a finite number of idempotents, then Ris finite. Finally, if Fis a field in which 2 is unit and the multiplicative group of all nonzero elenents in Fforms a cyclic group, then Fis finite.