Nilpotency Index Research Articles

On the degree of the inverse of an automorphism of the affine space

We study the degree of the inverse of an automorphism f of the affine n-space over a C -algebra k, in terms of the degree d of f and of other data. For n = 1, we obtain a sharp upper bound for deg ( f − 1) in terms of d and of the nilpotency index of the ideal generated by the coefficients of f ′'. For n = 2 and arbitrary d≥ 3, we construct a (triangular) automorphism f of Jacobian one such that deg( f − 1) > d. This answers a question of A. van den Essen (see [3]) and enables us to deduce that some schemes introduced by authors to study the Jacobian conjecture are not reduced. Still for n = 2, we give an upper bound for deg ( f − 1) when f is triangular. Finally, in the case d = 2 and any n, we complete a result of G. Meisters and C. Olech and use it to give the sharp bound for the degree of the inverse of a quadratic automorphism, with Jacobian one, of the affine 3-space.

Use of even grassmann variables to construct effective actions for mesons

In the framework of an approach to bosonization based on the use of fermionic composites as fundamental variables, a quadratic action in even Grassmann variables with the quantum numbers of the pions has been constructed. It includes the σ-field in order to be invariant under [ SU(2)] L ⊗ [ SU(2)] R tranfformations over the quarks. This action exhibits the Goldstone phenomenon reducing its symmetry to the O(3) isospin invariance. The model has been investigated in the Stratonovitch-Hubbard representation, in which form it is reminiscent of the Gell-Mann-Lévy model. By the saddle point method a renormalizable expansion in inverse powers of the index of nilpotency of the mesonic fields (which is 24), is generated. The way it might be used in a new perturbative approach to QCD is outlined.

A free action for pions as quark composites

In the framework of an approach to bosonization based on the use of fermionic composites as fundamental variables, a quadratic action in even Grassmann variables with the quantum numbers of the pions has been constructed. It includes the σ-field in order to be invariant under [ SU(2)] L ⊗ [ SU(2)] R transformations over the quarks. This action exhibits the Goldstone phenomenon reducing its symmetry to the O(3) isospin invariance. A squared mass for the pions is generated according to PCAC. The model has been investigated in the Stratonovitch-Hubbard representation, in which form it is reminiscent of the Gell-Mann-Lévy model. By the saddle-point method a renormalizable expansion in inverse powers of the index of nilpotency of the mesonic fields (which is 24), is generated. The way it might be used in a new perturbative approach to QCD is outlined.

Hierarchy of efficient generators of the symmetric inverse monoid

It is well-known that the symmetric inverse monoid on a set ofn elements can be generated as a semigroup by its group of units and a single element of rankn − 1. We show that the efficiency with which the semigroup is generated in this way depends solely on the index of nilpotence of the rankn − 1 generator. We also investigate the various ways of expressing elements of the semigroup most efficiently as a product of generators.

The p -Modular Descent Algebra of the Symmetric Group

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described. 1991 Mathematics Subject Classification 20F32.

Correction to: ``On $p$-radical groups $G$ and the nilpotency indices of $J(kG)$. II''

Correction to: ``On $p$-radical groups $G$ and the nilpotency indices of $J(kG)$. II''

On the structure of exchange rings

An associative ring R with identity is called an exchange ring if RR; has the exchange property introduced by Crawley and Jon- sson [5]. We prove, in this paper, that if R is an exchange ring with prime factors Artinian, then R is strongly π-regular. If R is an exchange ring with primitive factors Artinian and R/J(R) is homomorphically semipimitive, then R/J(R) is strongly π-regular and idempotents lift modulo J(R). Also, it is shown that for exchange rings, bounded index of nilpotence implies primitive factors Artinian. These are generalizations of the corresponding results in [16], [11], [8] and [2]. Examples are given showing that the generalizations are nontrivial.

Constructing Maximal Commutative Subalgebras of Matrix Rings in Small Dimensions

Let k denote an algebraically closed field of arbitrary characteristic. Let C denote the set of all commutative, finite dimensional, local k-algebras of the form (B, m, k) with i(m) ⋚2. Here i(m) denotes the index of nilpotency of the maximal ideal m. A Akalgebra (R, J,k)∈L is called a (c1-construction if there exists (B, m, k)∈ £ ≅ {(k, (0), k)} and a finitely generated, faithful B-module N such that R,≅B⋉(the idealization of N). (R.J.k) is called a (c2::-construction if there exist a (B,m k)∈ L, a positive integer p $ge;2 and a nonzero z £ SB(the socle of B) such that R≅B[x]/(mX, Xp- z). Let Mn×n(K) denote the set of all n x n matrices, over k with n≥2. Let .Mn(k) denote the set of all maximal, commutative A;-subalgebras of Mn×n(k). In this paper, we show any (R J, k) ∈£⋂Mn;(k) with n>5 is a C1 or C2 -construction except for one isomorphism class. The one exception occurs when n = 5.

On nilpotency of the barideal of a bernstein algebra

It is given a new bound for the index of nilpotency of the barideal of a finited generated Bernstein algebra. Also, it is found the index of solvability of the barideal of a Bernstein algebra and a bound of nilpotency for the square of the barideal.

On p-radical groups G and the nilpotency indices of J(kG). II

On p-radical groups G and the nilpotency indices of J(kG). II

Primal-Dual Combinatorial Relaxation Algorithms for the Maximum Degree of Subdeterminants

A primal-dual framework of combinatorial relaxation algorithms is proposed for computing the highest degree of a minor of order k of a rational function matrix. The algorithm can be used for computing the index of nilpotency of a matrix pencil (or the index of the associated differential algebraic equation). It is a linear algebraic version of the Hungarian method for the assignment problem. The proposed framework stands in contrast to the previous combinatorial relaxation algorithm based on weighted matchings, and may also be regarded as an extension of the Wolovich algorithm for row/column properness. Several algorithms are evaluated through computer experiments.

Ideal structure for superconformal semigroups

The role of univertible transformations in superstring theories is discussed. A new parametrization of superconformal groups is presented that permits their ideal extensions—superconformal semigroups—to be constructed adequately. These semigroups consist of a group part containing the standard superconformal transformations and an ideal part whose abstract structure is analyzed in detail. A classification of all elements in terms of different indices of nilpotency is performed. An ideal series is constructed, the generalized Green's “vector” and “tensor” relations are defined, and some types of quasi-characters separating the elements of superconformal semigroups are introduced. The need for similar constructions in other supersymmetric and superquantum models is pointed out.

Even Grassmann fields in momentum space

We extend previous results about fields whose Fourier components are even elements of a Grassmann algebra. Their main interest is related to the possibility that they describe fermionic composites. We evaluate the free propagators for arbitrary index of nilpotency and investigate to one loop a φ 4 model. Due to the nature of the integral over even Grassmann fields such a model exists for repulsive as well as attractive selfinteraction. In the first case the β -function is equal to that of the ordinary theory while in the second one the model is asymptotically free. The renormalized mass has a peculiar dependence on the cut off, being quadratically increasing/decreasing for attractive/repulsive selfinteraction.

On Nuclear Jordan-Bernstein Algebras

The aim of this paper is the construction of free nuclear Jordan-Bernstein algebras, that is, the largest nuclear Jordan-Bernstein algebra generated by a set of generators. In this way, it is proved that every nuclear Jordan-Bernstein algebra is a quotient of an adequate free nuclear Jordan-Bernstein algebra. The same construction may be made without the nuclear assumption and free Jordan-Bernstein algebras may be constructed and related to free nuclear Jordan-Bernstein algebras. The nilpotency index of these algebras is studied, which lets us get a good bound for the nilpotency index in every Jordan-Bernstein algebra and in every nuclear Bernstein algebra. Finally, we can apply the obtained results to get some information about inner derivations of Jordan-Bernstein algebras.

On Constants of Algebraic Derivations and Fixed Points of Algebraic Automorphisms

Let R be a semiprime algebra over a field F and d an algebraic derivation of R. We examine the relationship between R and the algebra of constants R d . We prove that: (1) The prime radical B ( R d ) is nilpotent with the index of nilpotency depending on the minimal polynomial of d; (2) R d is Artinian if and only if R is Artinian. Using these we obtain results about fixed subrings of algebraic automorphisms. For instance, we show that if σ is an automorphism of a prime order p of a semiprime ring R with pR = 0 then R is Artinian if and only if the fixed subring R σ is Artinian.

Restricted Lie Algebras and Their Envelopes

AbstractLet L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(L) the augmentation ideal of u(L). We give an explicit description for the dimension subalgebras of L, namely those ideals of L defined by Dn(L) - L∩ωu(L)n for each n ≥ 1. Using this expression we describe the nilpotence index of ωU(L). We also give a precise characterisation of those L for which ωu(L) is a residually nilpotent ideal. In this case we show that the minimal number of elements required to generate an arbitrary ideal of u(L) is finitely bounded if and only if L contains a 1-generated restricted subalgebra of finite codimension. Subsequently we examine certain analogous aspects of the Lie structure of u(L). In particular we characterise L for which u(L) is residually nilpotent when considered as a Lie algebra, and give a formula for the Lie nilpotence index of u(L). This formula is then used to describe the nilpotence class of the group of units of u(L).

Stable range one for exchange rings

An associative ring R with identity is called an exchange ring if R R has the exchange property introduced by Crawley and Jonsson (1964). We prove, in this paper, that an exchange ring R has stable range 1 in the following three cases: (1) primitive factors of R are Artinian; (2) R is of bounded index of nilpotence; (3) idempotents of R are central. Also it is shown that, for exchange rings, bounded index of nilpotence implies primitive factors are Artinian and that, for modules with the finite exchange property, cancellation from direct sums is equivalent to the fact that the endomorphism ring of the cancellable module has stable range 1.

A counterexample to a conjecture of Drużkowski and Rusek

Let F = X + H be a cubic homogeneous polynomial automorphism from $ℂ^n$ to $ℂ^n$. Let $p$ be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that $deg F^{-1} ≤ 3^{p-1}$. We show that the conjecture is true

Diagonally scaled permutations and circulant matrices

If R is an n × n matrix over the complex field which is the product of a diagonal matrix D and a permutation matrix P, then R is called a diagonally scaled permutation matrix. We present the eigenstructure of R by observing that R is permutation similar to the direct sum of diagonally scaled permutation matrices of the form DC where D is a diagonal matrix and C is the circulant permutation. 0 1 0 0 ⋯ 0 0 0 1 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 ⋯ 1 1 0 0 0 ⋯ 0 The matrix DC is called a scaled circulant permutation matrix. We consider two cases for R = DC: when the scaling matrix D is nonsingular, and when D is singular. In the singular case R is nilpotent, and we are able to obtain upper and lower bounds on the index of nilpotency of R. We conclude with information about matrices that commute with a scaled permutation matrix. We are also able to represent an arbitrary n × n Toeplitz matrix as a sum of matrices of the form D( k, α, β) C k for k = 1, …, n where D( k, α, β) is a diagonal matrix.

Nilpotency indices of the radicals of p-group algebras

Let k be a field of characteristic p>0. We classify all finite p-groups G satisfying the inequality p−2|G|≦t(G) < p−1|G|, where t(G) is the nilpotency index of the Jacobson radical of k[G].