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Let p be an odd prime, and let ∑n=0∞anXn∈Fp[[X]] be the reduction modulo p of the Artin–Hasse exponential series. We obtain a polynomial expression for akp in terms of those arp with r

Nottingham algebras are a class of just-infinite-dimensional, modular, N-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree 1, and the second occurs in degree q, a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to ∞. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra L can be assigned a type, in such a way that the degrees and types of the diamonds completely describe L. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals q - 1. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type ∞.

A thin Lie algebra is a Lie algebra L, graded over the positive integers, with its first homogeneous component L 1 of dimension two and generating L, and such that each nonzero ideal of L lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. If L 1 is the only diamond, then L is a graded Lie algebra of maximal class. We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is one less than the dimension of their largest metabelian quotient.

Let R(x)=g(x)/h(x) be a rational expression of degree three over the finite field Fq. We count the irreducible polynomials in Fq[x], of a given degree, that have the form h(x)deg⁡f⋅f(R(x)) for some f(x)∈Fq[x]. As an example of application of our results, we recover the number of irreducible transformation shift registers of order three, which were computed by Jiang and Yang in 2017.

We prove several structural results on Nottingham algebras, a class of infinitedimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree 1, and the second occurs in degree q , a power of the characteristic. Each diamond past the second is assigned a type, which either belongs to the underlying field or is ∞. Nottingham algebras with a variety of diamond patterns are known. In particular, some have diamonds of both finite and infinite type. We prove that each of those known examples is uniquely determined by a certain finite-dimensional quotient. Finally, we determine how many diamonds of type ∞ may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.

The algebras of the title are infinite-dimensional graded Lie algebras L = circle plus L-infinity(i=1)i, over a field of positive characteristic p, which are generated by an element of degree 1and an element of degree p, and satisfy [L-i, L-1] = Li+1 for i >= p. In case p = 2 such algebras were classified by Caranti and VaughanLee in 2003. We announce an extension of that classification to arbitrary prime characteristic, and prove several major steps in its proof. (C) 2021 Elsevier Inc. All rights reserved.

A thin Lie algebra is a Lie algebra, graded over the positive integers, with its first homogeneous component of dimension two and generating, and such that each non-zero ideal of lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of (that is, the next diamond past) occurs in degree. We prove that if 5$]]>, then for some non-zero element of. In characteristic different from two this means is a sandwich element of. We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

We introduce a generalization pounds_{d}^{(alpha)}(X) of the finite polylogarithms pounds_{d}^{(0)}(X)=pounds_d(X)=sum_{k=1}^{p-1}X^k/k^d, in characteristic p, which depends on a parameter alpha. The special case pounds_{1}^{(alpha)}(X) was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a {em grading switching} technique for non-associative algebras. Here, we extend such generalization to pounds_{d}^{(alpha)}(X) in a natural manner, and study some properties satisfied by those polynomials. In particular, we find how the polynomials pounds_{d}^{(alpha)}(X) are related to the powers of pounds_{1}^{(alpha)}(X) and derive some consequences.

We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation ex · ey = ex+y for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case.

We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. We identify two subclasses of Nottingham Lie algebras as loop algebras of finite-dimensional simple Lie algebras of Hamiltonian Cartan type. A property of Laguerre polynomials of derivations, which is related to toral switching, plays a crucial role in our constructions

We describe a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. This is inspired by a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. We trace the development of grading switching, from an early version based on taking the Artin-Hasse exponential of a nilpotent derivation, to a more general version which uses certain generalized Laguerre polynomials playing the role of generalized exponentials. Both versions depend on the existence of appropriate analogues of the functional equation exp(x).exp(y)=exp(x+y) for the classical exponential.

We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. We identify two subclasses of Nottingham Lie algebras as loop algebras of finite-dimensional simple Lie algebras of Hamiltonian Cartan type. A property of Laguerre polynomials of derivations, which is related to toral switching, plays a crucial role in our constructions

Borrowing some terminology from pro-p groups, thin Lie algebras are N-graded Lie algebras, generated in degree one, of width two and obliquity zero. In particular, their homogeneous components have degree one or two, and they are termed diamonds in the latter case. In one of the two main subclasses of thin Lie algebras the earliest diamond after that in degree one occurs in degree 2q-1, where q is a power of the characteristic. This paper is a contribution to an ongoing classification project of this subclass of thin Lie algebras. Specifically, we prove that the degree of the earliest diamond of finite type in such a Lie algebra can only attain certain values, which occur in explicit examples constructed elsewhere

We investigate a class of infinite-dimensional modular Lie algebras, graded over the positive integers, in which every homogeneous component has dimension one or two. We identify these Lie algebras with loop algebras of certain simple Lie algebras of Hamiltonian type. These Lie algebras are not finitely presented, but certain central extensions of them are, and we give an explicit construction for the latter.

We describe a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. This is inspired by a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. We trace the development of grading switching, from an early version based on taking the Artin-Hasse exponential of a nilpotent derivation, to a more general version which uses certain generalized Laguerre polynomials playing the role of generalized exponentials. Both versions depend on the existence of appropriate analogues of the functional equation e(x) . e(y) = e(x+y) for the classical exponential.

We prove a geometric property of the set A^{−1} of inverses of the nonzero elements of an F_q-subspace A of a finite field involving the size of its intersection with two-dimensional F_q-subspaces. We give some applications, including a new upper bound on |A^{−1}∩B| when A and B are F_q-subspaces of different dimension of a finite field, satisfying a suitable assumption.

We prove congruences, modulo a power of a prime $p$, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$, partly motivated by analogies with the well-known power series for $(\arcsin z)^2$ and $(\arcsin z)^4$. The right-hand sides of those congruences involve values of the finite polylogarithms $\pounds_d(x)=\sum_{k=1}^{p-1} x^k/k^d$. Exploiting the available functional equations for the latter we compute those values, modulo the required powers of $p$, in terms of familiar quantities such as Fermat quotients and Bernoulli numbers.

Thin Lie algebras are graded Lie algebras L = ⊕i = 1∞Li with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if Lk is the second diamond of L, then the quotient L/Lk is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/Lk is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/Lk need not be metabelian in characteristic two. We describe here all the possibilities for L/Lk up to isomorphism. In particular, we prove that k + 1 equals a power of two. © World Scientific Publishing Company.

We investigate when the sequence of binomial coefficients ((k; i)) modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0 ≤ i ≤ k. In particular, we prove that this cannot occur if 2 h ≤ k < p - h. This hypothesis can be weakened to 2 h ≤ k < p if we assume, in addition, that the characteristic polynomial of the relation does not have -1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.

Thin Lie algebras are Lie algebras over a field, graded over the positive integers and satisfying a certain narrowness condition. In particular, all homogeneous components have dimension one or two, and are called diamonds in the latter case. The first diamond is the component of degree one, and the second diamond can only occur in degrees 3, 5, q or 2 q - 1, where q is a power of the characteristic of the underlying field. Here we consider several classes of thin Lie algebras with second diamond in degree q. In particular, we identify the Lie algebras in one of these classes with suitable loop algebras of certain Albert-Zassenhaus Lie algebras. We also apply a deformation technique to recover other thin Lie algebras previously produced as loop algebras of certain graded Hamiltonian Lie algebras. © 2007 Elsevier Inc. All rights reserved.

We describe the additive subgroups of fields which are closed with respect to taking inverses, in particular, with characteristic different from two. Any such subgroup is either a subfield or the kernel of the trace map of a quadratic subextension of the field.

We exhibit normal subgroups of a free nilpotent group F of rank two and class three, which have isomorphic finite quotients but are not conjugate under any automorphism of F.

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. A study of the set -Rfnet-temp p of positive integers which occur as orders of nonsingular derivations of finite-dimensional nonnilpotent Lie algebras of characteristic p > 0 was initiated by Shalev and continued by the present author. In this paper we continue this study in the case of characteristic two. Among other results, we prove that any divisor n of 2 k - 1 with n 4 > (2k - n)3 belongs to N2. Our methods consist of elementary arguments with polynomials over finite fields and a little character theory of finite groups.

In this note we determine all power series F(X) ∈ 1 + X double-struck F sign p[[X]] such that (F(X + Y))-1 F(X)F(Y) has only terms of total degree a multiple of p. Up to a scalar factor they are all the series of the form F(X) = E p (cX)• G(X p ) for some c ∈ double-struck F sign p and G(X) ∈ 1 + X double-struck F sign p[[X]] where Ep(X)= exp (∑i=0 ∞ Xp i/pi) is the Artin-Hasse exponential.

We prove that knowledge of the character degrees of a finite group G and of their multiplicities determines whether G has a Sylow p-subgroup as a direct factor. An analogous result based on knowledge of the conjugacy class sizes is known. We prove variations of both results and discuss their similarities.

We show how Artin-Hasse exponentials of nilpotent derivations can be used to produce new cyclic gradings of non-associative algebras of prime characteristic from a given one. We discuss the connection with the technique of switching tori (and, consequently, the associated root space decompositions) in restricted modular Lie algebras. However, differently from total switching, our method can produce gradings over cyclic p-groups of order higher than p. In particular, we apply it to construct cyclic gradings of the Zassenhaus Lie algebra of dimension p^n, for p>3.

In her PhD thesis at Trento, Marina Avitabile had constructed certain infinite-dimensional, graded thin Lie algebras over finite fields, which share some common features with the graded Lie algebra of the celebrated Nottingham group. In this paper, we solve the uniqueness question for these algebra, as detailed below. The graded Lie algebra associated to the Nottingham group over the field with p elements, with respect to its lower central series, has been studied by the first author. This is a thin Lie algebra, with second diamond (homogeneous component of dimension 2) in weight p, and can be described as (the positive part of) a twisted loop algebra of the smallest Zassenhaus algebra W(1: 1). More generally, certain twisted loop algebras of the Zassenhaus algebrasW(1: n) over the field with p elements have been studied by the first author. These are thin Lie algebras, with diamonds occurring exactly in the weights congruent to 1 modulo q-1, where q = p^n. The goal of this paper is to give a complete description of a more general class of thin Lie algebras with diamonds (according a slightly more general definition) in these weights. These algebras we call Nottingham Lie algebras. We are naturally led to discuss some related features of thin Lie algebras with the second diamond in weight q.

We extend the results of Shalev [Sh] on the orders of nonsingular derivations of finite-dimensional non-nilpotent modular Lie algebras.

A pro-p-group is called thin if its lattice of closed normal subgroups contains no more than p + 1 pairwise incomparable elements. An investigation of certain (infinite) thin pro-p-groups was made in [A. Caranti et al. Quart. J. Math. Oxford 47 (1996), 279-296], by studying the graded Lie algebras associated with their lower central series. In particular, the main result of [A. Caranti et al. Quart. J. Math. Oxford 47 (1996), 279-296] asserts that those graded Lie algebras are, in certain cases, uniquely determined by their quotients of low dimension. In the present paper we construct pro-p-groups which have, associated with their lower central series, the graded Lie algebras of [A. Caranti et al. Quart. J. Math. Oxford 47 (1996), 279-296].

We investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type. Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours.