arXiv subjects

Marina Avitabile

Publications and source records attributed to Marina Avitabile.

13 recordsLinked to original sources

arXiv2023

Ideally $r$-constrained graded Lie subalgebras of maximal class algebras

Let $E\supseteq F$ be a field extension and $M$ a graded Lie algebra of maximal class over $E$. We investigate the $F$-subalgebras $L$ of $M$, generated by elements of degree $1$. We provide conditions for $L$ being either ideally $r$-constrained or not just infinite. We show by an example that those conditions are tight. Furthermore, we determine the structure of $L$ when the field extension $E\supseteq F$ is finite. A class of ideally $r$-constrained Lie algebras which are not $(r-1)$-constrained is explicitly constructed, for every $r\geq 1$.

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arXiv2023

On some coefficients of the Artin-Hasse series modulo a prime

Let $p$ be an odd prime, and let $\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ be the reduction modulo $p$ of the Artin-Hasse exponential. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even $k<p^2-1$. A conjectural analogue covering the case of odd $k<p$ can be stated in various polynomial forms, essentially in terms of the polynomial $γ(X) =\sum_{n=1}^{p-2}(B_{n}/n)X^{p-n}$, where $B_n$ denotes the $n$-th Bernoulli number. We prove that $γ(X)$ satisfies the functional equation $γ(X-1)-γ(X)=£_1(X)+X^{p-1}-w_p-1$ in $\mathbb{F}_p[X]$, where $£_1(X)$ and $w_p$ are the truncated logarithm and the Wilson quotient. This is an analogue modulo $p$ of a functional equation, in $\mathbb{Q}[[X]]$, established by Zagier for the power series $\sum_{n=1}^{\infty}(B_{n}/n)X^n$. Our proof of the functional equation establishes a connection with a result of Nielsen of 1915, of which we provide a fresh proof. Our polynomial framing allows us to derive congruences for certain numerical sums involving divided Bernoulli numbers.

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arXiv2023

The Artin-Hasse series and Laguerre polynomials modulo a prime

For an odd prime $p$, let $\mathrm{E}_{p}(X)=\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ denote the reduction modulo $p$ of the Artin-Hasse exponential series. It is known that there exists a series $G(X^p)\in \mathbb{F}_{p}[[X]]$, such that $L_{p-1}^{(-T(X))}(X)=\mathrm{E}_{p}(X)\cdot G(X^p)$, where $T(X)=\sum_{i=1}^{\infty}X^{p^{i}}$ and $L_{p-1}^{(α)}(X)$ denotes the (generalized) Laguerre polynomial of degree $p-1$. We prove that $G(X^p)=\sum_{n=0}^{\infty}(-1)^n a_{np}X^{np}$, and show that it satisfies $G(X^p)\,G(-X^p)\,T(X)=X^p. $

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arXiv2021

The earliest diamond of finite type in Nottingham algebras

We prove several structural results on Nottingham algebras, 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. 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 $\infty$. 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 $\infty$ may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.

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arXiv2021

Diamond distances in Nottingham algebras

Nottingham algebras are a class of just-infinite-dimensional, modular, $\mathbb{N}$-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 $\infty$. 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 $\infty$.

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arXiv2018

Generalized finite polylogarithms

We introduce a generalization $£_{d}^{(α)}(X)$ of the finite polylogarithms $£_{d}^{(0)}(X)=£_d(X)=\sum_{k=1}^{p-1}X^k/k^d$, in characteristic $p$, which depends on a parameter $α$. The special case $£_{1}^{(α)}(X)$ 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 $£_{d}^{(α)}(X)$ in a natural manner, and study some properties satisfied by those polynomials. In particular, we find how the polynomials $£_{d}^{(α)}(X)$ are related to the powers of $£_{1}^{(α)}(X)$ and derive some consequences.

math.NT↗

arXiv2018

A generalized truncated logarithm

We introduce a generalization $G^{(α)}(X)$ of the truncated logarithm $\mathcal{L}_1(X) = \sum_{k=1}^{p-1}X^k/k$ in characteristic $p$, which depends on a parameter $α$. The main motivation of this study is $G^{(α)}(X)$ being an inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential given by certain Laguerre polynomials. Such Laguerre polynomials play a role in a grading switching technique for non-associative algebras, previously developed by the authors, because they satisfy a weak analogue of the functional equation $\exp(X)\exp(Y)=\exp(X+Y)$ of the exponential series. We also investigate functional equations satisfied by $G^{(α)}(X)$ motivated by known functional equations for $\mathcal{L}_1(X)=-G^{(0)}(X)$.

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arXiv2013

Grading switching for modular non-associative algebras

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.

math.RA↗

arXiv2013

Nottingham Lie algebras with diamonds of finite and infinite type

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.

math.RA↗

arXiv2013

Laguerre polynomials of derivations

We introduce a 'grading switching' for arbitrary nonassociative 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 exp(x)*exp(y)=exp(x+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.

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arXiv2008

The structure of thin Lie algebras up to the second diamond

Thin Lie algebras are Lie algebras L, graded over the positive integers, with all homogeneous components of dimension at most two, and satisfying a more stringent but natural narrowness condition modeled on an analogous one for pro-p groups. The two-dimensional homogeneous components of L, which include that of degree one, 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 L_k is the second diamond of L, then the quotient L/L^k is a graded Lie algebras of maximal class. In characteristic not two, L/L^k 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. The quotient L/L^k need not be metabelian in characteristic two. We describe here all the possibilities for L/L^k up to isomorphism. In particular, we prove that k+1 equals a power of two.

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arXiv2006

Thin loop algebras of Albert-Zassenhaus algebras

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 2q-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.

math.RA↗

arXiv2005

Diamonds of finite type in thin Lie algebras

Borrowing some terminology from pro-p groups, thin Lie algebras are N-graded Lie algebras of width two and obliquity zero, generated in degree one. 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 a classification project of this subclass of thin Lie algebras. Specifically, we prove that, under certain technical assumptions, the degree of the earliest diamond of finite type in such a Lie algebra can only have a certain form, which does occur in explicit examples constructed elsewhere.

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