On Lie and associative duals
type: Item
instance of: scholarly article
title: On Lie and associative duals
zbMATH Open document ID: 0790.17010
DOI: 10.1016/0022-4049(93)90117-C
author: Warren D. Nichols
published in: Journal of Pure and Applied Algebra
publication date: 1994-06-26T00:00:00Z
review text: Let \(A\) be a commutative algebra over a field \(k\) of characteristic not 2 (some of the results also involve characteristic 2), \(D\) a derivation of \(A\). \(A_ D\) is \(A\) as a Lie algebra with bracket \([a,b]_ D= D(b)- D(a)b\). Let \(A^ 0\) be the coalgebra dual of \(A\), and let \((A_ D)^ 0\) be the Lie coalgebra dual of \(A_ D\). The dual \(D^*\) of \(D\) maps \(A^ 0\) to \(A^ 0\). Let \(D^ 0\) be restriction of \(A^ 0\). Let \((A^ 0)_{D^ 0}\) denote \(A^ 0\) as Lie coalgebra with cobracket \(\delta_{D^ 0}= (\text{id}\otimes D^ 0- D^ 0 \otimes \text{id})\Delta\), \(\Delta\) the comultiplication on the coalgebra \(A^ 0\). The author's main result is that if \(A\) is an algebra such that \(k[x]\subseteq A\subseteq k(x)\), then \((A^ 0)_{D^ 0} \cong (A_ D)^ 0 \cong(\text{Der } A)^ 0\) as Lie coalgebras, and the structure of this Lie coalgebra is analyzed, when \(k\) is algebraically closed and \(D=d/dx\), as a direct sum of divided power Lie coalgebras. For \(A=k[x]\), \textit{B. Peterson} and the reviewer [Aequationes Math. 20, 1-17 (1980; Zbl 0434.16008)] identified \(A^ 0\) as the coalgebra of linearly recursive sequences. It follows from the above result that if \(W_ 1= \text{Der } k[x]\), then \(W_ 1^ 0\) is the Lie coalgebra of linearly recursive sequences. This had been shown previously by the author by more calculational methods involving certain bases for linearly recursive sequences. His result also answers a question of the reviewer as to the relationship between the cobracket \(\delta_{D^ 0}\) on \(W_ 1^ 0\) and the comultiplication \(\Delta\) on \(A^ 0\) [J. Pure Appl. Algebra 87, 301-312 (1993; Zbl 0786.17015)]. An example is given by computing the cobracket of a Fibonacci sequence.
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 427771
zbMATH Keywords: derivation
zbMATH Keywords: Lie algebra
zbMATH Keywords: Lie coalgebra
zbMATH Keywords: linearly recursive sequences
MaRDI profile type: Publication
Wikidata QID: Q115364414
cites work: An example of a non-zero Lie coalgebra M for which \(Loc(M)=0\)
cites work: The structure of the dual Lie coalgebra of the Witt algebra
cites work: The Hopf algebra of linearly recursive sequences
cites work: Q5590089
cites work: Witt and Virasoro algebras as Lie bialgebras
cites work: A class of infinite-dimensional Lie bialgebras containing the Virasoro algebra
full work available at URL: 0022-4049(93)90117-c
OpenAlex ID: W2078774588
Recommended article: The structure of the dual Lie coalgebra of the Witt algebra
Recommended article: THE DUAL COALGEBRA OF CERTAIN INFINITE-DIMENSIONAL LIE ALGEBRAS
Recommended article: An example of a non-zero Lie coalgebra M for which \(Loc(M)=0\)
Recommended article: Q4405541
Recommended article: Divided power structures on Hopf algebras and embedding Lie algebras into special-derivation algebras
Recommended article: A Hopf algebra dual to a polynomial algebra over a commutative ring
Recommended article: Q4315525
Recommended article: Bialgebras of recursive sequences and combinatorial identities
Recommended article: Q3139278
Recommended article: Q5403780
description: scientific article; zbMATH DE number 427771