On Lie and associative duals (Q686113)

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.