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1. CMB 2011 (vol 54 pp. 297)
Lie Powers and Pseudo-Idempotents
We give a new factorisation of the classical Dynkin operator,
an element of the integral group ring of the symmetric group that
facilitates projections of tensor powers onto Lie powers.
As an application we show that the iterated Lie power $L_2(L_n)$ is
a module direct summand of the Lie power $L_{2n}$ whenever the
characteristic of the ground field does not divide $n$. An explicit
projection of the latter onto the former is exhibited in this case.
Categories:17B01, 20C30 |
2. CMB 2010 (vol 53 pp. 425)
Free Pre-Lie Algebras are Free as Lie Algebras We prove that the $\mathfrak{S}$-module $\operatorname{PreLie}$ is a free Lie algebra in
the category of $\mathfrak{S}$-modules and can therefore be written as the
composition of the $\mathfrak{S}$-module $\operatorname{Lie}$ with a new $\mathfrak{S}$-module
$X$. This implies that free pre-Lie algebras in the category of
vector spaces, when considered as Lie algebras, are free on
generators that can be described using $X$. Furthermore, we define a
natural filtration on the $\mathfrak{S}$-module $X$. We also obtain a
relationship between $X$ and the $\mathfrak{S}$-module coming from the
anticyclic structure of the $\operatorname{PreLie}$ operad.
Categories:18D50, 17B01, 18G40, 05C05 |
3. CMB 2005 (vol 48 pp. 445)
On the Garsia Lie Idempotent The orthogonal projection of the free associative algebra onto the
free Lie algebra is afforded by an idempotent in the rational group
algebra of the symmetric group $S_n$, in each homogenous degree
$n$. We give various characterizations of this Lie idempotent and show
that it is uniquely determined by a certain unit in the group algebra
of $S_{n-1}$. The inverse of this unit, or, equivalently, the Gram
matrix of the orthogonal projection, is described explicitly. We also
show that the Garsia Lie idempotent is not constant on descent classes
(in fact, not even on coplactic classes) in $S_n$.
Categories:17B01, 05A99, 16S30, 17B60 |
4. CMB 2000 (vol 43 pp. 3)
Resolutions of Associative and Lie Algebras Certain canonical resolutions are described for free associative and
free Lie algebras in the category of non-associative algebras. These
resolutions derive in both cases from geometric objects, which in turn
reflect the combinatorics of suitable collections of leaf-labeled
trees.
Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedron Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50 |