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Monomial bases and pre-Lie structure for free Lie algebras. (English) Zbl 1453.17019

A tree is a undirected connected finite graph, without cycles. A rooted tree is defined as a tree with one designated vertex called the root. The other remaining vertices are partitioned into \(k \geq 0\) disjoint subsets such that each of them in turn represents a rooted tree, and a subtree of the whole tree. A rooted tree is said to be planar, if it is endowed with an embedding in the plane. Otherwise, its called a non-planar rooted tree. Let \(E\) be a non-empty set. An \(E\)-decorated rooted tree is a pair \((t, d)\) of a rooted tree \(t\) together with a map \(d: V (t) \rightarrow E\), which decorates each vertex \(v\) of \(t\) by an element \(a\) of \(E\), i.e. \(d(v) = a\), where \(V (t)\) is the set of all vertices of \(t\).
Recall that a pre-Lie algebra is a vector space \(\mathcal{A}\) over a field \(K\), together with a bilinear operation \(\triangleright\) that satisfies the identity: \((x \triangleright y) \triangleright z - x \triangleright (y \triangleright z) = (y \triangleright x) \triangleright z - y \triangleright (x \triangleright z)\). The authors construct a pre-Lie structure on the free Lie algebra \(\mathcal{L}(E)\) generated by a set \(E\), giving an explicit presentation of \(\mathcal{L}(E)\) as the quotient of the free pre-Lie algebra \(\mathcal{T}^E\), generated by the (non-planar) \(E\)-decorated rooted trees, by some ideal \(I\). They describe a monomial bases in tree version for the free Lie (respectively pre-Lie) algebras using the procedures of Gr\(\ddot{o}\)bner bases, comparing with the one (i.e., the monomial basis) obtained for the free pre-Lie algebra in the work [M. J. H. Al-Kaabi, Sémin. Lothar. Comb. 71, B71b, 19 p. (2014; Zbl 1303.17023)].

MSC:

17D25 Lie-admissible algebras
17A50 Free nonassociative algebras
17A61 Gröbner-Shirshov bases in nonassociative algebras
17B01 Identities, free Lie (super)algebras
05C05 Trees

Citations:

Zbl 1303.17023

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References:

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