Key-recovery attack

2013-03-20T02:12:00Z

209.6.74.105:

In [[mathematics]], a '''pre-Lie algebra''' is an [[algebraic structure]] on a vector space, that describes some properties of objects such as [[Tree (graph theory)|rooted trees]] and [[vector fields]] on affine space.

The notion of pre-Lie algebra has been introduced by [[Murray Gerstenhaber]] in his work on deformations of algebras.

Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.

== Definition ==
A pre-Lie algebra <math>(V,\triangleleft)</math> is a vector space <math>V</math> with a bilinear map <math>\triangleleft : V \otimes V \to V</math>, satisfying the relation
<math>
(x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z) = (x \triangleleft z) \triangleleft y - x \triangleleft (z \triangleleft y).
</math>

This identity can be seen as the invariance of the [[associator]] <math>(x,y,z) = (x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z)</math> under the exchange of the two variables <math>y</math> and <math>z</math>.

Every [[associative algebra]] is hence also a pre-Lie algebra, as the associator vanishes identically.

== Examples ==
* [[Vector fields]] on the affine space

If we denote by <math>f(x)\partial_x</math> the vector field <math>x \mapsto f(x)</math>, and if we define <math>\triangleleft</math> as <math>f(x) \triangleleft g(x) = f'(x) g(x)</math>, we can see that the operator <math>\triangleleft</math> is exactly the application of the <math>g(x)\partial_x</math> field to <math>f(x)\partial_x</math> field.
<math>(g(x)\partial_x)(f(x)\partial_x) = g(x) \partial_x f(x) \partial_x = g(x) f'(x) \partial_x</math>

If we study the difference between <math>(x \triangleleft y) \triangleleft z</math> and <math>x \triangleleft (y \triangleleft z)</math>, we have
<math>(x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z) = (x' y)'z - x'y'z = x'y'z x''yz - z'y'z = x''yz</math>
which is symmetric on ''y'' and ''z''.

* Rooted trees

Let <math>\mathbb{T}</math> be the vector space spanned by all rooted trees.

One can introduce a bilinear product <math>\curvearrowleft</math> on <math>\mathbb{T}</math> as follows. Let <math>\tau_1</math> and <math>\tau_2</math> be two rooted trees.

<math>\tau_1 \curvearrowleft \tau_2 = \sum_{s \in \mathrm{Vertices}(\tau_1)} \tau_1 \circ_s \tau_2</math>

where <math>\tau_1 \circ_s \tau_2</math> is the rooted tree obtained by adding to the disjoint union of <math>\tau_1</math> and <math>\tau_2</math> an edge going from the vertex <math>s</math> of <math>\tau_1</math> to the root vertex of <math>\tau_2</math>.

Then <math>(\mathbb{T}, \curvearrowleft)</math> is a free pre-Lie algebra on one generator.

==References==
*{{citation
| last1 = Chapoton | first1 = F.
| last2 = Livernet | first2 = M.
| year = 2001
| mr = 1827084
| journal = [[International Mathematics Research Notices]]
| title = Pre-Lie algebras and the rooted trees operad
| doi = 10.1155/S1073792801000198
| pages = 395–408
| volume = 8
| issue = 8}}.
*{{citation
| last = Szczesny | first = M.
| year = 2010
| mr =
| journal =
| title = Pre-Lie algebras and incidence categories of colored rooted trees
| volume =1007
| bibcode = 2010arXiv1007.4784S
| pages = 4784
| arxiv = 1007.4784
| class = math.CO }}.

[[Category:Lie groups]]
[[Category:Lie algebras| ]]
[[Category:Non-associative algebra]]

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