209.6.74.105 at 02:12, 20 March 2013

2013-03-20T02:12:00Z

← Older revision		Revision as of 04:12, 20 March 2013
Line 1:		Line 1:
	~~Program~~ or ~~Task Administrator Calvin from Matachewan~~, ~~really likes piano~~, ~~Marvel~~ [~~http:~~//~~gamsolutions~~.~~edublogs~~.~~org~~/~~2014~~/10/16/~~marvel~~-~~puzzle~~-~~quest~~-~~hack-android-ios~~/ ~~Puzzle Quest Hack]~~ and ~~crafts~~. ~~Gains encouragement~~ by ~~planning~~ to ~~Jesuit Block~~ and ~~Estancias~~ of ~~Córdoba~~.		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]]

en>RJaguar3: add cryptography stub template

2011-01-28T04:32:13Z

add cryptography stub template

New page

Program or Task Administrator Calvin from Matachewan, really likes piano, Marvel [http://gamsolutions.edublogs.org/2014/10/16/marvel-puzzle-quest-hack-android-ios/ Puzzle Quest Hack] and crafts. Gains encouragement by planning to Jesuit Block and Estancias of Córdoba.

Key-recovery attack - Revision history

209.6.74.105 at 02:12, 20 March 2013

en>RJaguar3: add cryptography stub template