Leibniz algebra


Leibniz algebra

In mathematics, a (left) Leibniz algebra (sometimes called a Loday algebra) is a module "A" over a commutative ring or field "R" with a bilinear product [,] such that ["a", ["b","c"] = ["a","b"] ,"c"] + ["b", ["a","c"] . In other words, left multiplication by any element "a" is a derivation.

If in addition the bracket is alternating ( ["a","a"] = 0) then the Leibniz algebra is a Lie algebra.Conversely any Lie algebra is obviously a Leibniz algebra.

The Leibniz´s identity is also known by this formula: [a, [b,c] = [a,b] ,c] - [a,c] ,b] .

If in addition the bracket is anticonmutative (i.e. [a,b] = - [b,a] ; equiv ; [a,a] =0) then the Leibniz's identity is equivalent to Jacobi's identity ( [a, [b,c] + [c, [a,b] + [b, [c,a] = 0) and that's why in this case the Leibniz algebra is a Lie algebra.

References

* Yvette Kosmann-Schwarzbach, "From Poisson algebras to Gerstenhaber algebras". Annales de l'institut Fourier, 46 no. 5 (1996), p. 1243-1274.
* Jean-Louis Loday, "Une version non commutative des algèbres de Lie: les algèbres de Leibniz". Enseign. Math. (2) 39 (1993), no. 3-4, 269--293.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Leibniz (disambiguation) — Gottfried Leibniz (1646 – 1716) was a German philosopher and mathematician.In mathematics, several results and concepts are attributed to Leibniz:* Leibniz algebra, an algebraic structure * Leibniz formula for pi, a method for calculating pi; *… …   Wikipedia

  • Algebra — (arab., Ergänzung durch Vergleichung), in weiterer Bedeutung der Theil der Arithmetik, welcher Aufgaben durch Gleichungen löst, in der gewöhnlichen Bedeutung die Buchstabenrechnung. Die Alexandriner haben im 4. Jahrh. die A. zuerst aufgestellt… …   Herders Conversations-Lexikon

  • Leibniz operator — One of the cornerstone concepts in the theory of abstract algebraic logic is that of the Leibniz operator.MotivationThe Leibniz operator was introduced by Willem Blok and Don Pigozzi, two of the founders of the field, as a means to abstract the… …   Wikipedia

  • Leibniz-Formel — In der Linearen Algebra ist die Determinante eine spezielle Funktion, die einer quadratischen Matrix oder einem linearen Endomorphismus eine Zahl zuordnet. Zum Beispiel hat die Matrix die Determinante …   Deutsch Wikipedia

  • algebra — /al jeuh breuh/, n. 1. the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations. 2. any of… …   Universalium

  • Leibniz, Gottfried Wilhelm — ▪ German philosopher and mathematician Introduction born July 1 [June 21, old style], 1646, Leipzig died November 14, 1716, Hannover, Hanover  German philosopher, mathematician, and political adviser, important both as a metaphysician and as a… …   Universalium

  • Algebra — This article is about the branch of mathematics. For other uses, see Algebra (disambiguation). Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from… …   Wikipedia

  • Álgebra — Para los usos matemáticos de la palabra álgebra como estructura algebraica, véase álgebra no asociativa, álgebra asociativa, álgebra sobre un cuerpo. El álgebra es la rama de las matemáticas que estudia las estructuras, las relaciones y las… …   Wikipedia Español

  • Leibniz formula for determinants — In algebra, the Leibniz formula expresses the determinant of a square matrix A = (a {ij}) {i,j = 1, dots, n} in terms of permutations of the matrix elements. Named in honor of Gottfried Leibniz, the formula is:det(A) = sum {sigma in S n}… …   Wikipedia

  • Algèbre de Leibniz — Pour les articles homonymes, voir Algèbre (homonymie). En mathématiques, une algèbre de Leibniz (droite), ainsi nommée d après Gottfried Wilhelm Leibniz, et parfois appelée algèbre de Loday, d après Jean Louis Loday, est un module L sur un anneau …   Wikipédia en Français