mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Zinbiel algebra or dual Leibniz algebra is a

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

over a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

with a bilinear product satisfying the defining identity: :

(a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b).

Zinbiel algebras were introduced by . The name was proposed by Jean-Michel Lemaire as being "opposite" to

Leibniz algebra In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module ''L'' over a commutative ring ''R'' with a bilinear product _ , _ satisfying the Leibniz ident ...

. In any Zinbiel algebra, the symmetrised product :

a \star b = a \circ b + b \circ a

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over ''V'' is the

tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...

with product :

(x_0 \otimes \cdots \otimes x_p) \circ (x_ \otimes \cdots \otimes x_) =
x_0 \sum_ (x_1,\ldots,x_),

where the sum is over all

(p,q)

shuffles Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome. __TOC__ Techniques Overha ...

References

* * * * * {{DEFAULTSORT:Zinbiel Algebra Lie algebras Non-associative algebras Algebra of random variables