In
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:
:
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
:
is
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
:
where the sum is over all
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.
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Techniques
Overha ...
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References
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{{DEFAULTSORT:Zinbiel Algebra
Lie algebras
Non-associative algebras
Algebra of random variables