Zinbiel algebra - Alchetron, The Free Social Encyclopedia

In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:

( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) .

Zinbiel algebras were introduced by Jean-Louis Loday (1995). The name was proposed by J.-M. Lemaire as being "opposite" to Leibniz algebra.

The symmetrised product

a ⋆ b = a ∘ b + b ∘ a

is associative.

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

( x 0 ⊗ ⋯ ⊗ x p ) ∘ ( x p + 1 ⊗ ⋯ ⊗ x p + q ) = x 0 ∑ ( p , q ) ( x 1 , … , x p + q ) ,

where the sum is over all (p,q) shuffles.

References

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