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Zinbiel algebra

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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 ) . {\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).}

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

In any Zinbiel algebra, the symmetrised product

a ⋆ b = a ∘ b + b ∘ a {\displaystyle a\star b=a\circ b+b\circ 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 ) , {\displaystyle (x_{0}\otimes \cdots \otimes x_{p})\circ (x_{p+1}\otimes \cdots \otimes x_{p+q})=x_{0}\sum _{(p,q)}(x_{1},\ldots ,x_{p+q}),}

where the sum is over all ( p , q ) {\displaystyle (p,q)} shuffles.

References

  1. ^ Loday 2001, p. 45
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