Zinbiel algebra

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 \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 J.-M. Lemaire as being "opposite" to Leibniz algebra.[1]

The symmetrised product

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 \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) shuffles.[1]

References

  1. ↑ 1.0 1.1 Loday 2001, p. 45