title: Zinbiel algebra
text: # 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 Jean-Michel Lemaire as being "opposite" to Leibniz algebra.
  
  In any Zinbiel algebra, 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.
  
  