Free Zinbiel Algebras¶
AUTHORS:
- Travis Scrimshaw (2015-09): initial version
-
class
sage.algebras.free_zinbiel_algebra.
FreeZinbielAlgebra
(R, n, names)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
The free Zinbiel algebra on
generators.
Let
be a ring. A Zinbiel algebra is a non-associative algebra with multiplication
that satisfies
Zinbiel algebras were first introduced by Loday as the Koszul dual to Leibniz algebras (hence the name coined by Lemaire).
Zinbiel algebras are divided power algebras, in that for
we have
and
Note
This implies that Zinbiel algebras are not power associative.
To every Zinbiel algebra, we can construct a corresponding commutative associative algebra by using the symmetrized product:
The free Zinbiel algebra on
generators is isomorphic as
-modules to the reduced tensor algebra
with the product
where
is the set of
-shuffles.
The free Zinbiel algebra is free as a divided power algebra. Moreover, the corresponding commutative algebra is isomorphic to the (non-unital) shuffle algebra.
INPUT:
R
– a ringn
– (optional) the number of generatorsnames
– the generator names
Warning
Currently the basis is indexed by all words over the variables, incuding the empty word. This is a slight abuse as it is suppose to be the indexed by all non-empty words.
EXAMPLES:
We create the free Zinbiel algebra and check the defining relation:
sage: Z.<x,y,z> = algebras.FreeZinbiel(QQ) sage: (x*y)*z Z[xyz] + Z[xzy] sage: x*(y*z) + x*(z*y) Z[xyz] + Z[xzy]
We see that the Zinbiel algebra is not associative, nor even power associative:
sage: x*(y*z) Z[xyz] sage: x*(x*x) Z[xxx] sage: (x*x)*x 2*Z[xxx]
We verify that it is a divided powers algebra:
sage: (x*(x*x)) * (x*(x*(x*x))) 15*Z[xxxxxxx] sage: binomial(3+4-1,4) 15 sage: (x*(x*(x*x))) * (x*(x*x)) 20*Z[xxxxxxx] sage: binomial(3+4-1,3) 20 sage: ((x*x)*x)*x 6*Z[xxxx] sage: (((x*x)*x)*x)*x 24*Z[xxxxx]
REFERENCES:
[Loday95] Jean-Louis Loday. Cup-product for Leibniz cohomology and dual Leibniz algebras. Math. Scand., pp. 189–196 (1995). http://www.math.uiuc.edu/K-theory/0015/cup_product.pdf [LV12] Jean-Louis Loday and Bruno Vallette. Algebraic Operads. Springer-Verlag Berlin Heidelberg (2012). doi:10.1007/978-3-642-30362-3. -
algebra_generators
()¶ Return the algebra generators of
self
.EXAMPLES:
sage: Z.<x,y,z> = algebras.FreeZinbiel(QQ) sage: list(Z.algebra_generators()) [Z[x], Z[y], Z[z]]
-
gens
()¶ Return the generators of
self
.EXAMPLES:
sage: Z.<x,y,z> = algebras.FreeZinbiel(QQ) sage: Z.gens() (Z[x], Z[y], Z[z])
-
product_on_basis
(x, y)¶ Return the product of the basis elements indexed by
x
andy
.EXAMPLES:
sage: Z.<x,y,z> = algebras.FreeZinbiel(QQ) sage: (x*y)*z # indirect doctest Z[xyz] + Z[xzy]