Free algebra elements

AUTHORS:

  • David Kohel (2005-09)

TESTS:

sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: x == loads(dumps(x))
True
sage: x*y
x*y
sage: (x*y)^0
1
sage: (x*y)^3
x*y*x*y*x*y
class sage.algebras.free_algebra_element.FreeAlgebraElement(A, x)

Bases: sage.structure.element.AlgebraElement, sage.combinat.free_module.CombinatorialFreeModuleElement

A free algebra element.

to_pbw_basis()

Return self in the Poincare-Birkhoff-Witt (PBW) basis.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(ZZ, 3)
sage: p = x^2*y + 3*y*x + 2
sage: p.to_pbw_basis()
2*PBW[1] + 3*PBW[y]*PBW[x] + PBW[x^2*y] + PBW[x*y]*PBW[x] + PBW[y]*PBW[x]^2
variables()

Return the variables used in self.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(ZZ,3)
sage: elt = x + x*y + x^3*y
sage: elt.variables()
[x, y]
sage: elt = x + x^2 - x^4
sage: elt.variables()
[x]
sage: elt = x + z*y + z*x
sage: elt.variables()
[x, y, z]

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Free algebras

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Free associative unital algebras, implemented via Singular’s letterplace rings

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