PBW Deformations of smash products
General deformation functions
PBWDeformations.is_pbwdeformation
— Functionis_pbwdeformation(d::SmashProductLieDeform)
Check if d
is a Poincare-Birkhoff-Witt deformation of a smash product. Uses Theorem 3.1 of [WW14].
PBWDeformations.pbwdeform_eqs
— Functionpbwdeform_eqs(d::SmashProductLieDeform; disabled::Vector{Symbol}=Symbol[])
Returns the equations for d
being a PBW deformation of a smash product as in Theorem 3.1 of [WW14]. Subsets of the equations can be disabled by passing the corresponding symbols as keyword arguments, e.g. disabled = [:c, :d]
.
All PBW deformations
PBWDeformations.all_pbwdeformations
— Functionall_pbwdeformations(sp::SmashProductLie{C}, deform_basis::DeformBasis{C}; special_return=Nothing) where {C <: RingElem}
Computes a basis of all Poincare-Birkhoff-Witt deformations of sp
. deform_basis
specifies the basis to use for the space of deformation maps. If special_return
is SMat
, the function returns intermediate results.
Uses pbwdeform_eqs
and thus Theorem 3.1 of [WW14].
all_pbwdeformations(sp::SmashProductLie{C}, degs::AbstractVector{Int}, DeformBasisType::Type{<:DeformBasis{C}}=StdDeformBasis{C}; special_return=Nothing) where {C <: RingElem}
Computes a basis of all Poincare-Birkhoff-Witt deformations of sp
of degrees degs
. DeformBasisType
specifies the type of basis to use for the space of deformation maps. If special_return
is SMat
, the function returns intermediate results.
Uses pbwdeform_eqs
and thus Theorem 3.1 of [WW14].
all_pbwdeformations(sp::SmashProductLie{C}, deg::Int, DeformBasisType::Type{<:DeformBasis{C}}=StdDeformBasis{C}; special_return=Nothing) where {C <: RingElem}
The same as the other method, but only for a single degree deg
.
Bases of deformation map spaces
PBWDeformations.DeformBasis
— Typeabstract type DeformBasis{C <: RingElem} end
A basis for a deformation map space of a Lie algebra smash product. The constructor of a subtype should accept a SmashProductLie
and an AbstractVector{Int}
of degrees. It is required that Base.length
and Base.iterate
are implemented for subtypes, where iterating yields objects of type DeformationMap{C}
.
For a reference implementation, we refer to StdDeformBasis
.
Standard basis
PBWDeformations.StdDeformBasis
— TypeConcrete subtype of DeformBasis
that implements the standard basis. Each element of the basis is a skew-symmetric matrix with 2 non-zero entries, where one entry is a pure tensor power of degree ∈ degs
over the Lie algebra part of the smash product, and the other entry is its additive inverse.
Other bases
Please refer to Arc diagram induced bases and Pseudograph induced bases for more specialized bases.