PBW Deformations of smash products

is_pbwdeformation(d::SmashProductLieDeform)

Check if d is a Poincare-Birkhoff-Witt deformation of a smash product. Uses Theorem 3.1 of [WW14].

pbwdeform_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_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.

abstract 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.

Concrete 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.