diff --git a/dev/arc_diagrams/index.html b/dev/arc_diagrams/index.html index c1d57350..3e503cbd 100644 --- a/dev/arc_diagrams/index.html +++ b/dev/arc_diagrams/index.html @@ -1,2 +1,2 @@ -Arc diagrams · PBWDeformations.jl
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Arc diagrams

Arc diagram induced bases

Warning

The basis ArcDiagDeformBasis can currently only be used for exterior and symmetric powers of the standard module of special orthogonal Lie algebras.

Reverse direction

Given a basis element of an above basis, one can lookup all arc diagrams that induce it (up to a scalar).

PBWDeformations.lookup_data — Function
lookup_data(m::DeformationMap{T}, basis::DeformBasis{T}) where {T <: SmashProductLieElem}

Look up additional data that was used to generate the deformation map m in the basis basis. This can e.g. be an arc diagram or a pseudograph.

source
+Arc diagrams · PBWDeformations.jl
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Arc diagrams

Arc diagram induced bases

Warning

The basis ArcDiagDeformBasis can currently only be used for exterior and symmetric powers of the standard module of special orthogonal Lie algebras.

Reverse direction

Given a basis element of an above basis, one can lookup all arc diagrams that induce it (up to a scalar).

PBWDeformations.lookup_data — Function
lookup_data(m::DeformationMap{T}, basis::DeformBasis{T}) where {T <: SmashProductLieElem}

Look up additional data that was used to generate the deformation map m in the basis basis. This can e.g. be an arc diagram or a pseudograph.

source
diff --git a/dev/index.html b/dev/index.html index f2aa5881..d92a0373 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -PBWDeformations.jl · PBWDeformations.jl
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PBWDeformations

Introduction

The package PBWDeformations will provide both a general framework and specialized functions in order to

  • classify PBW deformations of certain smash products and
  • study their representations.

To solve classification problems efficiently, we use representation theoretic ideas.

Features

  • Construct Lie algebras and their modules.
  • Construct smash products of the form $TV \rtimes U(L)$ for a Lie algbra $L$ and a module $V$.
  • Construct deformations of such smash products.
  • Compute a normal form for elements of smash products and their deformations.
  • Check, if a given deformation is a PBW-deformation (using [WW14]).
  • For some smash product, compute a basis of all PBW-deformations up to a given degree (using [WW14]). It is possible to give a basis of the relevant part of the deformation space, which is then used in the computation.
  • For some modules of $\mathfrak{so}_n$, give an explicit basis using arc diagrams or pseudographs (cf. [FM22]).

Installation

As this package heavily relies on Oscar, it is recommended to install Oscar first (installation instructions). Then, install this package via the Julia package manager:

] add PBWDeformations

Outline

Index

+PBWDeformations.jl · PBWDeformations.jl
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PBWDeformations

Introduction

The package PBWDeformations will provide both a general framework and specialized functions in order to

  • classify PBW deformations of certain smash products and
  • study their representations.

To solve classification problems efficiently, we use representation theoretic ideas.

Features

  • Construct Lie algebras and their modules.
  • Construct smash products of the form $TV \rtimes U(L)$ for a Lie algbra $L$ and a module $V$.
  • Construct deformations of such smash products.
  • Compute a normal form for elements of smash products and their deformations.
  • Check, if a given deformation is a PBW-deformation (using [WW14]).
  • For some smash product, compute a basis of all PBW-deformations up to a given degree (using [WW14]). It is possible to give a basis of the relevant part of the deformation space, which is then used in the computation.
  • For some modules of $\mathfrak{so}_n$, give an explicit basis using arc diagrams or pseudographs (cf. [FM22]).

Installation

As this package heavily relies on Oscar, it is recommended to install Oscar first (installation instructions). Then, install this package via the Julia package manager:

] add PBWDeformations

Outline

Index

diff --git a/dev/pseudographs/index.html b/dev/pseudographs/index.html index f0845706..d60b416b 100644 --- a/dev/pseudographs/index.html +++ b/dev/pseudographs/index.html @@ -1,2 +1,2 @@ -Pseudographs · PBWDeformations.jl
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Pseudographs

Pseudograph induced bases

Warning

The basis PseudographDeformBasis can currently only be used for exterior and symmetric powers of the standard module of special orthogonal Lie algebras.

Reverse direction

Given a basis element of an above basis, one can lookup all pseudographs that induce it (up to a scalar). See lookup_data for more details.

+Pseudographs · PBWDeformations.jl
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Pseudographs

Pseudograph induced bases

Warning

The basis PseudographDeformBasis can currently only be used for exterior and symmetric powers of the standard module of special orthogonal Lie algebras.

Reverse direction

Given a basis element of an above basis, one can lookup all pseudographs that induce it (up to a scalar). See lookup_data for more details.

diff --git a/dev/references/index.html b/dev/references/index.html index 020eee84..baa111b7 100644 --- a/dev/references/index.html +++ b/dev/references/index.html @@ -6,4 +6,4 @@
C. Walton and S. Witherspoon. Poincaré-Birkhoff-Witt deformations of smash product algebras from Hopf actions on Koszul algebras. Algebra & Number Theory 8, 1701 – 1731 (2014).
- + diff --git a/dev/search/index.html b/dev/search/index.html index 92f6d262..d97752bc 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · PBWDeformations.jl

Loading search...

    +Search · PBWDeformations.jl

    Loading search...

      diff --git a/dev/smash_product_deform_lie/index.html b/dev/smash_product_deform_lie/index.html index b8f29e49..372c26ea 100644 --- a/dev/smash_product_deform_lie/index.html +++ b/dev/smash_product_deform_lie/index.html @@ -1,2 +1,2 @@ -Smash product deformations · PBWDeformations.jl
      Edit on GitHub

      Smash products deformations

      Constructors

      PBWDeformations.DeformationMap — Type
      DeformationMap{T} = MatElem{T} where {T <: SmashProductLieElem}

      The type for deformation maps of a Lie algebra smash product. The entry kappa[i,j] should be the image of $v_i \wedge v_j$ under the deformation map, i.e. $κ(v_i,v_j)$. Deformation maps are always assumed to be quadratic and skew-symmetric.

      source
      PBWDeformations.deform — Function
      deform(sp::SmashProductLie{C}, kappa::DeformationMap{elem_type(sp)}) where {C <: RingElem}

      Constructs the deformation of the smash product sp by the deformation map kappa.

      Returns a SmashProductLieDeform struct and a two-part basis.

      source

      SmashProductLieDeform struct

      Functions

      The SmashProductLieDeform struct can be used as an argument for the following functions:

      • gen
      • gens
      • ngens

      For gen, gens, and ngens, on can supply a symbol to choose the part of the smash product to use: :L for the Lie algebra, and :V for the module.

      +Smash product deformations · PBWDeformations.jl
      Edit on GitHub

      Smash products deformations

      Constructors

      PBWDeformations.DeformationMap — Type
      DeformationMap{T} = MatElem{T} where {T <: SmashProductLieElem}

      The type for deformation maps of a Lie algebra smash product. The entry kappa[i,j] should be the image of $v_i \wedge v_j$ under the deformation map, i.e. $κ(v_i,v_j)$. Deformation maps are always assumed to be quadratic and skew-symmetric.

      source
      PBWDeformations.deform — Function
      deform(sp::SmashProductLie{C}, kappa::DeformationMap{elem_type(sp)}) where {C <: RingElem}

      Constructs the deformation of the smash product sp by the deformation map kappa.

      Returns a SmashProductLieDeform struct and a two-part basis.

      source

      SmashProductLieDeform struct

      Functions

      The SmashProductLieDeform struct can be used as an argument for the following functions:

      • gen
      • gens
      • ngens

      For gen, gens, and ngens, on can supply a symbol to choose the part of the smash product to use: :L for the Lie algebra, and :V for the module.

      diff --git a/dev/smash_product_lie/index.html b/dev/smash_product_lie/index.html index 76c043c3..a74dddd5 100644 --- a/dev/smash_product_lie/index.html +++ b/dev/smash_product_lie/index.html @@ -1,2 +1,2 @@ -Smash products · PBWDeformations.jl
      Edit on GitHub

      Smash products

      Constructors

      General case

      PBWDeformations.smash_product — Function
      smash_product(L::LieAlgebra{C}, V::LieAlgebraModule{C}) where {C <: RingElem}

      Construct the smash product $TV \rtimes U(L)$.

      source
      smash_product(R::Ring, L::LieAlgebra{C}, V::LieAlgebraModule{C}) where {C <: RingElem}

      Construct the smash product $TV \rtimes U(L)$ and extend the coefficients to R.

      source

      Highest weight / GAP case

      SmashProductLie struct

      Functions

      The SmashProductLie struct can be used as an argument for the following functions:

      • gen
      • gens
      • ngens

      For gen, gens, and ngens, on can supply a symbol to choose the part of the smash product to use: :L for the Lie algebra, and :V for the module.

      +Smash products · PBWDeformations.jl
      Edit on GitHub

      Smash products

      Constructors

      General case

      PBWDeformations.smash_product — Function
      smash_product(L::LieAlgebra{C}, V::LieAlgebraModule{C}) where {C <: RingElem}

      Construct the smash product $TV \rtimes U(L)$.

      source
      smash_product(R::Ring, L::LieAlgebra{C}, V::LieAlgebraModule{C}) where {C <: RingElem}

      Construct the smash product $TV \rtimes U(L)$ and extend the coefficients to R.

      source

      Highest weight / GAP case

      SmashProductLie struct

      Functions

      The SmashProductLie struct can be used as an argument for the following functions:

      • gen
      • gens
      • ngens

      For gen, gens, and ngens, on can supply a symbol to choose the part of the smash product to use: :L for the Lie algebra, and :V for the module.

      diff --git a/dev/smash_product_pbwdeform_lie/index.html b/dev/smash_product_pbwdeform_lie/index.html index da12984b..fe206b6b 100644 --- a/dev/smash_product_pbwdeform_lie/index.html +++ b/dev/smash_product_pbwdeform_lie/index.html @@ -1,2 +1,2 @@ -PBWDeformations · PBWDeformations.jl
      Edit on GitHub

      PBW Deformations of smash products

      General deformation functions

      PBWDeformations.pbwdeform_eqs — Function
      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].

      source

      All PBW deformations

      PBWDeformations.all_pbwdeformations — Function
      all_pbwdeformations(sp::SmashProductLie{C}, deform_basis::DeformBasis{elem_type(sp)}; 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].

      source
      all_pbwdeformations(sp::SmashProductLie{C}, degs::AbstractVector{Int}, DeformBasisType::Type{<:DeformBasis}=StdDeformBasis; 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].

      source
      all_pbwdeformations(sp::SmashProductLie{C}, deg::Int, DeformBasisType::Type{<:DeformBasis}=StdDeformBasis; special_return=Nothing) where {C <: RingElem}

      The same as the other method, but only for a single degree deg.

      source

      Bases of deformation map spaces

      PBWDeformations.DeformBasis — Type
      abstract type DeformBasis{T <: SmashProductLieElem} 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{T}.

      For a reference implementation, we refer to StdDeformBasis.

      source

      Standard basis

      PBWDeformations.StdDeformBasis — Type

      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.

      source

      Other bases

      Please refer to Arc diagram induced bases and Pseudograph induced bases for more specialized bases.

      +PBWDeformations · PBWDeformations.jl
      Edit on GitHub

      PBW Deformations of smash products

      General deformation functions

      PBWDeformations.pbwdeform_eqs — Function
      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].

      source

      All PBW deformations

      PBWDeformations.all_pbwdeformations — Function
      all_pbwdeformations(sp::SmashProductLie{C}, deform_basis::DeformBasis{elem_type(sp)}; 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].

      source
      all_pbwdeformations(sp::SmashProductLie{C}, degs::AbstractVector{Int}, DeformBasisType::Type{<:DeformBasis}=StdDeformBasis; 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].

      source
      all_pbwdeformations(sp::SmashProductLie{C}, deg::Int, DeformBasisType::Type{<:DeformBasis}=StdDeformBasis; special_return=Nothing) where {C <: RingElem}

      The same as the other method, but only for a single degree deg.

      source

      Bases of deformation map spaces

      PBWDeformations.DeformBasis — Type
      abstract type DeformBasis{T <: SmashProductLieElem} 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{T}.

      For a reference implementation, we refer to StdDeformBasis.

      source

      Standard basis

      PBWDeformations.StdDeformBasis — Type

      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.

      source

      Other bases

      Please refer to Arc diagram induced bases and Pseudograph induced bases for more specialized bases.

      diff --git a/dev/util/index.html b/dev/util/index.html index 59122a0e..1faa2d06 100644 --- a/dev/util/index.html +++ b/dev/util/index.html @@ -1,2 +1,2 @@ -Util functions · PBWDeformations.jl
      Edit on GitHub

      Utility functions

      This section contains some utility functions that are used in other parts of the package.

      +Util functions · PBWDeformations.jl
      Edit on GitHub

      Utility functions

      This section contains some utility functions that are used in other parts of the package.