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
- Smash products
- Smash products deformations
- PBW Deformations of smash products
- Arc diagrams
- Utility functions
Index
PBWDeformations.ArcDiagDeformBasis
PBWDeformations.DeformBasis
PBWDeformations.DeformationMap
PBWDeformations.PseudographDeformBasis
PBWDeformations.SmashProductLie
PBWDeformations.SmashProductLieDeform
PBWDeformations.StdDeformBasis
PBWDeformations.all_pbwdeformations
PBWDeformations.deform
PBWDeformations.is_pbwdeformation
PBWDeformations.lookup_data
PBWDeformations.pbwdeform_eqs
PBWDeformations.smash_product
PBWDeformations.symmetric_deformation