Differential geometric algebra: compute using Grassmann.jl, Cartan.jl

Differential geometric algebra: compute using Grassmann.jl, Cartan.jl

https://vixra.org/abs/2505.0100

Initially, Grassmann.jl and Cartan.jl introduced the DirectSum formalism into computational language design for differential geometric algebra, thus enabling the construction of custom algebras through metaprogramming. Grassmann.jl pioneered a novel type system design along with its syntax and semantics, which has undergone significant refinement through many years of development and continuous improvement. Cartan.jl represents a groundbreaking extension of capabilities offered by Grassmann.jl, marking a pioneering fully realized implementation of numerical differential geometric algebra, based on TensorField representations over a FrameBundle and the ImmersedTopology of the FiberBundle. Grassmann.jl and Cartan.jl build on Julia's multiple dispatch and metaprogramming capabilities, presenting a new computational language design approach to interfacing with differential geometric algebra based on a new sector integral theorem. This pioneering design not only actualizes but also elevates computational language syntax to new heights using the foundations of Grassmann.jl and Cartan.jl. The Grassmann.jl and Cartan.jl packages introduce pioneering computational language designs, having inspired imitation projects and thereby validating the project's relevance as significant advance in computational mathematics.

https://github.com/chakravala

The Grassmann.jl package provides tools for computations based on multi-linear algebra and spin groups using the extended geometric algebra known as Grassmann-Clifford-Hodge algebra. Algebra operations include exterior, regressive, inner, and geometric, along with the Hodge star and boundary operators. Code generation enables concise usage of the algebra syntax. DirectSum.jl multivector parametric type polymorphism is based on tangent vector spaces and conformal projective geometry. Additionally, the universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built. The design is based on TensorAlgebra{V} abstract type interoperability from AbstractTensors.jl with a K-module type parameter V from DirectSum.jl. Abstract vector space type operations happen at compile-time, resulting in a differential geometric algebra of multivectors.

Principal Differential Geometric Algebra (Hardcover, 2025)
https://www.lulu.com/shop/michael-ree...

Principal Differential Geometric Algebra (Paperback, 2025)
https://www.lulu.com/shop/michael-ree...

Mathematical foundations and definitions specific to the Grassmann.jl implementation provide an extensible platform for computing with a universal language for finite element methods based on a discrete manifold bundle. Tools built on these foundations enable computations based on multi-linear algebra and spin groups using the geometric algebra known as Grassmann algebra or Clifford algebra. This foundation is built on a DirectSum.jl parametric type system for tangent bundles and vector spaces generating the algorithms for local tangent algebras in a global context. With this unifying mathematical foundation, it is possible to improve efficiency of multi-disciplinary research using geometric tensor calculus by relying on universal mathematical principles.

Cartan.jl introduces a pioneering unified numerical framework for comprehensive differential geometric algebra, purpose-built for the formulation and solution of partial differential equations on manifolds with non-trivial topological structure and Grassmann.jl algebra. Written in Julia, Cartan.jl unifies differential geometry, geometric algebra, and tensor calculus with support for fiber product topology; enabling directly executable generalized treatment of geometric PDEs over grids, meshes, and simplicial decompositions.

The system supports intrinsic formulations of differential operators (including the exterior derivative, codifferential, Lie derivative, interior product, and Hodge star) using a coordinate-free algebraic language grounded in Grassmann-Cartan multivector theory. Its core architecture accomodates numerical representations of principal G-fiber bundles, product manifolds, and submanifold immersion, providing native support for PDE models defined on structured or unstructured domains.

Cartan.jl integrates naturally with simplex-based finite element exterior calculus, allowing for geometrical discretizations of field theories and conservation laws. With its synthesis of symbolic abstraction and numerical execution, Cartan.jl empowers researchers to develop PDE models that are simultaneously founded in differential geometry, algebraically consistent, and computationally expressive, opening new directions for scientific computing at the interface of geometry, algebra, and analysis.