Grassmann.jl

⟨Grassmann-Clifford-Hodge⟩ multilinear differential geometric algebra

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 $\mathbb{K}$-module type parameter $V$ from DirectSum.jl. Abstract vector space type operations happen at compile-time, resulting in a differential geometric algebra of multivectors.

• TensorAlgebra{V} design and code generation
• • Direct sum parametric type polymorphism
  • Higher dimensions with SparseBasis and ExtendedBasis
  • Interoperability for TensorAlgebra{V}
• Grassmann elements and geometric algebra Λ(V)
• • Grassmann.jl API design overview
  • Grassmann-Hodge complement
  • Symbolic coefficients by declaring algebra
• Grassmann.jl videos
• Grassmann.jl Library
• References

• Michael Reed, Principal Differential Geometric Algebra: compute using Grassmann.jl, Cartan.jl (Hardcover, 2025)
• Michael Reed, Principal Differential Geometric Algebra: compute using Grassmann.jl, Cartan.jl (Paperback, 2025)

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