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A note on subspaces of fixed grades in Clifford algebras
Ch. 060001. P. 060001-1–060001-4.
In this note, we present basis-free definitions of subspaces of fixed grades of real Clifford algebras of arbitrary dimension. We do not use fixed basis of Clifford algebra and use only the properties of commutators and anticommutators.
Yu Z., Wang J., Wang Z. et al., Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 2026 Vol. 384 No. 2326 Article 20250103
The integration of data-driven and knowledge-driven approaches in generative geospatial modelling (GGM) is often hindered by their mathematical incompatibilities. Here, we propose a geometric algebra (GA)-based framework that employs a unified multi-vector representation to fuse heterogeneous data and diverse knowledge. The framework facilitates structured reasoning and hypothesis generation through a task-adaptable, five-stage cycle: representation, reasoning, ...
Added: August 13, 2026
Abdulkhaev K., Shirokov D., , in: 10th International Workshop, ENGAGE 2025, Hong Kong, China, July 14, 2025, Proceedings. Empowering Novel Geometric Algebra for Graphics and Engineering, (LNCS, volume 16510)Vol. 16510.: Cham: Springer, 2026. P. 189–200.
In this paper, we present an explicit formula for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension . We generalize the concept of conjugation to basis conjugation operations, allowing us to express the determinant formula independently of any specific algebra isomorphism. This construction provides a practical computational tool for determining invertibility ...
Added: August 13, 2026
Filimoshina E., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 16
This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that these Lie groups can be equivalently defined using norm functions of multivectors applied in the theory of spin groups. We also study the ...
Added: January 12, 2026
Sofia Rumyantseva, Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 5
This paper investigates the Lorentz invariance of the multidimensional Dirac–Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac–Hestenes equation, comparing both transformation approaches. These results are subsequently ...
Added: December 19, 2025
Sharma H., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 9
We investigate commutative analogues of Clifford algebras - algebras whose generators square to ±1 but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces - we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space ...
Added: December 2, 2025
Filimoshina E., Shirokov D., , in: 2025 International Joint Conference on Neural Networks (IJCNN).: IEEE, 2025. P. 1–8.
This work is devoted to construction and implementation of new equivariant neural networks based on geometric (Clifford) algebras. We propose, implement, test, and compare with competitors a new architecture of equivariant neural networks, which we call Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations. We introduce ...
Added: November 15, 2025
Filimoshina E., Shirokov D., , in: Volume 267: International Conference on Machine Learning, 13-19 July 2025, Vancouver Convention Center, Vancouver, CanadaVol. 267.: [б.и.], 2025. P. 17153–17188.
We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization ...
Added: October 28, 2025
Sharma H., Shirokov D., Advances in Applied Clifford Algebras 2025 Vol. 35 Article 44
Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper, we solve the problem of finding multiplicative inverses in commutative analogues of Clifford algebras by introducing a matrix representation ...
Added: October 2, 2025
Shirokov D., , in: Hypercomplex Analysis and Its Applications.Extended Abstracts of the International Conference Celebrating Paula Cerejeiras’ 60th Birthday. ICHAA 2024. Trends in Mathematics (TM, volume 9)Vol. 9.: Birkhäuser, 2025. P. 143–150.
For the first time, we introduce a grade automorphism in ternary Clifford algebras and discuss a number of its properties. This operation is not an involution, but naturally generalizes the grade involution (or the main involution) in ordinary (quadratic) Clifford algebras. The new operation can be used in different applications of ternary Clifford algebras in ...
Added: July 6, 2025
Filimoshina E., Shirokov D., Advances in Applied Clifford Algebras 2025 Vol. 35 Article 29
This paper introduces and studies generalized degenerate Clifford and Lipschitz groups in geometric (Clifford) algebras. These Lie groups preserve the direct sums of the subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations in degenerate geometric algebras. We prove that the generalized degenerate Clifford and Lipschitz groups can be ...
Added: May 29, 2025
Shirokov D., , in: Advances in Computer Graphics: 41st Computer Graphics International Conference, CGI 2024, Geneva, Switzerland, July 1–5, 2024, Proceedings, Part IIIVol. 15340.: Springer, 2025. P. 336–348.
Added: April 1, 2025
Filimoshina E., Dmitry Shirokov, , in: Advances in Computer Graphics: 41st Computer Graphics International Conference, CGI 2024, Geneva, Switzerland, July 1–5, 2024, Proceedings, Part IIIVol. 15340.: Springer, 2025. P. 364–376.
This paper introduces generalized Clifford and Lipschitz groups in degenerate geometric (Clifford) algebras. These groups preserve the direct sums of the subspaces determined by the grade involution and the reversion under the adjoint and twisted adjoint representations. We prove that the generalized degenerate Clifford and Lipschitz groups can be defined using centralizers and twisted centralizers ...
Added: April 1, 2025
Rumiantseva S., Shirokov D., , in: Advances in Computer Graphics: 41st Computer Graphics International Conference, CGI 2024, Geneva, Switzerland, July 1–5, 2024, Proceedings, Part IIIVol. 15340.: Springer, 2025. P. 323–335.
It is easier to investigate phenomena in particle physics geometrically by exploring a real solution to the Dirac–Hestenes equation instead of a complex solution to the Dirac equation. The present research outlines the 2d-dimensional Dirac–Hestenes equation. In the geometric algebra G_{1,3}, there is a lemma on the unique decomposition of an element of the minimal left ...
Added: February 28, 2025
Shirokov D., International Journal of Geometric Methods in Modern Physics 2026 Vol. 23 No. 5 Article 2540031
In this paper, we present a method for calculation of spin groups elements for known pseudo-orthogonal group elements with respect to the corresponding two-sheeted coverings. We present our results using the Clifford algebra formalism in the case of arbitrary dimension and signature, and then explicitly using matrices, quaternions, and split-quaternions in the cases of all ...
Added: December 5, 2024
Dmitry Shirokov, Mathematical Methods in the Applied Sciences 2025 Vol. 48 No. 11 P. 11095–11102
We introduce the notion of rank of multivector in Clifford geometric algebras of arbitrary dimension without using the corresponding matrix representations and using only geometric algebra operations. We use the concepts of characteristic polynomial in geometric algebras and the method of SVD. The results can be used in various applications of geometric algebras in computer ...
Added: December 4, 2024