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On solutions of the Yang-Mills equations in the algebra of h-forms
Ch. 012015. P. 1–7.
We study the Yang-Mills equations in the algebra of h-forms, which is developed in the works of N. G. Marchuk and the author. The algebra of h-forms is a special geometrization of the Clifford algebra and is a generalization of the Atiyah-Kahler algebra. We discuss an invariant subspace of the constant Yang-Mills operator in the algebra of h-forms and present particular classes of solutions of the Yang-Mills equations.
In book
Vol. 2099: International Conference «Marchuk Scientific Readings 2021» (MSR-2021) 4-8 October 2021, Novosibirsk, Russian Federation. , IOP Publishing, 2021.
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
Alexander S. Rabinowitch, Annals of Physics 2025 Vol. 480 Article 170149
In the present paper, transverse progressive waves in Yang-Mills fields with SU(2) symmetry propagating in the direction of the Cartesian z-axis are considered. In this case, the considered Yang-Mills equations are reduced to a system of six nonlinear partial differential equations. Field potentials satisfying them are sought in a special form. Substituting it in the ...
Added: July 14, 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
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
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
Sofia Rumyantseva, Shirokov D., Advances in Applied Clifford Algebras 2025 Vol. 35 Article 24
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 multidimensional Dirac-Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra ℂ⊗Cℓ1,n depends on a parity of n, we explore even and odd ...
Added: November 8, 2024
Filimoshina E., Shirokov D., Advances in Applied Clifford Algebras 2024 Vol. 34 Article 50
This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in ...
Added: November 8, 2024
Shirokov D., Advances in Applied Clifford Algebras 2024 Vol. 34 Article 23
In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related ...
Added: August 23, 2024
Alexander S. Rabinowitch, Nuclear Physics B 2024 Vol. 1001 Article 116505
In the present paper, a new class of wave solutions to the Yang-Mills equations with SU(2) symmetry is considered. They describe the propagation of non-Abelian transverse progressive waves. In the case under consideration, the problem is reduced to six nonlinear partial differential equations. Their solutions are sought in a special form that allows them to be ...
Added: March 6, 2024
Filimoshina E., Shirokov D., , in: Advanced Computational Applications of Geometric Algebra: First International Conference, ICACGA 2022, Denver, CO, USA, October 2-5, 2022, ProceedingsVol. 13771.: Springer, 2024. P. 186–198.
In this paper, we introduce and study several Lie groups in degenerate (Clifford) geometric algebras. These Lie groups preserve the even and odd subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups are interesting for the study of spin groups and their generalizations in degenerate case. ...
Added: February 3, 2024
P. M. Akhmet'ev, Dvornikov M. S., Journal of Geometry and Physics 2024 Vol. 198 Article 105102
We construct a new Yang-Mills 3D-solution on the space of negative scalar curvature. We
discuss a problem of non-abelian gauge symmetry is broken with the assumption that
a scalar curvature of the domain is a negative small parameter. In this case we use the
following fact: a geometrical scale related with Vassiliev’s discriminant of magnetic lines
coincides with a ...
Added: January 23, 2024
Marchuk N., М.: Издательская группа URSS, 2023.
В книге изучаются релятивистские уравнения теории поля, в частности рассматриваются свойства ковариантности и симметрии уравнений Дирака—Максвелла и Дирака—Янга—Миллса. Вводится ряд новых систем уравнений, называемых модельными уравнениями теории поля. Эти системы уравнений воспроизводят основные свойства стандартных систем уравнений теории поля. В то же время модельные уравнения имеют ряд отличий от стандартных уравнений теории поля, в частности обладают новой внутренней симметрией ...
Added: January 3, 2024
Shirokov D., , in: Advances in Computer Graphics: 40th Computer Graphics International Conference, CGI 2023, Shanghai, China, August 28 – September 1, 2023, Proceedings, Part IV* 4. Vol. 14498.: Springer, 2024. P. 391–401.
This paper is a brief note on the natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real (Clifford) geometric algebras of arbitrary dimension and signature. We naturally define these and other related structures (operation of Hermitian conjugation, Euclidean space, and Lie groups) in geometric algebras. The results ...
Added: December 25, 2023
Shirokov D., Modern Physics Letters A 2023 Vol. 38 No. 20n21 Article 2350096
We present a classification and an explicit form of all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in pseudo-Euclidean space ℝp,q of arbitrary finite dimension n=p+q. Using hyperbolic singular value decomposition and two-sheeted covering of orthogonal group by spin group, we solve the nontrivial system for constant solutions of the Yang-Mills ...
Added: October 5, 2023
Shirokov D., Mathematics 2023 Vol. 11 No. 16 Article 3607
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Open AccessFeature PaperArticle
Development of the Method of Averaging in Clifford Geometric Algebras
by
Dmitry Shirokov
1,2
1
HSE University, Myasnitskaya Str. 20, Moscow 101000, Russia
2
Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny Per. 19, Moscow 127051, Russia
Mathematics 2023, 11(16), 3607; https://doi.org/10.3390/math11163607
Received: 29 June 2023 / Revised: 15 August 2023 / Accepted: 17 August 2023 / Published: 21 August 2023
(This article belongs to the ...
Added: October 5, 2023
Dmitry Shirokov, , in: Empowering Novel Geometric Algebra for Graphics and Engineering. 7th International Workshop, ENGAGE 2022, Virtual Event, September 12, 2022, Proceedings.: Cham: Springer, 2023. P. 28–37.
In this paper, we discuss a generalization of Vieta theorem (Vieta’s formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta’s formulas with the ordinary Vieta’s formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand – Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. ...
Added: August 19, 2023
Ekaterina Filimoshina, Dmitry Shirokov, Advances in Applied Clifford Algebras 2023 Vol. 33 Article 44
In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in ...
Added: August 19, 2023