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June 5, 2026
Neural Network Maps as a Method for Constructing Mathematical Models
Scientists from HSE University–Nizhny Novgorod and the Institute of Physics Belgrade, Serbia, are jointly exploring the application of machine learning techniques and neural networks to the study of nonlinear dynamics. Natalya Stankevich, Leading Research Fellow at the Laboratory of Topological Methods in Dynamics of the Faculty of Informatics, Mathematics, and Computer Science at HSE University–Nizhny Novgorod, spoke to the HSE News Service about this international project.
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‘In the Age of Technology, It Is Interesting to Look into the Past and Think about What We Can Take from It
Polina Tabakova decided to apply for a Philology degree at HSE in Nizhny Novgorod because she grew up in Mari El and did not want to move far away from the Russian forests. In an interview for the Young Scientists of HSE University project, she spoke about the genre of the campus novel, the existential drama of Kolobok, and a blackout version of Eugene Onegin.
June 5, 2026
HSE Scientists Develop Method to Compress Large Language Models Without Losing Quality
Researchers from the AI and Digital Science Institute at the HSE Faculty of Computer Science have developed a new compression method for large language models such as GPT and LLaMA that reduces their size by 25–36% without additional training or significant loss of accuracy. This is the first approach to use mathematical transformations—specifically, rotations of model weights—to make models more amenable to compression with structured matrices. The study results have been published in ACL Findings 2025. The code is available on GitHub.

 

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On SU(3) in Ternary Clifford Algebra

P. 336–348.
Shirokov D.
Language: English
DOI
Text on another site
Keywords: characteristic polynomialdeterminantgeometric algebraunitary groupinverseternary Clifford algebraSU(3)

In book

Advances in Computer Graphics: 41st Computer Graphics International Conference, CGI 2024, Geneva, Switzerland, July 1–5, 2024, Proceedings, Part III
Vol. 15340. , Springer, 2025.
Similar publications
On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras
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
Lorentz Invariance of the Multidimensional Dirac–Hestenes Equation
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
Equivariant Neural Networks with Geometric Algebras: A New Approach
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
GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras
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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 ...
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Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras
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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
On Grade Automorphism in Ternary Clifford Algebras
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 ...
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Generalized Degenerate Clifford and Lipschitz Groups in Geometric Algebras
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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
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We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic form in ordinary Clifford algebras. We present a natural realization of unitary Lie groups, which are important in physics and ...
Added: May 20, 2025
Generalized Degenerate Clifford and Lipschitz Groups
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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
On Multidimensional Dirac–Hestenes Equation in Geometric Algebra
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
Calculation of Spin Group Elements Revisited
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
On Rank of Multivectors in Geometric Algebras
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
Introducing Multidimensional Dirac–Hestenes Equation
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
A Note on Centralizers and Twisted Centralizers in Clifford Algebras
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 ...
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Maximal generalized rank in graphical matrix spaces
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Added: October 3, 2024
On SVD and Polar Decomposition in Real and Complexified Clifford Algebras
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
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