Three papers by laboratory staff members are published in international book series
The works of the laboratory staff members were published in the international book series Lecture Notes in Computer Science (Scopus Q2):
- S. V. Rumyantseva, D. S. Shirokov, 'On Multidimensional Dirac-Hestenes Equation in Geometric Algebra', Advances in Computer Graphics. CGI 2024. Lecture Notes in Computer Science (Geneva, Switzerland, July 1–5, 2024), 15340, eds. Magnenat-Thalmann, N., Kim, J., Sheng, B., Deng, Z., Thalmann, D., Li, P., Springer, Cham, 2025, 323–335, https://doi.org/10.1007/978-3-031-82024-3_25
Abstract: 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 ideal into the product of the idempotent and an element of the real even subalgebra. The lemma is used to construct the four-dimensional Dirac–Hestenes equation. The analogous lemma is not valid in the multidimensional case, since the dimension of the real even subalgebra of G_{1,2d−1} is bigger than the dimension of the minimal left ideal for d>2. Hence, we consider the auxiliary real subalgebra of G_{1,2d−1} to prove a similar statement. We present the multidimensional Dirac–Hestenes equation for the case G_{1,2d−1}. We prove that one might obtain a solution to the multidimensional Dirac–Hestenes equation using a solution to the multidimensional Dirac equation and vice versa. We also show that the multidimensional Dirac–Hestenes equation has gauge invariance.
- D. S. Shirokov, 'On SU(3) in Ternary Clifford Algebra', Advances in Computer Graphics. CGI 2024. Lecture Notes in Computer Science (Geneva, Switzerland, July 1–5, 2024), 15340, eds. Magnenat-Thalmann, N., Kim, J., Sheng, B., Deng, Z., Thalmann, D., Li, P., Springer, Cham, 2025, 336–348, https://doi.org/10.1007/978-3-031-82024-3_26
Abstract: We discuss a generalization of geometric algebras known as ternary Clifford algebras. In these objects, we have a fixed ternary form instead of a quadratic form as in ordinary geometric algebras. Basis-free definitions of the determinant, trace, and characteristic polynomial in ternary Clifford algebra are introduced. Explicit formulas are presented for all coefficients of the characteristic polynomial and inverse in ternary Clifford algebra. The operation of Hermitian transpose (Hermitian conjugation) in ternary Clifford algebra is introduced without using the corresponding matrix representation. We present a natural realization of the unitary Lie group SU(3), which is important for physical applications, using only operations in ternary Clifford algebra. An explicit basis of the corresponding Lie algebra su(3) is presented. We present an explicit connection with the well-known Gell-Mann basis of su(3). The results can be used in physics, computer science, and engineering.
- E. R. Filimoshina, D. S. Shirokov, 'Generalized Degenerate Clifford and Lipschitz Groups', Advances in Computer Graphics. CGI 2024. Lecture Notes in Computer Science (Geneva, Switzerland, July 1–5, 2024), 15340, eds. Magnenat-Thalmann, N., Kim, J., Sheng, B., Deng, Z., Thalmann, D., Li, P., Springer, Cham, 2025, 364–376, https://doi.org/10.1007/978-3-031-82024-3_28
Abstract: 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 of the fixed grades subspaces and the norm functions that are widely used in the theory of spin groups. The presented groups are interesting for the study of generalized degenerate spin groups and for applications in computer science, physics, and engineering.