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## On Some Lie Groups in Degenerate Clifford Geometric Algebras

Advances in Applied Clifford Algebras. 2023. Vol. 33. Article 44.

Ekaterina Filimoshina, Dmitry Shirokov

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 the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science.

Publication based on the results of:

Ekaterina Filimoshina, Dmitry Shirokov, Mathematical Methods in the Applied Sciences 2024 Vol. 47 No. 3 P. 1375-1400

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and ...

Added: October 11, 2022

Kamron Abdulkhaev, Shirokov D., Advances in Applied Clifford Algebras 2022 Vol. 32 No. 5 Article 57

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras Gp,q of vector space of dimension 𝑛=𝑝+𝑞. We present basis-free formulas for all characteristic polynomial coefficients in the cases 𝑛≤6, alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas ...

Added: October 11, 2022

Shirokov D., Advances in Applied Clifford Algebras 2021 Vol. 31 Article 30

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras — subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted ...

Added: May 10, 2021

Shirokov D., Advances in Applied Clifford Algebras 2012 Vol. 22 No. 1 P. 243-256

We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements. ...

Added: June 16, 2015

Filimoshina E., Shirokov D., , in : Advanced Computational Applications of Geometric Algebra: First International Conference, ICACGA 2022, Denver, CO, USA, October 2-5, 2022, Proceedings. Vol. 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

Shirokov D., Computational and Applied Mathematics 2021 Vol. 40 P. 1-29

In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only ...

Added: July 15, 2021

Shirokov D., Linear and Multilinear Algebra 2018 Vol. 66 No. 9 P. 1870-1887

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group ...

Added: September 29, 2017

Shirokov D., Advances in Applied Clifford Algebras 2021 Vol. 31 P. 1-19

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and ...

Added: September 19, 2021

Shirokov D., Advances in Applied Clifford Algebras 2019 Vol. 29 No. 50 P. 1-12

We present a method of computing elements of spin groups in the case of arbitrary dimension. This method generalizes Hestenes method for the case of dimension 4. We use the method of averaging in Clifford’s geometric algebra previously proposed by the author. We present explicit formulas for elements of spin group that correspond to the ...

Added: July 22, 2019

Dmitry Shirokov, Mathematical Methods in the Applied Sciences 2023 P. 1-16

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 formulas with the ordinary Vieta 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. We introduce the ...

Added: April 2, 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

Shirokov D., Advances in Applied Clifford Algebras 2010 Vol. 20 No. 2 P. 411-425

In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudo-unitary groups. Our main techniques are Clifford algebras. We have found 12 types of subalgebras of Lie algebras of pseudo-unitary groups. ...

Added: June 16, 2015

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

Sheina K., / Cornell University. Series arXiv "math". 2020. No. 04348v1.

The basic automorphism group of a Cartan foliation (M, F) is the quotient group of the automorphism group of (M, F) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates ...

Added: December 9, 2020

Covolo T., Journal of Noncommutative Geometry 2015 Vol. 9 No. 2 P. 543-565

We develop the theory of linear algebra over a (Z2)n-commutative algebra (n∈N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduce analogues of the notions of trace and determinant. Our construction ...

Added: September 28, 2015

Popov V., Известия РАН. Серия математическая 2022 Т. 86 № 5 С. 73-96

We explore to what extent the group variety of a connected algebraic group or the group manifold of a real Lie group determines its group structure. ...

Added: June 9, 2022

Shirokov D., Advances in Applied Clifford Algebras 2018 Vol. 28 No. 3 P. 1-16

We present a new class of covariantly constant solutions of the Yang–Mills equations. These solutions correspond to the solution of the field equation for the spin connection of the general form. ...

Added: July 6, 2018

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2018. No. 1804.00323v1.

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of cha\-racte\-ristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan. ...

Added: April 3, 2018

Shirokov D., Marchuk N., Advances in Applied Clifford Algebras 2008 Vol. 18 No. 2 P. 237-254

For the complex Clifford algebra <img /> (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These ...

Added: June 16, 2015

Shirokov D., Marchuk N., Reports on Mathematical Physics 2016 Vol. 78 No. 3 P. 305-326

We find general solutions of some field equations (systems of equations) in pseudo-Euclidian spaces (so-called primitive field equations). These equations are used in the study of the Dirac equation and Yang-Mills equations. These equations are invariant under orthogonal O(p,q) coordinate transformations and invariant under gauge transformations, which depend on some Lie groups. In this paper ...

Added: September 27, 2016

Shirokov D., Наноструктуры. Математическая физика и моделирование 2013 Т. 9 № 1 С. 93-104

В работе доказаны утверждения, которые обобщают так называемую фундаментальную теорему Паули о гамма-матрицах. Рассмотрены алгебры Клиффорда над полем вещественных и комплексных чисел произвольной размерности. Для произвольных двух наборов из четного или нечетного числа элементов, удовлетворяющих определяющим антикоммутационным соотношениям алгебры Клиффорда, доказаны обобщения теоремы Паули. Предъявлены алгоритмы для вычисления элемента, осуществляющего связь между двумя наборами. ...

Added: July 22, 2019

Shirokov D., Advances in Applied Clifford Algebras 2015 Vol. 25 No. 3 P. 707-718

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras. ...

Added: March 12, 2015

Shirokov D., Advances in Applied Clifford Algebras 2015 Vol. 25 No. 1 P. 227-244

We formulate generalizations of Pauli’s theorem on the cases of real and complex Clifford algebras of even and odd dimensions. We give analogues of these theorems in matrix formalism. Using these theorems we present an algorithm for computing elements of spin groups that correspond to elements of orthogonal groups as double cover. ...

Added: March 11, 2015

V. L. Popov, Mathematical notes 2018 Vol. 103 No. 5 P. 811-819

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan. ...

Added: April 13, 2018