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## Classification of Lie algebras of specific type in complexified Clifford algebras

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 for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.

Publication based on the results of:

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

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 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

Shirokov D., P-Adic Numbers, Ultrametric Analysis, and Applications 2011 Vol. 3 No. 3 P. 212-218

In this article we consider Clifford algebras over the field of real numbers of finite dimension. We define the operation of Hermitian conjugation for the elements of Clifford algebra. This operation allows us to define the structure of Euclidian space on the Clifford algebra. We consider pseudo-orthogonal group and its subgroups – special pseudo-orthogonal, orthochronous, ...

Added: June 16, 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

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 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

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., Journal of Geometry and Symmetry in Physics 2016 Vol. 42 P. 73-94

In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups (symplectic, orthogonal, linear, unitary) in the cases of arbitrary dimension and arbitrary signature. Also we obtain isomorphisms of corresponding Lie algebras ...

Added: December 14, 2016

N. I. Zhukova, Journal of Geometry and Physics 2018 Vol. 132 P. 146-154

We present a new method of investigation of G-structures on orbifolds.
This method is founded on the consideration of a G-structure on an
n-dimensional orbifold as the corresponding transversal
structure of an associated foliation. Using this method we prove the
existence and the uniqueness of a finite dimensional Lie group structures
on the full automorphism group of an elliptic G-structure ...

Added: April 4, 2017

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

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

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

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.

Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...

Added: May 28, 2021

Zhukova N. I., Mathematical notes 2013 Vol. 93 No. 5-6 P. 928-931

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...

Added: October 19, 2014

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

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

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

Switzerland : Birkhauser/Springer, 2019

Lie theory, inaugurated through the fundamental work of Sophus Lie during the late
nineteenth century, has proved central in many areas of mathematics and theoretical
physics. Sophus Lie’s formulation was originally in the language of analysis and
geometry; however, by now, a vast algebraic counterpart of the theory has been
developed. As in algebraic geometry, the deepest and most ...

Added: October 26, 2019

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

Kharchev S. M., Khoroshkin S. M., Advances in Mathematics 2020 Vol. 375 No. 107368 P. 1-56

We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups ...

Added: October 18, 2020

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

Arzhantsev I., Liendo A., Stasyuk T., Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499

Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...

Added: July 29, 2020

Shirokov D., Theoretical and Mathematical Physics 2013 Vol. 175 No. 1 P. 454-474

We discuss a generalized Pauli theorem and its possible applications for describing n-dimensional (Dirac, Weyl, Majorana, and Majorana–Weyl) spinors in the Clifford algebra formalism. We give the explicit form of elements that realize generalizations of Dirac, charge, and Majorana conjugations in the case of arbitrary space dimensions and signatures, using the notion of the Clifford ...

Added: March 11, 2015