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## Clifford algebras and their applications to Lie groups and spinors

Ch. 1. P. 11-53.

We discuss some well-known facts about Clifford algebras: matrix representations, Cartan’s periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in <span data-mathml="nn dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

### In book

Vol. 19. , Sofia: Avangard Prima, 2018

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

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

For the complex Clifford algebra (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 representations ...

Added: June 16, 2015

Marchuk N., Shirokov D., Azerbaijan Journal of Mathematics 2020 Vol. 10 No. 1 P. 38-56

Generalized Pauli’s theorem, proved by D. S. Shirokov for two sets of anticommuting elements of a real or complexified Clifford algebra of dimension 2n, is extended to the case, where both sets of elements depend smoothly on points of Euclidean space of dimension r. We prove that in the case of even n there exists ...

Added: February 3, 2020

Soldatenkov A. O., Verbitsky M., Journal of Geometry and Physics 2014

Let (M,I,J,K) be a hyperkahler manifold, and Z⊂(M,I) a complex subvariety in (M,I). We say that Z is trianalytic if it is complex analytic with respect to J and K, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures (M,I,J′,K′) containing I. For a generic complex structure I ...

Added: December 26, 2014

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 2012 Vol. 22 No. 2 P. 483-497

In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous papers. On the basis of new classification of Clifford algebra elements, it is possible to reveal and prove a number of new properties of Clifford algebras. We use k-fold commutators and anticommutators. In ...

Added: June 16, 2015

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

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 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., Method of generalized Reynolds operators and Pauli's theorem in Clifford algebras / Cornell University. Series math "arxiv.org". 2016.

We consider real and complex Clifford algebras of arbitrary even and odd dimensions and prove generalizations of Pauli's theorem for two sets of Clifford algebra elements that satisfy the main anticommutative conditions. In our proof we use some special operators - generalized Reynolds operators. This method allows us to obtain an algorithm to compute elements ...

Added: October 9, 2016

Shirokov D., Nikolay Marchuk, 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., Advances in Applied Clifford Algebras 2017 Vol. 27 No. 1 P. 149-163

In this paper we consider different operators acting on Clifford algebras. We consider Reynolds operator of Salingaros’ vee group. This operator “average” an action of Salingaros’ vee group on Clifford algebra. We consider conjugate action on Clifford algebra. We present a relation between these operators and projection operators onto fixed subspaces of Clifford algebras. Using ...

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

Н.И. Жукова, Mathematical Notes (Rusian Federation) 2013 Т. 93 № 6 С. 994-996

In this paper a unified method for studying foliations with transversal psrsbolic 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: September 28, 2014

Danilov V., European Journal of Mechanics - B/Fluids 2019 Vol. 74 P. 152-158

A stratified liquid with two layers separated by a fast oscillating interface in the case of Raleigh--Taylor instability
is considered. The averaged equations are derived, and it is shown that a mushy region of a certain density appears after averaging. The similarity between this fact and the case of unstable jump decay is discussed. ...

Added: October 21, 2018

Bezrukavnikov R., Ivan Losev, On dimension growth of modular irreducible representations of semisimple Lie algebras / Cornell University. Series arXiv "math". 2017. No. 1708.01385.

Added: October 9, 2017

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

Feigin E., Selecta Mathematica, New Series 2012 Vol. 18 No. 3 P. 513-537

Let Fλ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration Fλa, which is a GaM variety. Moreover, there exists a larger group Ga acting on Fλa, which is a degeneration of the group G. The group Ga contains ...

Added: August 31, 2012

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