# Clifford algebras and their applications to Lie groups and spinors

@article{Shirokov2017CliffordAA, title={Clifford algebras and their applications to Lie groups and spinors}, author={Dmitry Shirokov}, journal={arXiv: Mathematical Physics}, year={2017} }

In these lectures, 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 $n$ 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…

## 21 Citations

### On inner automorphisms preserving subspaces of Clifford algebras

- Mathematics
- 2020

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…

### On Inner Automorphisms Preserving Fixed Subspaces of Clifford Algebras

- MathematicsAdvances in Applied Clifford Algebras
- 2021

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…

### Calculation of Elements of Spin Groups Using Method of Averaging in Clifford’s Geometric Algebra

- MathematicsAdvances in Applied Clifford Algebras
- 2019

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…

### Calculation of Elements of Spin Groups Using Method of Averaging in Clifford’s Geometric Algebra

- MathematicsAdvances in Applied Clifford Algebras
- 2019

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…

### Pro-p Subgroups of Spin Groups and Quaternion Algebras

- Mathematics
- 2021

The main objective of this thesis is to classify pro-p subgroups of the spin groups of Clifford algebras defined on modules over p-adic rings using the method developed by Richard Pink. To accomplish…

### Clifford Algebra, Lorentz Transformation and Unified Field Theory

- Mathematics
- 2018

According to a framework based on Clifford algebra $$C\ell (1,3)$$Cℓ(1,3), this paper gives a classification for elementary fields, and then derives their dynamical equations and transformation laws…

### On generalization of Lipschitz groups and spin groups

- MathematicsMathematical Methods in the Applied Sciences
- 2022

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…

### Application of Clifford Algebra in Solving the Eigen Equations of Quantum Mechanics

- MathematicsAlgebras Groups and Geometries
- 2020

Clifford algebra is unified language and efficient tool for geometry and physics. In this paper, we introduce this algebra to derive the integrable conditions for Dirac and Pauli equations. This…

### Generalization of Dirac Conjugation in the Superalgebraic Theory of Spinors

- MathematicsTheoretical and Mathematical Physics
- 2019

In the superalgebraic representation of spinors using Grassmann densities and the corresponding derivatives, we introduce a generalization of Dirac conjugation, and this generalization yields…

### A Note on the Representation of Clifford Algebra

- Mathematics
- 2020

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant…

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