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**On some Lie groups containing spin group in Clifford algebra.**
*(English)*
Zbl 1379.15016

This algebraic paper is of interest for graduate students and researchers of theoretical physics and hypercomplex algebra.

The introduction gives a thorough overview of related work, including the previous work of the authors. A number of fundamental definitions of the Clifford algebra, the complexified Clifford algebra, and the \(Z_2\)-grading is given. A definition of real Clifford algebras with quaternion type subspaces is given, as well as a corresponding complexified notion.

Section 2 explains a recurrent method for the construction of isomorphic matrix represetations of real Clifford algebras of a general signature \(\mathrm{Cl}(p,q)\), together with a set of representative low-dimensional examples. Section 3 describes the relations between various conjugations (and involutions) in a Clifford algebra and matrix operations, like transposition, Hermitian conjugation, and conjugate transpose of quaternion matrices, including the case of complexified Clifford algebras. Section 4 introduces the helpful notion of additional signature of real (and complexified) Clifford algebras, which means the number of symmetric isomorphic matrix basis vectors (generators), and the number of skew-symmetric isomorphic matrix basis vectors (generators). The application of the notion of additional signature for relating matrix transposition and Clifford algebra involutions is demonstrated.

Section 5 includes a number of theorems for five types of subsets of Clifford algebras, isomorphic to Lie groups. For example, the subset \(G^2_{p,q} = \{U \in \mathrm{Cl}^{(0)}(p,q): \mathrm{reverse}(U)U = 1\}\), where \(\mathrm{Cl}^{(0)}(p,q)\) denotes the even subalgebra of \(\mathrm{Cl}(p,q)\). The isomorphic Lie groups are stated, their Lie algebras, and their dimensions. Theorem 1 includes a collection of isomorphisms between these five Lie groups. Theorems 2 and 3 provide, dependent on the Clifford algebra signature, isomorphisms of the five Clifford algebra represented Lie groups with unitary, symplectic, orthogonal, and general linear classical matrix groups.

Section 6 first states that \(\mathrm{Spin}_+(p,q)\) is a subgroup of all previously considered five types of Lie groups, and for underlying vector space dimension \(n=p+q<6\) it coincides with \(G^2_{p,q}\). Furthermore, Section 6 presents a table with isomorphisms of \(G^2_{p,q}\) with classical groups (or subgroups of classical groups, listing for \(n>6\) the corresponding classical super groups) for all values of \(p\) and \(q\), ranging from zero to seven. Section 7 represents group by group (for all five types) isomorphisms with classical groups, taking the various signatures into account. This also leads to isomorphisms between the five types of Lie algebras in Clifford algebra representation and classical matrix Lie algebras.

The paper is quite comprehensive, but for the readers it might have been interesting to e.g. give the following reference as well: [C. Doran et al., J. Math. Phys. 34, No. 8, 3642–3669 (1993; Zbl 0810.15014)].

The introduction gives a thorough overview of related work, including the previous work of the authors. A number of fundamental definitions of the Clifford algebra, the complexified Clifford algebra, and the \(Z_2\)-grading is given. A definition of real Clifford algebras with quaternion type subspaces is given, as well as a corresponding complexified notion.

Section 2 explains a recurrent method for the construction of isomorphic matrix represetations of real Clifford algebras of a general signature \(\mathrm{Cl}(p,q)\), together with a set of representative low-dimensional examples. Section 3 describes the relations between various conjugations (and involutions) in a Clifford algebra and matrix operations, like transposition, Hermitian conjugation, and conjugate transpose of quaternion matrices, including the case of complexified Clifford algebras. Section 4 introduces the helpful notion of additional signature of real (and complexified) Clifford algebras, which means the number of symmetric isomorphic matrix basis vectors (generators), and the number of skew-symmetric isomorphic matrix basis vectors (generators). The application of the notion of additional signature for relating matrix transposition and Clifford algebra involutions is demonstrated.

Section 5 includes a number of theorems for five types of subsets of Clifford algebras, isomorphic to Lie groups. For example, the subset \(G^2_{p,q} = \{U \in \mathrm{Cl}^{(0)}(p,q): \mathrm{reverse}(U)U = 1\}\), where \(\mathrm{Cl}^{(0)}(p,q)\) denotes the even subalgebra of \(\mathrm{Cl}(p,q)\). The isomorphic Lie groups are stated, their Lie algebras, and their dimensions. Theorem 1 includes a collection of isomorphisms between these five Lie groups. Theorems 2 and 3 provide, dependent on the Clifford algebra signature, isomorphisms of the five Clifford algebra represented Lie groups with unitary, symplectic, orthogonal, and general linear classical matrix groups.

Section 6 first states that \(\mathrm{Spin}_+(p,q)\) is a subgroup of all previously considered five types of Lie groups, and for underlying vector space dimension \(n=p+q<6\) it coincides with \(G^2_{p,q}\). Furthermore, Section 6 presents a table with isomorphisms of \(G^2_{p,q}\) with classical groups (or subgroups of classical groups, listing for \(n>6\) the corresponding classical super groups) for all values of \(p\) and \(q\), ranging from zero to seven. Section 7 represents group by group (for all five types) isomorphisms with classical groups, taking the various signatures into account. This also leads to isomorphisms between the five types of Lie algebras in Clifford algebra representation and classical matrix Lie algebras.

The paper is quite comprehensive, but for the readers it might have been interesting to e.g. give the following reference as well: [C. Doran et al., J. Math. Phys. 34, No. 8, 3642–3669 (1993; Zbl 0810.15014)].

Reviewer: Eckhard Hitzer (Tokyo)

### MSC:

15A66 | Clifford algebras, spinors |

22E60 | Lie algebras of Lie groups |

11E57 | Classical groups |

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |

51F25 | Orthogonal and unitary groups in metric geometry |

15B57 | Hermitian, skew-Hermitian, and related matrices |