Symplectic, Orthogonal and Linear Lie Groups in Clifford Algebra

2016

JGSP

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 which are direct sums of subspaces of quaternion types. Spin group is a subgroup of all considered groups and it coincides with one of them in the cases n ≤ 5. We present classical matrix Lie groups that contain spin group in the case of arbitrary dimension. 1 Note that e a is not exponent. We use notation with upper indices [4]. 2 We use Einstein's summation convention: there is a sum over index a.

Section: Introductionmentioning

confidence: 64%

Section: Recurrent Methods Of Construction Of Matrix Representations Omentioning

confidence: 99%

“…Clifford algebra has faithful and irreducible representations over R, R ⊕ R, C, H, H ⊕ H in different cases. However, items 1-4 are the same as in [20].…”

Section: Recurrent Methods Of Construction Of Matrix Representations Omentioning

confidence: 99%

“…In the Section 5 of the present paper we discuss connection between groups Spin + (p, q) and G 2 p,q . In the Section 6 we summarize results of [20] and the present paper.…”

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

On Some Lie Groups Containing Spin Group in Clifford Algebra

2016

JGSP

“…These Lie algebras are isomorphic to the linear, orthogonal, symplectic, and unitary classical Lie algebras in different cases (see papers [4] and [5]). Now we are interested in Lie algebras with numbers 1-11 in Table 1. 3.…”

Section: Lie Groupmentioning

confidence: 99%

Classification of Lie algebras of specific type in complexified Clifford algebras

Linear and Multilinear Algebra

2017

Self Cite

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These sixteen 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 sixteen Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.

On generalization of Lipschitz groups and spin groups

Math Methods in App 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 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 spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras.