Classification of Lie algebras of specific type in complexified Clifford algebras

Abstract

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 the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics

“…Choosing a Clifford algebra structure is equivalent to choosing a particular subgroup of the algebra to call 'the' spin group, so that one can also approach the problem from the point of view of studying subgroups [32][33][34] of the Clifford algebra. It is intimately connected with the question of choosing a particular real form of the Clifford algebra and the associated spin group.…”

Clifford Algebras and Their Applications to Lie Groups and Spinors

“…The notion of quaternion type was introduced by the author in the brief report [48] and the paper [56]. Further development of this concept is given in [52], [57], [61], [66], see also books [35], [55]. Subspaces of quaternion types are useful in different calculations (see [52], [57], [61]).…”

“…Using the classification of Clifford algebra elements based on the notion of quaternion type, we present a number of Lie algebras in C ⊗ Cℓ p,q (see Section 5.4 and [66]).…”

“…See also [7]. We study the connection between matrix operations (transpose, matrix complex conjugation) and operations in Clifford algebra (reverse, complex conjugation, grade involution), we introduce the notion of additional signature of the Clifford algebra (for more details, see [59], also [60], [63], [66], where we develop and use these results). Let us consider chirality operator (pseudoscalar) in C ⊗ Cℓ p,q : ω = e 1...n , p − q = 0, 1 mod 4 ie 1...n , p − q = 2, 3 mod 4.…”

Covariantly Constant Solutions of the Yang–Mills Equations

“…Taking into account U(2 n 2 ) ⊂ GL(2 n 2 , C) and using operation of Hermitian conjugation in Clifford algebra [10], we can reformulate theorems of the current paper with the use of unitary Lie groups (and unitary Lie algebras) of corresponding dimensions. We can also use another classical Lie groups and corresponding Lie algebras in the complexified Clifford algebra C ⊗ Cℓ p,q (see papers [17], [18], [19]).…”