Classification of Five-Dimensional Lie Algebras of Class T2

Cyclic metric Lie groups

“…, 4, with α 1 + α 2 + α 3 + α 4 = 0 because g is not unimodular. In accordance with Example 5.6, we have the metric Lie groups in (2) in the statement with the condition 4 i=1 α i = 0. Third case.…”

“…The classification of unimodular Lie algebras of dimension five corresponding to traceless cyclic metric Lie groups was given by Bieszk [2]. We add the corresponding nonunimodular Lie groups in the next theorem.…”

“…Finally, from the classification of unimodular Lie algebras of dimension five with a cyclic left-invariant metric, given by Bieszk [2], we also have the metric direct product Lie group SL(2, R) × R 2 in (1) in the statement, and the metric Lie group G 5 (α 1 , α 2 , α 3 , α 4 ) in (2) Remark 6.5 Many decomposable nonunimodular metric Lie groups appear as particular cases of the metric Lie groups H 5 (ρ, σ, τ ; λ, μ). For example, if ρ + σ = 0, τ = 0, λ = 0, and μ = −λ, we have the direct product H 4 (ρ, σ ; λ) × R. Moreover, one can prove that H 2 (c) × G 3 (α, β) is isometrically isomorphic to H 5 (ρ, σ, τ ; λ, μ), where…”

“…and a = c 2 + (α + β) 2 . In particular, if α = β this metric Lie group corresponds to H 2 (c) × H 3 (α).…”

“…The classification of connected, simply-connected nonabelian Lie groups G admitting a (nontrivial) traceless cyclic homogeneous structure, that is, for unimodular G, was given for dimensions three and four by Kowalski and Tricerri [8, Theorems 2.1, 3.1] and for (Lie algebras of) dimension five by Bieszk [2]. We extend their results, adding the corresponding nonunimodular Lie groups, in Theorems 6.1, 6.2 and 6.4.…”