Keppeler's matrix Lie algebra convention

In his course Groups and Representations, Stefan Keppeler uses a slightly unorthodox convention of multiplying tangential matrices by $- 𝑖$ when forming the Complex matrix Lie algebra¹, so given a Lie group $𝐺 \subseteq G L 𝑛 (ℂ)$

𝔤 = {- 𝑖 𝑐' (0) : 𝑐 \in 𝖬 𝖺 𝗇 𝜔 (ℝ, 𝐺), 𝑐 (0) = 𝟏 𝑛} \subseteq ℂ 𝑛 \times 𝑛

Then the Lie bracket becomes the matrix commutator premultiplied by $- 𝑖$ . When used in these notes, I will reserve the brackets for the commutator and explicitly prepend the $- 𝑖$ .

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2023, Groups and representations, §6.4, p. 82 ↩

Keppeler's matrix Lie algebra convention

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