Timeline for Question about the notation of Lie algebra valued differential forms

6 events

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Dec 28, 2021 at 18:03	comment	added	Daniel Waters		Okay thanks for clarifying
Dec 28, 2021 at 18:00	comment	added	Mozibur Ullah		@Daniel Waters: Yes, but I wouldn't say that this is only 'a visual difference'. It in fact relies on the isomorphism of $gl(\mathbb{R}^m)$ with $m \times m$ matrices. You can consider either $(\omega^i_j)$ which is the whole collection and is matrix valued or individually, $\omega^i_j$, which are each individually real valued. This is no different in considering a matrix as a whole or individually as a matrix of entries.
Dec 28, 2021 at 17:41	comment	added	Daniel Waters		Thank you for your detailed answer. So essentially, the indices are only there to indicate that the image of the form is a matrix? I.e. it is only a visual difference. The notes I linked above say that the forms can be considered as $(\dim M)^2$ $\mathbb R$-valued functions $\omega ^i_j$. What is the difference between the two approaches?
Dec 28, 2021 at 17:39	vote	accept	Daniel Waters
Dec 28, 2021 at 9:28	history	edited	Mozibur Ullah	CC BY-SA 4.0	added 893 characters in body
Dec 28, 2021 at 9:16	history	answered	Mozibur Ullah	CC BY-SA 4.0