Timeline for Goldman symplectic form vs Weil–Petersson symplectic form

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Feb 27 at 16:15	comment	added	AMath91		Ok, you're right. But still, if with Goldman's conventions, Goldman symplectic form is (-8) times the Weil-Petersson symplectic form (as he claims in the paper), then when you take the standard killing form on the Lie algebra, you divide Goldman symplectic form by $4$, which means that the multiplicative factor should become $-2$.
Feb 27 at 15:54	comment	added	Carlo Beenakker		Goldman defines the Killing form on page 214; that definition is a factor of 4 larger than the conventional definition (see for example math.stackexchange.com/a/2833737/87355 ), as pointed out in the first reference I cited.
Feb 27 at 15:28	comment	added	AMath91		I know the reference you are referring to. The problem is that I don't really agree with what they say. They say that Goldman uses $\mathrm{trace}(XY)$ as coupling in the Lie algebra but at page 211 of "The symplectic nature of fundamental groups of surfaces", Adv. in Math. 54 (1984), no. 2, 200-225, Goldman says that he is considering the Killing form on the Lie algebra. Moreover, the second paragraph of what the authors say is very unsatisfactory.
Feb 27 at 15:14	history	edited	Carlo Beenakker	CC BY-SA 4.0	deleted 25 characters in body
Feb 27 at 14:34	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 2 characters in body
Feb 27 at 14:27	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 202 characters in body
Feb 27 at 14:07	history	answered	Carlo Beenakker	CC BY-SA 4.0