Tracelike from an $R$ -matrix

Let $H$ be an autonomous Hopf $𝕂$ -algebra, quasitriangular with $R$ -matrix and inverse

𝑅 = \sum 𝑖 𝛼 𝑖 \otimes 𝛽 𝑖, 𝑅 - 1 = \sum 𝑖 𝛾 𝑖 \otimes 𝛿 𝑖 .

Then the element

𝑢 = \sum 𝑖 𝜎 (𝛽 𝑖) 𝛼 𝑖

has inverse

𝑢 - 1 = \sum 𝑖 𝑆 - 1 (𝛿 𝑖) 𝛾 𝑖,

moreover $𝑢$ makes $H$ a Tracelike, #m/thm/hopf i.e.

𝑢 𝑥 = 𝜎 2 (𝑥) 𝑢

for all $𝑥 \in H$ .¹

Proof

#missing/proof

#state/develop | #lang/en | #SemBr

Tracelike from an R-matrix