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Results tagged with yang-baxter-equations
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user 498826
In the classical equation, one looks for $R\in\Lambda^2\mathfrak g$ such that $$[R,R]=0,$$ where the bracket is Schouten's bracker in $\Lambda^\bullet\mathfrak g$, the exterior algebra on a Lie algebra $\mathfrak g$. In the quantum one (in its non-parametric form...), one looks for endomorphisms $R:V\otimes V\to V\otimes V$ of tensor squares of vector spaces $V$ such that $$R_{12} \ R_{13} \ R_{23} = R_{23} \ R_{13} \ R_{12},$$
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Why does the definition of a braided monoidal category not mention the braid equation?
Indeed, the axioms of a braided monoidal category are enough to derive the Yang-Baxter equation. See Braided monoidal categories by Joyal and Street (diagram B7), or 1Lab for a formalised proof.
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