Lie coalgebra

Remarks

Example: let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $E = \mathfrak{g}^*$ its dual space. Identify $E \otimes E$ with the bilinear forms on $\mathfrak{g}$, and set $\delta(\xi)(x, y) = \xi([x, y])$. Antisymmetry of $\delta$ follows from $[y, x] = -[x, y]$, and the co-Jacobi identity is the Jacobi identity of $\mathfrak{g}$ read through $\xi$. Conversely, for a finite-dimensional Lie coalgebra $E$, the transpose of $\delta$ is a Lie bracket on $E^*$, so the two notions are dual to each other.

Non-example: $E = k$ with $\delta(1) = 1 \otimes 1$ fails clause 1, since $\tau(1 \otimes 1) = 1 \otimes 1 \neq -1 \otimes 1$ when the characteristic of $k$ is not $2$.

Since $\delta$ is antisymmetric, one can view it as a map $E \to \Lambda^2 E$ in characteristic not $2$.