skew-symmetric linear map

Question

We all know the definition in the following

Definition. Let $\frak{g}$ be a Lie algebra. A Lie bialgebra structure on $\frak{g}$ is a skew-symmetric linear map $\delta_{\frak{g}}: \frak{g}\rightarrow \frak{g}\otimes \frak{g},$ called the cocommutator, such that

1. $\delta_{\frak{g}}^{*}: \frak{g}^{*}\otimes \frak{g}^{*}\rightarrow \frak{g}^{*}$ is a Lie bracket on $\frak{g}^{*}$,
2. $\delta_{\frak{g}}$ is a $1$-cocycle of $\frak{g}$ with values in $\frak{g}\otimes \frak{g}.$

My question is that ``what is a skew-symmetric linear map'' in the definition?

Answer 1

The definition you posted is also OK. First, we need a flip map $$\tau:\mathfrak{g} \otimes \mathfrak{g}\rightarrow \mathfrak{g} \otimes \mathfrak{g}, x \otimes y \mapsto y\otimes x, \forall x,y\in \mathfrak{g}.$$ Then the skew symmetry of $\delta_{\mathfrak{g}}$ is given by $$\tau \delta_{\mathfrak{g}}(x)=-\delta_{\mathfrak{g}}(x), \forall x\in \mathfrak{g}.$$