Does $\sum_i e_i \otimes e_i$ in $V\otimes V$ have a name?

Question

If $V$ is a vector space with orthogonal basis $(e_i)$, consider the element $\sum_i e_i \otimes e_i \in V\otimes V$. It can be shown that this element is independent of the choice of orthogonal basis.

Does this element have a name?

Answer 1

You could call it the Casimir element. If $V$ is finite-dimensional, then under the identification $V \otimes V \cong V \otimes V^{\ast} \cong \text{End}(V)$ it corresponds to the identity

$$V \otimes V^{\ast} \ni \sum e_i \otimes e_i^{\ast} \mapsto \text{id}_V \in \text{End}(V)$$

and can also be called the unit.