1. Context.
Let $A$ be a $\mathbb K$-Frobenius algebra with Frobenius form $\kappa$.
Let $(l_i)_{i \in 1,..., N}$ be a basis of $A$.
Choose $(r_i)_{i \in 1,..., N}$ such that $\kappa (l_i, r_j)=\delta_{ij}$. Because $\kappa$ is non-degenerate, the familiy $(r_i)_{i \in 1,..., N}$ is $\mathbb K$-linearly independent, hence a basis. We call this pair of bases a pair of dual bases.
Define $C := \sum\limits_{i=1}^N l_i \otimes r_i \in A \otimes A$. My lecture notes state:
$C$ is a Casimir element, i.e. $aC=Ca$ for all $a \in A$.
The notes go on to mention that the notion of Casimir element is related to the Casimir effect in physics. However, they do not specify how.
2. Questions
- How is a Casimir element defined in the above context?
I read the wikipedia and the Encyclopedia of Mathematics article on the Casimir element. I can see similarities between the above hints at a definition of $C$ and the construction presented under the subheading "Quadratic Casimir element" on wikipedia. However, on wikipedia a Casimir element is defined as "a certain distinguished element of the center of the universal enveloping algebra of a Lie algebra." In the above context, however, $A \otimes A$ is not a universal enveloping algebra in general. - How is the above Casimir element related to the Casimir effect? I could not find anything online. Even a reference would be appreciated.