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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.
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  • $\begingroup$ An intuition that I find helpful: Consider an embedded circular string moving smoothly in some manifold. If you regard the $l_i$ as states that the circular string might be in, then the elements of $A$ describe superpositions of those states. If the circular string pinches into a figure of eight and splits into two circular strings, then multiplying by $C$ describes the resulting superpositions of pairs of states on the two new circular strings. That is, if the initial string is in superposition of states $a$, then after splitting, the pair will be in superpositions of state $aC=Ca$. $\endgroup$ – tkf Aug 6 at 16:34
  • $\begingroup$ @MatthewTowers Sorry for the imprecision. I changed it to "How is a Casimir element defined?" $\endgroup$ – M.C. Aug 6 at 16:44
  • $\begingroup$ I am sure your notes mention that $C$ is independent of the choice of basis elements. One way to define it independently of a basis is to use $\kappa$ to identify $A\otimes A$ with ${\rm End}_{\mathbb{K}}(A)$. That is regard $a\otimes b$ as the linear map sending $c\mapsto a\kappa(b,c)$. Then $C$ is just $1_A$. $\endgroup$ – tkf Aug 6 at 16:45
  • $\begingroup$ @tkf The intuition might prove to be helpful, thanks. But at the moment I am still looking for a definition of a Casimir element. Something like "A Casimir element $C$ in a Frobenius algebra $A$ is ..." Is it just an element $C$ such that $aC=Ca$ for all $a \in A$? Or is it the tensor product of a pair of dual bases in the above manner? $\endgroup$ – M.C. Aug 7 at 17:07
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    $\begingroup$ I don't know about the terminology, but certainly the element $C$ you described is fundamental to $1+1$ dimensional topological quantum field theory for the reason I described, so it deserves its own name. Also it certainly results from tensoring any pair of dual basis', so is canonical to the algebra, and has the property $aC=Ca$. I have also heard $C$ referred to as "the canonical element". Hopefully someone will be able to help more. $\endgroup$ – tkf Aug 7 at 19:41

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