×

On Casimir’s ghost. (English) Zbl 0888.17002

Let \(L\) be the orthosymplectic Lie superalgebra \(\text{osp}(1|2n)\), and \(U(L)\) its universal enveloping algebra. Both algebras are \({\mathbb{Z}}_2\) graded, i.e. having even and odd elements. Apart from the usual adjoint action, it is shown that a nonstandard adjoint action can be defined by \(g\cdot x \equiv g x - (-1)^{\eta\xi+\eta} x g\), for \(g\in L\), \(x\in U(L)\), with \(\deg(g)=\eta\) and \(\deg(x)=\xi\). This action turns \(U(L)\) into an \(L\)-module, which is completely decomposable, and the decomposition contains a one-dimensional module. The basis element of this one-dimensional module is called the Scasimir. It is an even element of \(U(L)\), which commutes with all even elements of \(L\) and anti-commutes with all odd elements of \(L\). The square of the Scasimir is a Casimir operator (commuting with all elements of \(L\)). A procedure for calculating explicit expressions for the Scasimir is given.

MSC:

17A70 Superalgebras
17B35 Universal enveloping (super)algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv