6
$\begingroup$

If $R$ is a commutative ring, then the Cayley-Hamilton theorem states that any endomorphism $\phi: R^{n} \rightarrow R^{n}$ of a rank $n$ free module satisfies its own characteristic polynomial, in particular satisfies a monic degree $n$ polynomial.

Now suppose that instead $R = R_{0} \oplus R_{1}$ is a commutative super ring (that is, it is $\mathbb{Z}/2$-graded, with even degree elements in the center, and odd degree elements anticommuting with each other).

Does every endomorphism $\phi: R^{p, q} \rightarrow R^{p, q}$ of a free module on $p$ even and $q$ odd variables satisfy a monic polynomial over $R$ of degree $p+q$? Does it satisfy some monic polynomial?

$\endgroup$

1 Answer 1

6
$\begingroup$

The case when $R$ is purely even but the module has a $Z/2$-grading was studied before, see for example

"On the Cayley-Hamilton equation in the supercase" by Issai Kantor and Ivan Trishin, Comm. in Algebra 27:1 (1999), 233-259

"Berezinians, Exterior Powers and Recurrent Sequences", by H. M. Khudaverdian and TH. TH. Voronov, Letters in Mathematical Physics, 74 (2005), 201–228

Already in this case it seems that you should not expect the polynomial to be monic.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.