I have the following questions: Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra $U(\mathfrak{gl}(m,n))$.

Recall that one always consider the usual notion: "The graded representations" are those modules which have $\mathbb{Z}_2-$gradation compatible with the action of $\mathfrak{gl}(m,n)$. A representation is said to be non-graded if it is not a graded representation.

$\bf My$ $\bf Questions:$ Is there any finite-dimensional, non-graded representations over $\mathfrak{gl}(m,n)$? How can we construct this representations in general?

Thanks very much!

I have the following questions: Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra $U(\mathfrak{gl}(m,n))$.

Recall that one always considers the usual notion: "The graded representations" are those modules which have $\mathbb{Z}_2-$gradation compatible with the action of $\mathfrak{gl}(m,n)$. A representation is said to be non-graded if it is not a graded representation.

$\bf My$ $\bf Questions:$ Is there a finite-dimensional, non-graded representation over $\mathfrak{gl}(m,n)$? How can we construct such representation in general?

Thanks very much!