Three Examples of Graded Lie Groups
Jan Vysoky
TL;DR
This work constructs explicit graded Lie groups from a graded vector space $V$ equipped with a degree-ℓ metric or symplectic form: the general linear group $\mathrm{GL}(V)$, the graded orthogonal group $\mathrm{O}(V,g)$, and the graded symplectic group $\mathrm{Sp}(V,\omega)$. It develops both a direct geometric construction and a functor-of-points viewpoint (via the diamond functor ${V}_{\diamond}$ and the module-theoretic perspective), and identifies their Lie algebras with canonical subalgebras of $\mathfrak{gl}(V)$ under graded commutators. The paper also analyzes isomorphisms, degree shifts, and the effect of isomorphisms on graded structures, and provides a detailed set of examples and applications including standard forms of metrics, fiber-bundle contexts, and representations. These graded groups serve as natural analogues of classical groups in the $\ Z$-graded setting and offer groundwork for graded geometry and potential generalizations like Berezinian-based notions. Overall, the results give concrete, coordinate-free models of graded Lie groups and clarify how classical structures extend to the graded context.
Abstract
Lie theory is, beyond any doubt, an absolutely essential part of differential geometry. It is therefore necessary to seek its generalization to $\mathbb{Z}$-graded geometry. In particular, it is vital to construct non-trivial and explicit examples of graded Lie groups and their corresponding graded Lie algebras. Three fundamental families of graded Lie groups are developed in this paper: the general linear group associated with any graded vector space, the graded orthogonal group associated with a graded vector space equipped with a metric, and the graded symplectic group associated with a graded vector space equipped with a symplectic form. We provide both a direct geometric construction and a functor-of-points perspective. It is shown that their corresponding Lie algebras are isomorphic to the anticipated subalgebras of the graded Lie algebra of linear endomorphisms. Isomorphisms of graded Lie groups induced by linear isomorphisms, as well as possible applications, are also discussed.