I am working on non commutative homogeneus Lie groups and I was wandering if there is a generalisation of the concept of multilinear map (tensors, as we know are defined on the tangent). I define multiomomorphism as a map:
$\phi:G^k=G\times G\times\ldots\times G\longrightarrow(\mathbb{R},+)$
for which $\phi$ is a homomorphism in each component and it is $k$ homogeneus wrt dilations of the group. Are these objects studied? How are they called in literature? I can find almost nothing on the web.
In particular I would like to understand if it is possible to characterise $k$ homogeneus maps $\psi:G\to\mathbb{R}$ which are possible to write as:
$\psi(a)=\phi(a,a,\ldots,a)$.
I particuarly appreciate indications for references.