Background
I am trying to learn about relativity. One way to define the Poincaré Group is that its elements commute with the spacetime metric. The spacetime metric is an example of inner product, but could also be viewed as an algebraic form. I'm interested in how this concept generalizes to other algebraic forms.
Question
Suppose that you have some vector space $V$ along with an algebraic form (in tensor notation)
$$\mathbf{\Phi x} = A + B_{\mu}x^\mu + C_{\mu \nu}x^\mu x^\nu + D_{\mu \nu \gamma}x^\mu x^\nu x^\gamma + ...$$
Suppose also that have a family $\Lambda^\mu_\nu \in \bf\Lambda$ of linear automorphisms of $V$ that preserves $\mathbf{\Phi}$, i.e. for which
$$\mathbf{\Phi} \Lambda^\mu_\nu = \Lambda^\mu_\nu \mathbf{\Phi}$$
How can you find a basis for $\bf\Lambda$ in terms of the of $A$, $B_\mu$, $C_{\mu \nu}$, etc?
Thoughts
We can view $\bf \Lambda$ as a subspace of $V^*$, the dual space of $V$. My intuition tells me there should be some "$\mathbf{\Phi}^*$" defined on $V^*$ and that the algebraic expression of this $\mathbf{\Phi}^*$ would be homomorphic to the algebraic form of $\mathbf{\Phi}$ in $V$. I.e., something like "if $C$ is anti-symmetric, then $C^*$ is symmetric".
Update
I think the formulation of question has some conceptual errors. The key property of the Poincaré Group is that it preserves a certain relationship between two points in Minkowski space. So there are actually two generalizations:
- Increase the number of points under consideration. Then $\mathbf{x}$ would be more aptly interpreted as a collection of vectors
- Generalize the relationship to some function of the coordinates of both points rather than a quadratic equation like the Minkowski metric
I think I somewhat mashed the two up in the original question.