For this discussion, $G$ is a compact semisimple Lie Group.

For many of the common representations of compact groups, there is a realization of the representation as the automorphisms of some algebraic structure. For example, the adjoint representation of any group $G$ is realized as the automorphisms of $LG$, the standard representation of $SO(n)$ is as the stabilizer of the volume form on $\mathbb{R}^n$, the 7-dimensional representation of $G_2$ is realized as the automorphisms of the imaginary octonions or as the stabilizer of a certain 3-form on $\mathbb{R}^7$, and so forth.

My particular interest for this question is in the automorphisms of algebras over $\mathbb{R}$ (although for Question 1, I don't care about the ground field).

Question 1:Given a representation of $G$ (not necessarily irreducible) is there a general method of constructing an algebra (over the appropriate ground field) such that $G$ is the group of automorphisms of this algebra acting by the prescribed representation?

My gut feeling is that a solution to this question in general would be related to the construction of Cartan Products, but even given the irreducible summands of the representation I don't see how to make an algebra which is not a direct sum of algebras (the problem with a direct sum of algebras is that it can have larger automorphism group than $G$).

Since this is such a general question, I understand it may have no easy answer. However, I do have a specific example in mind; some of my recent experimentation has suggested that if a representation has certain properties, any algebra on which this representation acts as the full set of automorphisms should also have certain nice properties. One specific case I would like to know about is a 16-dimensional representation of $G_2$ which contains two trivial summands and one copy of the 14-dimensional adjoint representation.

Question 2:Does anyone have an example of a 16-dimensional $\mathbb{R}$-algebra whose automorphism group is $G_2$ with the decomposition of the automorphic action given by $1\oplus1\oplus 14$. For this it would be enough to describe the bilinear multiplication.

Note that I do not care if the algebra is a unital ring, is commutative, associative, or any other particular property; any 16-dimensional algebra structure whose automorphism group is provably $G_2$ with the given representation will suffice for me to test my ideas.

anysuch maps... $\endgroup$ – Mariano Suárez-Álvarez Feb 12 '11 at 1:53