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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.

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  • $\begingroup$ Also, as this is my first time using the gray boxes to highlight my questions, if anyone can tell me how to correct all the math formatting in them that would be much appreciated so I know how to do it in the future. $\endgroup$
    – ARupinski
    Commented Feb 11, 2011 at 18:21
  • $\begingroup$ I fixed the grey boxes. To make them (and the math formatting) display properly, begin each paragraph with the > character. $\endgroup$
    – Tom Church
    Commented Feb 11, 2011 at 23:30
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    $\begingroup$ For question 2, you need to pick the 14-dimensional $G_2$-module, add two trivial ones, call it $V$, and see if there is a trivial summand in $\hom(V\otimes V,V)$. That gives you all possibly non-associative, possibly non-unitary multiplications which are $G_2$-equivariant; this computation you can do with LiE, for example. I wonder if there are any such maps... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 12, 2011 at 1:53
  • $\begingroup$ «you need to» should really be «a good start would be to»... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 12, 2011 at 1:54
  • $\begingroup$ @Mariano: If $V$ is the 16-dimensional representation in question then I calculate that $hom(V\otimes V,V)$ is 15 dimensional, so it seems like there should actually be lots of such algebraic structures. In light of this, what exactly do you mean by "see if there is a trivial summand in $hom(V\otimes V,V)$"? If this method works, it suggests a similar method for answering Question 1 as well. $\endgroup$
    – ARupinski
    Commented Feb 12, 2011 at 3:39

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