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Let $\mathbb{H}$ denote the quaternions. Let $G$ be a group, and define a representation of $G$ on $\mathbb{H}^n$ in the natural way; that is, its a map $\rho:G\rightarrow Hom_{\mathbb{H}-}(\mathbb{H}^n,\mathbb{H}^n)$ such that $\rho(gg')=\rho(g)\rho(g')$ (where $Hom_{\mathbb{H}-}$ denotes maps as left $\mathbb{H}$-modules). Representations of algebras and Lie algebras can be defined in a similar way.

Any quaternionic representation of $G$/$R$/$\mathfrak{g}$ can restrict to a complex representation by choosing a $\gamma\in \mathbb{P}^1$ such that $\gamma^2=-1$, and using $\gamma$ to define a map $\mathbb{C}\hookrightarrow \mathbb{H}$. In this way, any quaternionic representation gives a $\mathbb{C}\mathbb{P}^1$-family of complex representations which parametrize the choice of $\gamma$. Furthermore, since every element in $\mathbb{H}$ is in the image of some inclusion $\mathbb{C}\hookrightarrow \mathbb{H}$, this family of complex representations determines the quaternionic representation.

This observation almost seems to imply that there is nothing interesting to say about quaternionic representations that wouldn't come up while studying complex representations. However, this is neglecting the fact that there might be interesting information in how the $\mathbb{C}\mathbb{P}^1$-family of complex representations is put together.

For instance, any finite group will have a discrete set of isomorphism classes of complex representations, and so any quaternionic representation will have all complex restrictions isomorphic. However, the quaternion group $\mathbf{Q}:= ( \pm1,\pm i, \pm j,\pm k)$ has an 'interesting' quaternionic representation on $\mathbb{H}$ (more naturally, it is a representation of the opposite group $\mathbf{Q}^{op}$ by right multiplication, but $\mathbf{Q}^{op}\simeq \mathbf{Q}$).

My question broadly is: What other groups, rings and Lie algebras have quaternionic representations that are interesting (in some non-specific sense)?

This question came up when I was reading a paper of Kronheimer's, where he describes a non-canonical hyper-Kahler structure on a coadjoint orbit of a complex semisimple Lie algebra. At any point in such a coadjoint orbit determines a representation of $\mathfrak{g}$ on the tangent space to the coadjoint orbit, which by the hyper-Kahler structure is naturally a quaternionic vector space. I wondered if this representation could be 'interesting', and then realized I had no real sense of what an interesting quaternionic representation would be.

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I think it would be better to have your quaternions always act on the right. It's awkward to have both the noncommutative group G and the noncommutative ring H both acting on the same side. –  Noah Snyder Mar 8 '10 at 18:21
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up vote 10 down vote accepted

Recall that if G is a group, k a field, and V_k an irreducible representation of G over k then End_G(V_k) is a division algebra D over k. For example, if $V_{\mathbb{R}}$ is a real representation then the endomorphism ring is $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. Such representations are typically called real, complex, or quaternionic.

This usage is complimentary to your definition. Suppose that $V_{\mathbb{R}}$ is a quaternionic representation, then $\mathbb{H}$ acts on $V_{\mathbb{R}}$ on the right. Since $\mathbb{H}$ is a division ring, it follows that $V_{\mathbb{R}} \cong \mathbb{H}^k$. Hence a real representation which is quaternionic is canonically a quaternionic representation in your sense.

Now Frobenius-Schur indicator theory explains how to determine which representations over $\mathbb{C}$ "come from" which kind of real representations. Explicitly, if $V_{\mathbb{C}}$ is not selfdual then it plus its dual is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{C}$; if $V_{\mathbb{C}}$ orthogonal (i.e. has an invariant bilinear form) then it is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{R}$; finally, if $V_{\mathbb{C}}$ is symplectic then it plus itself is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{H}$.

In summary, given any symplectic representation $V_{\mathbb{C}}$ the representation $V_{\mathbb{C}} \oplus V_{\mathbb{C}}$ is the complexification of a representation $W_\mathbb{R}$ which is itself a module over $\mathbb{H}$. In particular, there is an "interesting" representation over $\mathbb{H}$ whose quaternionic dimension is half the complex dimension of $V_{\mathbb{C}}$. Your example fits into this scheme, where you start with the 2-dimensional irrep of $\mathbf{Q}$. Another great example involves the symplectic Lie group which can also be realized as a unitary group over the quaternions (see wikipedia).

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I don't think it's really accurate to say the symplectic Lie group can also be realized as a unitary group over the quaternions. There's a real form of the symplectic group which is the unitary matrices over the quaternions, but "symplectic group" has a meaning and that is not it. –  Ben Webster Mar 9 '10 at 1:31
    
Fair enough. So what's the right way to fix it? If I take the complex symplectic group then it has a quaternionic representation. So what's going on, is Sp(2n,C) something like GL(n, H)? –  Noah Snyder Mar 9 '10 at 4:24
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