Higgs bundles, spectral curves, and low-dimensional Lie group
isomorphisms
-- Steve Bradlow, February 13, 2015

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We will explore some interesting relations among Higgs bundles,
from the point of view of spectral data, that result from
special isomorphisms among low-dimensional Lie algebras and Lie
groups.

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Higgs bundles provide an algebro-geometric description of
surface group representations into complex reductive Lie
groups, and also into their real forms, say G. The defining
data sets for such G-Higgs bundles include a Riemann surface
(&Sigma;), a holomorphic principal bundle
(E&#8594;&Sigma;), and a Higgs field (&Phi;) which is a
holomorphic section of an associated vector
bundle. Alternatively, the defining data can be encoded in a
ramified cover S&#8594;&Sigma; (the spectral curve) and a
line bundle in the Jacobian of the spectral curve.  If two
groups, say G<sub>1</sub> and G<sub>2</sub>, are related by a
group homomorphism, one can expect the corresponding Higgs
bundles and their spectral data sets to inherit induced
relationships. We will explore this phenomenon in the case of
isogenies resulting from accidental isomorphisms among
low-dimensional Lie algebras.