Description
For a possibly noncommutative finite-dimensional algebra $A$ with an anti-involution $\sigma$ many classical Lie groups can be realized as certain symplectic or orthogonal Lie groups over the pair $(A, \sigma)$. When the involutive algebra $(A, \sigma)$ is Hermitian, one can define and study the associated Riemannian symmetric space of such Lie groups. We will describe different geometric interpretations of such symmetric spaces that generalize the various models of the hyperbolic plane viewed as the symmetric space associated to the group $\mathrm{SL}_2(\mathbb{R}) = \mathrm{Sp}_2{\mathbb{R}}$. Moreover, we will explore implications of this theory in the realm of non-abelian Hodge theory using these new geometric models of the symmetric space. This is joint work with Pengfei Huang, Eugen Rogozinnikov and Anna Wienhard.