For a commutative ring $R$, I can consider $\operatorname{GL}_n(R)$ as a group scheme over $\operatorname{Spec} R$. Are there analogs of this notion when $R$ is non-commutative, say $R = \operatorname{End}_k V$ (for $V$ a finite-dimensional $k$-vector space), which retain any useful part of the theory of group schemes?
Analogs of group schemes over non-commutative rings
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algebraic-groups
noncommutative-geometry
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0Well, it's easy to define a group-valued functor on the category of not-necessarily-commutative rings. But group schemes in the usual sense are very special group-valued functors, and it is not so easy to generalise that part of the definition to the non-commutative setting. – 2012-11-03
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0I know it's tough; that's why I asked. I was thinking in terms of modern notions of spec of a noncommutative ring (which I don't really understand). – 2012-11-03
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0One possibility is to consider possibly non-commutative Hopf algebras. Representable group functors on the category of commutative rings (i.e. affine group schemes) are represented by commutative Hopf algebras. When replacing the category of commutative algebras by the the category of algebras (say over a fixed commutative ring), then an analogue of an affine group scheme might be a possibly non-commutative Hopf algebra. But the analogy is not perfect as the tensor product of rings is not the coproduct in general. – 2017-01-03