In this talk I will introduce noncommutative loops over Lie algebras as a tool for studying algebraic groups over noncommutative rings.
Given a Lie algebra $g$ sitting inside an associative algebra $A$ and any associative algebra $F$, the $F$-loop algebra is the Lie subalgebra of tensor product $F\otimes A$ generated by $F \otimes g$.
For a large class of Lie algebras $g$, including semisimple ones, an explicit description of all $F$-loop algebras will be presented. This description has a striking resemblance to the commutator expansions of $F$ used by M. Kapranov in his approach to noncommutative geometry.
I will also define and study Lie groups associated with $F$-loop algebras.
This is a joint paper with A. Berenstein (Univ. of Oregon).