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Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.
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Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra
I don't have a full answer, but the following calculation may be useful.
Consider the example $G = SL_3$ so that $N$ is the Heisenberg group on three generators and in characteristic zero, $\bar{U}(N …