Exotic and inverted Kloosterman sums over semisimple algebras

Abstract: We introduce exotic Kloosterman sums and exotic inverted Kloosterman sums attached to non-commutative finite-dimensional semisimple algebras over a finite field $\mathbf{F}_q$, and prove their reduction formulae to exotic Kloosterman and exotic inverted Kloosterman sums over commutative étale $\mathbf{F}_q$ algebras. We then obtain square-root estimates for these sums; for inverted sums an explicit correction term may appear.