Differentiable functions in associative and alternative algebras and smooth surfaces in projective spaces over these algebras}

Boris Rosenfeld

Abstract: The following assertions are proved: (1) in simple noncommutative associative and alternative algebras only linear functions $y=ax+b$ and $y=xa+b$ have left and right derivatives, and (2) in the space over all commutative associative algebras smooth $m$-surfaces (lines for $m=1$) have tangent $m$-planes depending on the same number of parameters as points in surfaces. In the spaces over simple noncommutative associative and alternative algebras only $m$-planes (straight lines for $m=1$) are smooth $m$-surfaces. In the spaces over nonsemisimple noncommutative algebras smooth $m$-surfaces have tangent $m$-planes depending on the number of parameters less than points in surfaces.

Classification (MSC2000): 17A99; 53A40

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