#Nicholls edu math how to
As an example of the geometrical aspects of our construction, we show how to construct differential forms and involutions on iterated products starting from the corresponding structures on the factors, and give some examples of algebras that can be described within our theory. We find conditions for constructing an iterated product of three factors, and prove that they are enough for building an iterated product of any number of factors. We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. Finally, we apply these results to classification of twists in group algebras and of cosemisimple triangular finite-dimensional Hopf algebras in positive characteristic, generalizing the previously known classification in characteristic zero. This bijection generalizes to the non-commutative case the procedure of inverting a non-degenerate bilinear form on a vector space.
Next, we define the notion of a non-degenerate twist for a Hopf algebra H, and set up a bijection between such twists for H and H ∗ if both are semisimple. We also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular Hopf algebra by two different twists on two sides, and describe the representation theory of its dual. We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finite-dimensional Hopf algebras are stable under twisting.
We determine how the integral of the dual to a unimodular Hopf algebra H changes under twisting of H. In this paper we study the properties of Drinfeld’s twisting for finite-dimensional Hopf algebras.
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