Abstract:

I will start with  a pedagogical survey of symmetries starting with the work of Galois and Abel on the solvability of algebraic equations. The resulting axiomatic structure of groups can be recast in terms of certain algebras and their dual objects (co-algebras); groups correspond to the special case where the algebra is non-commutative by the co-algebra is co-commutative. Allowing both to be non-commuting allows for a more general structure, the Hopf algebra. This axiomatic formalization was found  by Drinfeld based on physical examples of Yang, Baxter and Faddeev at. al.

I will illustrate this idea  with the simplest case, the Taft algebra.

An application to physics is in discretizing Ordinary Differential Equations. (One Dimensional lattice). I will then outline my recent work( joint with G. Landi) on extending this to two and three dimensions.

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