Universal deformation formulas and breaking symmetry
Let \(A\) be an algebra over an algebraically closed field of characteristic zero (not necessarily associative or Lie). Let \(\varphi\) and \(\psi\) be two commuting derivations of \(A\). The authors define a new multiplication on the algebra \(At\) of formal power series on \(A\) by using the following formula: \[ a*_t b= ab+ t\varphi (a) \psi(b)+ {\textstyle {t^2 \over {2!}}} \varphi^2 (a) \psi^2 (b)+ \cdots\;. \] If \(A\) is associative this gives a one-parameter family of associative multiplications. The authors then observe that if \(A\) has a Lie group of automorphisms of dimension at least two, then \(A\) may be deformed by using the above formula in such a way that the full group no longer acts on \(A\). They call this phenomenon ``breaking symmetry''.
scientific article; zbMATH DE number 500583
scientific article