arXiv subjects
Publications and source records attributed to Hugo Mathevet.
We introduce a $q$-deformation of the Pythagoras equation $a^2 + b^2 = c^2$, which is a polynomial version of it different from the standard one. We construct a polynomial analogue, or ``$q$-analogue'', of every primitive Pythagorean triple. We also construct such analogue for a larger class of Pythagorean triples called standard. Our approach is based on the notion of $q$-deformed rational numbers and the modular group $\mathrm{PSL}(2,\mathbb{Z})$.
Given a natural number m and a Lie algebra g, the m th generalized Takiff Lie algebra of g is the Lie algebra gm\,:= g $\otimes$ C[T ]/T m+1 . For n $\ge$ m, we define the (m, n)-modality of an adjoint orbit $\Omega$m in gm to be the minimum codimension of an adjoint orbit in the pullback of $\Omega$m in gn. In this paper, we study this family of invariants in generalized Takiff Lie algebras associated to a quadratic Lie algebra g. We show that this family of invariants satisfies some concavity and hereditary properties. From which we deduce that (n -m)$\chi$(g) is a lower bound, where $\chi$(g) is the index of g. We prove that this lower bound is in fact an equality for a dense set of orbits, and that if g is reductive, it is always an equality when m = 0 (and also some special orbits). We conjecture that equality holds for all m when g is reductive.