We start from the motivating toy example. For a finite dimensional vector space V, consider the graded algebras A=Sym(V) and A^!= Lambda(V*) - the symmetric and exterior algebra respectively. Their Poincare series satisfy the identity f(t) f^!(-t)=1. This is a special case of a general statement, a numeric incarnation for a pair of Koszul dual algebras.
Next, given a polytope P, we consider its incidence matrix M_P(t). Specialists in combinatorics know that it satisfies the matrix equation M_P(t) M_P(-t)= Id. Polo in mid-nineties explained this by proving that the incidence algebra for a polytope is Koszul self dual.
Recently Daria Poliakova generalized this. Given a polytope, she introduced an automorphism I_P(t) of a non-commutative power series algebra whose linear part is precisely M_P. She proved in her thesis that in some cases the automorphism is almost an involution. This result is a numeric incarnation of a surprising Koszul self-duality for a certain colored operad associated with the polytope P.
We show that automorphisms of a similar nature appear naturally in representation theory. In particular, given a finite dimensional associative algebra with a non-degenerate scalar product, we obtain a non-trivial example similar to the construction of I_P(t).