Lattices of covariant quadratic forms (Q1769628)

The author studies the \({\mathbb Z}\)-lattice \(\text{ Bil}_\Lambda(L)\) of all bilinear forms on a lattice \(L\) that are covariant with respect to a positive involution on the order \(\Lambda \) of the lattice. He studies the discriminant of \(\text{ Bil}_\Lambda(L)\), the autoequivalence group \(\text{ Aut}^e(\text{ Bil}_\Lambda(L))\), its subgroup fixing the center of \(\text{ End}_\Lambda(L)\), the group of inner automorphisms, and the outer group of equivalences. The notions of e-depth and e-\(*\)-depth for \(\text{ Bil}_\Lambda(L)\) are defined. They measure how far \(\text{ End}_\Lambda(L)\) and \(\text{ End}_\Lambda(L\oplus L^*)\) are away from being hereditary. The \(*\)-depth zero situations are often classifiable. In the depth zero situation structural results on the outer group of autoequivalences are given. Some examples are studied in detail, e. g. if \(\text{ End}_\Lambda(L)\) is a \(\mathbb Z\)-order in the algebra \({\mathbb Q}^{2\times 2}\).