Canonical matrices of isometric operators on indefinite inner product spaces

Abstract

We give canonical matrices of a pair (A, B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax, Ay) = B(x, y) on a vector space over F in the following cases:• F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric;• F is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F.These classification problems are wild if B may be degenerate. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].

Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form

“…Proof. By Lemma 2.1, there exists a decomposition V = V 1 ⊕ ⋅ ⋅ ⋅ ⊕ V t that ensures (20). It remains to prove that the summands in (20) are uniquely determined, up to permutations and isomorphisms.…”

“…(c 2 ) For every a ∈ P determined up to replacement by ζa and every nonzero b ∈ P determined up to replacement by −b, Note that the problems of classifying isometric operators and selfadjoint operators on a vector space with a possibly degenerate symmetric or Hermitian form are wild; see [20,Theorem 6.1]. Recall that a classification problem is wild if it contains the problem of classifying pairs of linear operators, and hence (see [2]) the problem of classifying arbitrary systems of linear operators.…”

Existence of an invariant form under a linear map

“…Suppose that the linear map T admits an invariant non-degenerate hermitian, resp skew-hermitian, form H. Then the necessary condition follows from existing literatures, for example see [Wal63,Ser87,Ser08].…”

Problems of classifying associative or Lie algebras over a field of characteristic not two and finite metabelian groups are wild

“…One of them is about matrix triples up to congruence; its proof is based on the method of reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings, which was developed in [11] and was presented in detail in [12,Section 3]. In Section 6, we recall this reduction, restricting ourselves to the problem of classifying triples of bilinear forms.…”

“…Replace each triple in ind(Q) that is equivalent to a selfadjoint triple by one that is actually selfadjoint, and denote the set of these selfadjoint triples by ind 0 (Q) (7) in [11], whose representations are pairs of symmetric or skew-symmetric bilinear forms). The "quiver with involution" of G (see [12,Section 3] …”