We extend Witt’s theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces φ : E → E′ to an isometry φV : V → V ′ that also sends a given subspace to another, or a given self-dual flag to another, or a Witt’s decomposition to another and a special self-dual flag to another. We also determine s...

In low dimensional gravity on anti-de sitter space there exists a possibility that the asymptotic isometry is raised to a Virasoro symmetry living at spatial infinity. We discuss in detail asymptotic isometry of the most general anti-de sitter dilaton gravity in two dimensions. From analysis of the boundary dynamics it turns out that the Virasoro symmetry is not preserved in all the models. Mor...

Every ε-isometry u between real normed spaces of the same finite dimension which maps the origin to the origin may by uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces.

Hyperbolic area is characterized as the unique continuous isometry invariant simple valuation on convex polygons in H. We then show that continuous isometry invariant simple valuations on polytopes in H for n ≥ 1 are determined uniquely by their values at ideal simplices. The proofs exploit a connection between valuation theory in hyperbolic space and an analogous theory on the Euclidean sphere...

A finite set W ⊂ R is said to realize the group G if the isometry group of W is isomorphic to G. The isometry dimension of a group is the minimum dimension of a realization. It is known that the isometry dimension of G is less than |G| [1]. We show that the isometry dimension of Z2 is n. The orbit number of a group is the minimum number of orbits in a realization. We show that the groups Z2 are...

In 1964, Michael Edelstein presented an amazing affine isometry acting on the space of square-summable sequences. This operator has no fixed points, but a suborbit that converges to 0 while another...

We explore the structure of compact Riemann surfaces by studying their isometry groups. First we give two constructions due to Accola [1] showing that for all g ≥ 2, there are Riemann surfaces of genus g that admit isometry groups of at least some minimal size. Then we prove a theorem of Hurwitz giving an upper bound on the size of any isometry group acting on any Riemann surface of genus g ≥ 2...

By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C∗-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C∗-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplica...

We show how the data of a finite dimensional weak C∗-Hopf algebra can be encoded into a pair (H, V ) where H is a finite dimensional Hilbert space and V :H⊗H → H⊗H is a partial isometry satisfying, among others, the pentagon equation. In case of V being unitary we recover the Baaj-Skandalis multiplicative unitary of the discrete compact type. Relation to the pseudomultiplicative unitary approac...