Commutators originated over 100 years ago as a by-product of computing group characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in their own right soon after their introduction. In particular, the phenomenon that the set of commutators does not necessarily form a subgroup has been well documented ...

We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces. Many moduli space...

Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz-ers, are all defined from the binary operation [x, y] = x −1 y −1 xy. Each of these notions, except centralizers of elements, may also be defined in terms of the commutator of normal subgroups. The commutator [M, N] (where M and N are normal subgrou...

We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered R-categories, thus answering questions of what might be ‘commutative versions’ of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theory here does not satisfy the Leibniz identity. We also discuss potential applic...

Comtrans algebras are unital modules over a commutative ring R, equipped with two basic trilinear operations: a commutator [x, y, z] satisfying the left alternative identity [x, x, y] = 0, (1.1) and a translator 〈x, y, z〉 satisfying the Jacobi identity 〈x, y, z〉+ 〈y, z, x〉+ 〈z, x, y〉 = 0, (1.2) such that together the commutator and translator satisfy the comtrans identity [x, y, x] = 〈x, y, x〉....

This work studies how certain problems in quantum theory have motivated some recent reseach in pure Mathematics in matrix and operator theory. The mathematical key is that of a commutator or a generalized commutator, that is, find an operator X ∈ B(H) satisfying the operator equation AX − XB = C. By this we will show how and why to solve the operator equation AX − XB = C. Some problems are stud...

Basic ideas about noncommuting coordinates are summarized, and then coordinate noncommutativity, as it arises in the Landau problem, is investigated. I review a quantum solution to the Landau problem, and evaluate the coordinate commutator in a truncated state space of Landau levels. Restriction to the lowest Landau level reproduces the well known commutator of planar coordinates. Inclusion of ...

We consider the sum rule proposed by one of us (SLA), obtained by taking the expectation value of an axial vector commutator in a state with one pion. The sum rule relates the pion decay constant to integrals of pion-pion cross sections, with one pion off the mass shell. We remark that recent data on pionpion scattering allow a precise evaluation of the sum rule. We also discuss the related Adl...

It has been shown that there is a Hamilton cycle in every connected Cayley graph on any group G whose commutator subgroup is cyclic of prime-power order. This note considers connected, vertex-transitive graphs X of order at least 3, such that the automorphism group of X contains a vertex-transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs...

For smooth domains, Liu et al. (Comm. Pure Appl. Math. 60: 1443-1487, 2007) used optimal estimates for the commutator of the Laplacian and the Leray projection operator to establish well-posedness of an extended Navier-Stokes dynamics. In their work, the pressure is not determined by incompressibility, but rather by a certain formula involving the Laplace-Leray commutator. A key estimate of Liu...