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We survey some known facts and open questions concerning the global properties of 3+1 dimensional space–times containing a compact Cauchy surface. We consider space–times with an l–dimensional Lie algebra of space–like Killing fields. For each l ≤ 3, we give some basic results and conjectures on global existence and cosmic censorship. For the case of the 3+1 dimensional Einstein equations witho...

We consider the space of linear maps from a coassociative coalgebra C into a Lie algebra L. Unless C has a cocommutative coproduct, the usual symmetry properties of the induced bracket on Hom(C,L) fail to hold. We define the concept of twisted domain (TD) algebras in order to recover the symmetries and also construct a modified ChevalleyEilenberg complex in order to define the cohomology of suc...

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with m arbitrary time-dependent coefficients are obtained possessing symmetries involving m arbitrary functions of time. A particular grade...

Fix a manifold M , and let V be an infinite dimensional simple Lie subalgebra of the Lie algebra VecM of vector fields on M . Assume that V contains a finite dimensional simple maximal subalgebra a(V). We define an a(V)-quantization of a V-module of differential operators on M to be a decomposition of the module into irreducible a(V)-modules. In this article we survey some recent results and op...

The concept of symmetry plays an important role in solving equations and systems of equations. We will present solutions for the (constant and spectral-parameter) Yang-Baxter equations and Yang-Baxter systems arising from algebra structures and discuss about their symmetries. In the last section, we present enhanced versions of Theorem 1 (from [20]), solutions for the classical Yang-Baxter equa...

t to )b x2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to a different time-dependent quadratic Schrödinger equation (TQ) and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state 〈x〉 and 〈p〉 (with their classical mo...

We describe the normalizers for all non-conjugate maximal Abelian subgroups of diagonalizable automorphisms of sl(3, C) and show their relation to the symmetries of equations related to the graded contraction. Introduction — Admissible gradings of a simple Lie algebra L over the complex or real number field are basic structural properties of each L. Examples of exploitation of coarse gradings l...

The Dirac equation can be derived using the relativistic relation between mass and energy E = m (normalized units) and the symmetries of the universe, so it is necessary to have a firm grasp on the latter. The relativistic symmetry group is the orthochronous Poincare group SO(1, 3) n R, though we shall mainly use just the orthochronous Lorentz group SO(1, 3). Its Lie algebra is so(1, 3). We fir...

Picard groups of tensor categories play an important role in rational conformal field theory. The Picard group of the representation category C of a rational vertex algebra can be used to construct examples of (symmetric special) Frobenius algebras in C. Such an algebra A encodes all data needed to ensure the existence of correlators of a local conformal field theory. The Picard group of the ca...