First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section 6 we generalize the Maurer-Cartan equation to strongly homotopy Li...

It is proved that the Lie groups E (5) 7 and E (7) 7 represented in R 56 and the Lie group E C 7 represented in R 112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension. §1 Introduction The ...

First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section 6 we generalize the Maurer-Cartan equation to strongly homotopy Li...

A Cartan subalgebra of a finite-dimensional Lie algebra L is a nilpotent subalgebra H of L that coincides with its normalizer NL H . Such subalgebras occupy an important place in the structure theory of finite-dimensional Lie algebras and their properties have been explored in many articles (see, e.g., Barnes, 1967; Benkart, 1986; Wilson, 1977; Winter, 1969). In general (more precisely, when th...

In the repetitive approach to the theory of jet bundles there are three methods of repetition, which yield non-holonomic, semi-holonomic, and holonomic jet bundles respectively. However the classical approach to holonomic jet bundles failed to be truly repetitive, for it must resort to such a non-repetitive coordinatedependent construction as Taylor expansion. The principal objective in this pa...

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E.Cartan. Especially, the E.Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and CartanEhresmann connection theory on fibered spaces. General structu...

Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A,A, . . . ,A are pairwise complementa...

Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A,A, . . . ,A are pairwise complementa...

We present a new approach to the covariant canonical formulation of EinsteinCartan gravity that preserves the full Lorentz group as the local gauge group. The method exploits lessons learned from gravity in 2+1 dimensions regarding the relation between gravity and a general gauge theory. The dynamical variables are simply the frame field and the spin-connection pulled-back to the hypersurface, ...

The bi-Hamiltonian structure of the two known vector generalizations of the mKdV hierarchy of soliton equations is derived in a geometrical fashion from flows of nonstretching curves in Riemannian symmetric spaces G/SO(N). These spaces are exhausted by the Lie groups G = SO(N + 1), SU(N). The derivation of the bi-Hamiltonian structure uses a parallel frame and connection along the curve, tied t...