Triangular de Rham Cohomology of Compact Kahler Manifolds

We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with values in a Lie group G. By definition, this is the quotient of the set of flat connections in the trivial principle bundle $M\times G$ by the so-called gauge equivalence. We consider the case when M is a compact K\"ahler manifold and G is a solvable complex linear algebraic group of a special class which contains the Borel subgroups of all complex classical groups and, in particular, the group $T_n(\Bbb C)$ of all triangular matrices. In this case, we get a description of the set H^1_{DR}(M,G) in terms of the 1-cohomology of M with values in the (abelian) sheaves of flat sections of certain flat Lie algebra bundles with fibre $\frak g$ (the Lie algebra of G) or, equivalently, in terms of the harmonic forms on M representing this cohomology

Classification of double flag varieties of complexity 0 and 1

A classification of double flag varieties of complexity 0 and 1 is obtained. An application of this problem to decomposing tensor products of irreducible representations of semisimple Lie groups is considered

Dressing Symmetries of Holomorphic BF Theories

We consider holomorphic BF theories, their solutions and symmetries. The equivalence of Cech and Dolbeault descriptions of holomorphic bundles is used to develop a method for calculating hidden (nonlocal) symmetries of holomorphic BF theories. A special cohomological symmetry group and its action on the solution space are described.Comment: 14 pages, LaTeX2

Affine algebraic groups with periodic components

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.Comment: 20 page

Local invariants of stabilizer codes

In [Phys. Rev. A 58, 1833 (1998)] a family of polynomial invariants which separate the orbits of multi-qubit density operators $\rho$ under the action of the local unitary group was presented. We consider this family of invariants for the class of those $\rho$ which are the projection operators describing stabilizer codes and give a complete translation of these invariants into the binary framework in which stabilizer codes are usually described. Such an investigation of local invariants of quantum codes is of natural importance in quantum coding theory, since locally equivalent codes have the same error-correcting capabilities and local invariants are powerful tools to explore their structure. Moreover, the present result is relevant in the context of multipartite entanglement and the development of the measurement-based model of quantum computation known as the one-way quantum computer.Comment: 10 pages, 1 figure. Minor changes. Accepted in Phys. Rev.

A derivation of quantum theory from physical requirements

Quantum theory is usually formulated in terms of abstract mathematical postulates, involving Hilbert spaces, state vectors, and unitary operators. In this work, we show that the full formalism of quantum theory can instead be derived from five simple physical requirements, based on elementary assumptions about preparation, transformations and measurements. This is more similar to the usual formulation of special relativity, where two simple physical requirements -- the principles of relativity and light speed invariance -- are used to derive the mathematical structure of Minkowski space-time. Our derivation provides insights into the physical origin of the structure of quantum state spaces (including a group-theoretic explanation of the Bloch ball and its three-dimensionality), and it suggests several natural possibilities to construct consistent modifications of quantum theory.Comment: 16 pages, 2 figures. V3: added alternative formulation of Requirement 5, extended abstract, some minor modification

On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras

We define the twisted loop Lie algebra of a finite dimensional Lie algebra $\mathfrak g$ as the Fr\'echet space of all twisted periodic smooth mappings from $\mathbb R$ to $\mathfrak g$. Here the Lie algebra operation is continuous. We call such Lie algebras Fr\'echet Lie algebras. We introduce the notion of an integrable $\mathbb Z$-gradation of a Fr\'echet Lie algebra, and find all inequivalent integrable $\mathbb Z$-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.Comment: 26 page