Deformations of representations

A connection between deformation of Lie group representations and deformations of associated Lie algebra representations is established. Applications are given to the theory of analytic continuation of K-finite quasi-simple representations of semi-simple Lie groups. A construction process of all TCI representations of SL(2,R) is obtained.     

Deformations of the embedded Einstein spaces

We discuss a method of studying the stability of solutions of Einstein's equations, which can be outlined as follows: Consider an embedding of a given Einstein spaceV ₄ into a pseudo-Euclidean spaceE _p,q ^N (N > 4,p + q =N) (p,q) describing the signature of the spaceE _p,q ^N . Then all the geometrical objects ofV ₄ can be expressed in terms of the embedding functions,Z ^A (x ⁱ ),A = 1, 2,...,N, i = 0, 1, 2, 3. Then let us deform the embedding:Z ^A Z ^A + ^A , being an infinitesimal parameter. The Einstein equations can be developed then in the powers of; we study the equations arising by requirement of the vanishing of the first- or second-order terms. Some partial results concerning the de Sitter, Einstein, and Minkowskian spaces are given.     

Deformations of the canonical commutation relations

Deformations of the canonical commutation relations which have the effect of altering the spectrum of a standard Hamiltonian, bilinear in creation and annihilation operators are described. The problem of going over from an eigenvalue situation, as is the case in the vast majority of papers in the literature, to a theory with time evolution is discussed, and a special example with deformation parameter an Nth root of unity is constructed which possesses a consistent time evolution. This work is an account of some recent studies of associative deformations of the Heisenberg algebra of several creation and annihilation algebras, with Jean Nuyts of the University of Mons, Hainaut, together with some observations of my own concerning the difficulty of implementing time evolution in a quantum group context. It builds on earlier work with Cosmas Zachos (Argonne National Laboratory, USA), which in turn is re;ated to work of Manin, and Wess, Zumino and collaborators. The main idea is that, if quantum groups have any role in physics, then they must manifest themselves at the level of the basic rules of quantisation.     

Deformations of quantum hyperplanes

We consider the quantum hyperplanex ⁱ x ^j =q _ij x ^j x ⁱ (i,j = 1..n) and define and consider deformations of the formx ⁱ x ^j =q _ij x ^j x ⁱ + _{k k} ^ij x ^k + ^ij , where _k ^ij and ^ij are complex numbers. We prove that for genericq _ij no nontrivial deformations exist forn 3.     

Deformations of duality-symmetric theories

We prove that a sum of free non-covariant duality-symmetric actions does not allow consistent, continuous and local self-interactions that deform the gauge transformations. For instance, non-abelian deformations are not allowed, even in 4 dimensions where Yang–Mills type interactions of 1-forms are allowed in the non-manifestly duality-symmetric formulation. This suggests that non-abelian duality should require to leave the standard formalism of perturbative local field theories. The analyticity of self-interactions for a single duality-symmetric gauge field in four dimensions is also analyzed.     

Deformations of super-KMS functionals

We investigate the stability of the super-KMS property under deformations. We show that a family of continuous deformations of the super-derivation in the quantum algebra yields a continuous family of deformed super-KMS functionals. These functionals define a family of cohomologous, entire cocycles.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065Visiting from the Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland     

Deformations of Dispersionless KdV Hierarchies

The obstructions to the existence of a hierarchy of hydrodynamic conservation laws are studied for a multicomponent dispersionless KdV system. It is proved that if the lowest order obstruction vanishes then all higher obstructions automatically vanish, if and only the underlying algebra is a Jordan algebra. Deformations of these multicomponent dispersionless KdV-type equations are also studied. It is shown that no new obstructions appear and, hence, that the existence of a fully deformed hierarchy depends only on the existence of a single purely hydrodynamic conservation law.     

Deformations of generalized holomorphic structures

A deformation theory of generalized holomorphic structures in the setting of (generalized) principal fibre bundles is developed. It allows the underlying generalized complex structure to vary together with the generalized holomorphic structure. We study the related differential graded Lie algebra, which controls the deformation problem via the Maurer–Cartan equation. As examples, we check the content of the Maurer–Cartan equation in detail in the special cases where the underlying generalized complex structure is symplectic or complex. A deformation theorem, together with some non-obstructed examples, is also included.     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Integrable Rosochatius Deformations of the Restricted cKdV Flows

Three novel finite-dimensional integrable Hamiltonian systems of Rosochatius type and their Lax representations are presented. We make a deformation for the Lax matrbces of the Neumann type, the Bargmann type and the high-order symmetry type of restricted cKdV flows by adding an additional term and then prove that this kind of deformation does not change the r-matrix relations. Finally the new integrable systems are generated from these deformed Lax matrices.     

Formal and Analytic Deformations of the Witt Algebra

When studying gauge theories (e.g. with finite energy conditions), attention is traditionally restricted to the subset of irreducible connections, which is open and dense in the full space of connections. We point out that generally the residual set of reducible connections contains critical points of the gauge functionals which, moreover, are the only ones common to all theories with a given symmetry, i.e. those determined by the symmetry and geometry of the problem alone, and not by the specific choice of functional.     

Deformations of the Hodge map and optical geometry

A simple formula is derived for the infinitesimal change of the Hodge dual of a k-form, induced by a deformation of the scalar product in the underlying vector space. By considering deformations due to a flow generated by a vector field on a differential manifold, one obtains an expression for the commutator of the Hodge dual with the Lie derivation with respect to the vector field, acting on differential forms. This formula is useful in proving theorems on optical solutions of Maxwell's and Yang-Mills equations. The optical geometry underlying such solutions is defined as a restriction of the bundle of linear frames of a 4-dimensional manifold to a 9-dimensional optical group. This geometry provides a natural framework for the study of shearfree, optical and geodesic congruences and of the associated fields.     

Deformations of Modules of Differential Forms

Abstract

We study non-trivial deformations of the natural action of the Lie algebra Vect(?ⁿ) on the space of differential forms on ?ⁿ. We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in the sense of [3].     

Morphisms Cohomology and Deformations of Hom-algebras

The purpose of this paper is to define cohomology complexes and study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.     

Deformations and renormalisations ofW ∞

Deformations of the infiniteN limit of the ZamolodchikovW _N algebra are discussed. A recent one, due to Pope, Romans and Shen with non-zero central extensions for every conformal spin is shown to be formally renormalisable to one representable in Moyal bracket form. Another deformation is discovered which, like the algebra of Pope et al. possesses automatic closure, but has non-zero central extension only in the Virasoro subalgebra.On Research Leave from the University of Durham UK; research supported in part by the Department of Energy under Grant DE[FG02/88/ER25065, and by a grant from the Alfred P. Sloan Foundation