Noncommutative differential forms on the kappa-deformed space

Consider any representation ϕ \boldsymbol {\phi } of a finite-dimensional Lie algebra g \mathfrak {g} by derivations of the completed symmetric algebra S ^ ( g ∗ ) \hat {S}(\mathfrak {g}^*) of its dual. Consider the tensor product of S ^ ( g ∗ ) \hat {S}(\mathfrak {g}^*) and the exterior algebra Λ ( g ) \Lambda (\mathfrak {g}) . We show that the representation ϕ \boldsymbol {\phi } extends canonically to the representation ϕ ~ \tilde {\boldsymbol {\phi }} of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the universal enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the graded Leibniz rule is deformed.

The basic concepts of exterior calculus for space–time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space–time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms.

In this work we investigate generalized kappa-deformed spaces. We develop a systematic method for constructing realizations of noncommutative (NC) coordinates as formal power series in the Weyl algebra. All realizations are related by a group of similarity transformations, and to each realization we associate a unique ordering prescription. Generalized derivatives, the Leibniz rule and coproduct, as well as the star product are found in all realizations. The star product and Drinfel'd twist operators are given in terms of the coproduct, and the twist operator is derived explicitly in special realizations. The theory is applied to a Nappi–Witten type of NC space.

For a field theory that is invariant under diffeomorphisms there is a subtle\ninterplay between symmetries, conservation laws and the phase space of the\ntheory. The natural language for describing these ideas is that of differential\nforms and both differential forms on space-time and differential forms on the\ninfinite dimensional space of solutions of the equations of motion of the field\ntheory play an important role. There are exterior derivatives on both spaces\nand together they weave a double differential complex which captures the\ncohomology of the theory. This is important in the definition of invariants in\ngeneral relativity, such as mass and angular momentum and is also relevant to\nthe study of quantum anomalies in gauge theories.\n We derive the structure of this double complex and show how it relates to\nconserved quantities in gravitational theories. One consequence of the\nconstruction is that conserved quantities can be calculated exactly at finite\ndistance --- for example it is not necessary to go to asymptotic regimes to\ncalculate the mass or angular momentum of a stationary solution of Einstein's\nequations, they can be obtained exactly by an integration over any sphere\noutside the mass even at finite radius.\n

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.

Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms.

The purpose of the present paper is to prove that any germ of strongly pseudoconvex CR structure (on a manifold of real dimension ? 9) is realized by an embedding in the complex euclidean space of real codimension 1. It is well known that the above is true as formal power series. Hence we find an embedding such that the induced structure is very close to the given one on a small neighborhood of the reference point. We correct our embedding by Newton's method. We then apply the Moser-Nash procedure (cf. [4])' to prove that the sequence of embeddings obtained by the successive correction converges on a sufficiently small neighborhood of the reference point to an embedding which realizes the given structure. To explain how the correction is made, let D be the differential operator in the complex associated with the given structure. When f = (f..... ,f) is an approximating embedding defined on U, we set Ox = Dfx. If we can find gx such that (*) DgA = ON and f g is still an embedding, f is completely corrected. Unfortunately, we do not know how to find such a g.. However, the Neumann boundary value problem in Parts I and II for the differential operator Df (which is associated with the induced structure) is a tool to handle such question. By Newton's method we mean that we make the best correction using Df instead of D. This is done as follows: OA is not a differential form in the complex of Df. Hence we replace (01,. OJ) by its projection (O',... , On) to the complex of Df. Note that 0A may no longer be in the kernel of Df and our domain may not be distinguished. We first construct a distinguished distance function tf on U and consider a ball U= {tf ? r}

For a given Lagrangian $L$ a differential form ${\ensuremath{\theta}}_{L}^{l}$ is introduced which generalizes the classical notion of the Cartan form ${\ensuremath{\theta}}_{L}^{l}$. This extended Cartan form ${\ensuremath{\theta}}_{L}^{l}$ still gives rise to the same field equations (Euler-Lagrange equations), but has the added feature that $d{\ensuremath{\theta}}_{L}^{l}=0$ if and only if the Euler-Lagrange equations for $L$ vanish identically. This has the important consequence that the symmetries of a given Lagrangian $L$ are precisely those diffeomorphisms which transform $d{\ensuremath{\theta}}_{L}^{l}$ into itself, a property not shared by the classical form $d{\ensuremath{\theta}}_{L}^{l}$.

Formal analysis is the study of formal power series, formal Laurent series, formal root series, and other formal series or formal functionals. This book is the first comprehensive presentation of the topic that systematically introduces formal analysis, including its algebraic, analytic, and topological structure, along with various applications.

We relate the recent derivation of the linearised Einstein equation on an AdS background from holographic entanglement entropy arguments to the Sparling construction: we derive the differential form whose exterior derivative gives the Einstein equation from the Sparling formalism. We develop the study of perturbations within the context of the Sparling formalism and find that the Sparling form vanishes for linearised perturbations on flat space.

A program, written for the IBM 7090, which defines a model for a finite collection of associative algebras for the computer, is described and illustrated. The program contains a structure broad enough in scope to allow one to perform operations on such diverse mathematical concepts as differential equations, infinite series and differential forms in a simple yet comprehensive manner, while also serving in a foundation upon which a variety of higher level symbol manipulation languages can be developed. The development of the structure of this model is given. In particular, it is assumed that a family of not necessarily commutative algebras defined over a collection of commutative rings is given, where the algebras are algebras of modules generated by recursive sets. A brief discussion is given on how the vectors and scalars of these algebras should be represented, and how their basic algebraic operations (addition and multiplication) can be handled in a systematic manner without regard to the specific underlying scalar rings involved. Functional expressions are treated by developing a procedure for assigning an arbitrary list of operators to an algebra, and then assigning to each operator a list of assumptions that it is to satisfy. The latter is accomplished through the use of what are called "axiom" or "partial evaluation" functions. The resulting structure, which is called ALGEBRA, is shown to serve as a model for a family of associative algebras, where the resulting families of equivalence classes of representations for the vectors are uniquely representable. Various related notions, such as topological and decision considerations, are also briefly discussed. After having outlined the development of the system ALGEBRA, polynomial, exterior and series algebras are given as examples to illustrate how the model can be used. It is observed that a wide variety of mathematical data can be handled fairly simply, depending only on the formulation of the problem under consideration. The possibility of constructing various higher level symbol manipulation languages that have a mathematical structure, broad enough in scope to allow an analyst to manipulate various problems in a variety of ways, is also discussed. A programming language called FLAP has already been developed, demonstrating that such a language can indeed be studied and formulated in a consistent manner.

We study a Lie algebra type κ-deformed space with an undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. The space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.

The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator $L$ in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator $M$ to be in the centralizer of $L$. Whenever the centralizer equals the algebra generated by $L$ and $M$, we call $L$, $M$ a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order $4$ in the first Weyl algebra. Moreover, for true rank $r$ pairs, by means of differential subresultants, we effectively compute the fiber of the rank $r$ spectral sheaf over their spectral curve.

We investigate the translation equation (T) , and the cocycle equations (Co1) , (Co2) in [ x ], the ring of formal power series over . Here we restrict ourselves to iteration groups of type I, i. e. to solutions of (T) of the form , where c 1 ? 1 is a generalized exponential function. It is easy to prove that the coefficient functions c n (s), a n (s), s n (s) of are polynomials, c 1(s), polynomials or rational functions in one more generalized exponential function and, in a special case, polynomials in an additive function. For example, we obtain . For this we do not need any detailed information about the polynomials P n . Under some conditions on the exponential and additive functions it is possible to replace the exponential and additive functions by independent indeterminates. In this way we obtain formal versions of the translation equation and the cocycle equations in rings of the form [ x ], [ x ], [ x ], [ x ], and [ x ]. We solve these equations in a completely algebraic way, by deriving formal differential equations or Aczel�Jabotinsky type equations for these problems. It is possible to get the structure of the coefficients in great detail which are now polynomials or rational functions. We prove the universal character (depending on certain parameters) of these polynomials or rational functions. And we deduce the canonical form S -1(yS(x)) for iteration groups of type I. This approach seems to be more clear and more general than the original one. Some simple substitutions allow us to solve these problems in rings of the form ( [ u ]) [ x ], i. e. where the coefficient functions are formal series.

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.