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A. Chakrabarti

Publications and source records attributed to A. Chakrabarti.

At least 55 records · Page 3Linked to original sources
arXiv2001

Gravitation with superposed Gauss--Bonnet terms in higher dimensions: Black hole metrics and maximal extensions

Our starting point is an iterative construction suited to combinatorics in arbitarary dimensions d, of totally anisymmetrised p-Riemann 2p-forms (2p\le d) generalising the (1-)Riemann curvature 2-forms. Superposition of p-Ricci scalars obtained from the p-Riemann forms defines the maximally Gauss--Bonnet extended gravitational Lagrangian. Metrics, spherically symmetric in the (d-1) space dimensions are constructed for the general case. The problem is directly reduced to solving polynomial equations. For some black hole type metrics the horizons are obtained by solving polynomial equations. Corresponding Kruskal type maximal extensions are obtained explicitly in complete generality, as is also the periodicity of time for Euclidean signature. We show how to include a cosmological constant and a point charge. Possible further developments and applications are indicated.

hep-th↗
arXiv2001

Modified Braid Equations for SO_q (3) and noncommutative spaces

General solutions of the $\hat{R}TT$ equation with a maximal number of free parameters in the specrtal decomposition of vector $SO_q (3)$ $\hat{R}$ matrices are implemented to construct modified braid equations (MBE). These matrices conserve the given, standard, group relations of the nine elements of T, but are not constrained to satisfy the standard braid equation (BE). Apart from q and a normalisation factor our $\hat{R}$ contains two free parameters, instead of only one such parameter for deformed unitary algebras studied in a previous paper [1] where the nonzero right hand side of the $MBE$ had a linear term proprotional to $(\hat{R}_{(12)} - \hat{R}_{(23)})$. In the present case the r.h.s. is, in general, nonliear. Several particular solutions are given (Sec.2) and the general structure is analysed (App.A). Our formulation of the problem in terms of projectors yield also two new solutions of standard (nonmodified) braid equation (Sec.2) which are further discussed (App.B). The noncommutative 3-spaces obtained by implementing such generalized $\hat{R}$ matrices are studied (Sec.3). The role of coboundary $\hat{R}$ matrices (not satisfying the standard BE) is explored. The MBE and Baxterization are presented as complementary facets of the same basic construction, namely, the general solution of $\hat{R}TT$ equation (Sec.4). A new solution is presented in this context. As a simple but remarkable particular case a nontrivial solution of BE is obtained (App.B) for q=1. This solution has no free parameter and is not obtainable by twisting the identity matrix. In the concluding remarks (Sec.5), among other points, generalisation of our results to $SO_{q}(N)$ is discussed.

math.QA↗
arXiv2001

Duality for Exotic Bialgebras

In the classification of Hietarinta, three triangular $4\times 4$ $R$-matrices lead, via the FRT formalism, to matrix bialgebras which are not deformations of the trivial one. In this paper, we find the bialgebras which are in duality with these three exotic matrix bialgebras. We note that the $L-T$ duality of FRT is not sufficient for the construction of the bialgebras in duality. We find also the quantum planes corresponding to these bialgebras both by the Wess-Zumino R-matrix method and by Manin's method.

math.QA↗
arXiv2001

RTT relations, a modified braid equation and noncommutative planes

With the known group relations for the elements $(a,b,c,d)$ of a quantum matrix $T$ as input a general solution of the $RTT$ relations is sought without imposing the Yang - Baxter constraint for $R$ or the braid equation for $\hat{R} = PR$. For three biparametric deformatios, $GL_{(p,q)}(2), GL_{(g,h)}(2)$ and $GL_{(q,h)}(1/1)$, the standard,the nonstandard and the hybrid one respectively, $R$ or $\hat{R}$ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter $K$,such that only for two values of $K$, given explicitly for each case, one has the braid equation. Arbitray $K$ corresponds to a class (conserving the group relations independent of $K$) of the MQYBE or modified quantum YB equations studied by Gerstenhaber, Giaquinto and Schak. Various properties of the triparametric $\hat{R}(K;p,q)$, $\hat{R}(K;g,h)$ and $\hat{R}(K;q,h)$ are studied. In the larger space of the modified braid equation (MBE) even $\hat{R}(K;p,q)$ can satisfy $\hat{R}^2 = 1$ outside braid equation (BE) subspace. A generalized, $K$- dependent, Hecke condition is satisfied by each 3-parameter $\hat{R}$. The role of $K$ in noncommutative geometries of the $(K;p,q)$,$(K;g,h)$ and $(K;q,h)$ deformed planes is studied. K is found to introduce a "soft symmetry breaking", preserving most interesting properties and leading to new interesting ones. Further aspects to be explored are indicated.

math.QA↗
arXiv2000

On Combined Standard-Nonstandard or Hybrid (q,h)-Deformations

Combined $(q,h)$-deformations proposed by Kupershmidt and Ballesteros-Herranz-Parashar are studied. In each case a transformation is shown to lead to an equivalent, standard $q$-deformation. We briefly indicate that appropriate singular limits of the same type of transformations can however lead from standard biparametric $(p,q)$-deformations to non-hybrid but biparametric nonstandard $(g,h)$ ones. Finally a case of hybrid $(q,h)$-deformation is recalled, related to the superalgebra $GL(1|1)$.

math.QA↗
arXiv2000

Finite energy/action solutions of $p_1$ Yang-Mills equations on $p_2$ Schwarzschild and deSitter backgrounds for dimension $d \ge 4$

Physically relevant gauge and gravitational theories can be seen as special members of hierarchies of more elaborate systems. The Yang-Mills (YM) system is the first member of a hierarchy of Lagrangians which we will index by $p_1$, and the Einstein-Hilbert (EH) system of general relativity is the first member of another hierarchy which we index by $p_2$. In this paper, we study the classical equations of the $p_1 = 1,2$ YM hierarchy considered in the background of special geometries (Schwarzschild, deSitter,anti-deSitter) of the $p_2=1,2,3$ EH hierarchy. Solutions are obtained in various dimensions and lead to several examples of non-self-dual YM fields. When $p_1=p_2$ self-dual solutions exist in addition. Their action is equal to the Chern-Pontryagin charge and can be compared with that of the non-self-dual solutions.

hep-th↗
arXiv2000

The Gervais-Neveu-Felder equation for the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra

Using a contraction procedure, we construct a twist operator that satisfies a shifted cocycle condition, and leads to the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra. The corresponding universal ${\cal R}_{h}(y)$ matrix obeys a Gervais-Neveu-Felder equation associated with the U_{h;y}(sl(2)) algebra. For a class of representations, the dynamical Yang-Baxter equation may be expressed as a compatibility condition for the algebra of the Lax operators.

math.QA↗
arXiv1999

Antisymmetrised 2p-forms generalising curvature 2-forms and a corresponding p-hierarchy of Schwarzschild type metrics in dimensions d>2p+1

Starting with the curvature 2-form a recursive construction of totally antisymmetrised 2p-forms is introduced, to which we refer as p-Riemann tensors. Contraction of indices permits a corresponding generalisation of the Ricci tensor. Static, spherically symmetric ``$p$-Ricci flat'' Schwarzschild like metrics are constructed in this context for d>2p+1, d being the spacetime dimension. The existence of de Sitter type solutions is pointed out. Our 2p-forms vanish for $d<2p$ and the limiting cases d=2p and d=2p+1 exhibit special features which are discussed briefly. It is shown that for d=4p our class of solutions correspond to double-selfdual Riemann 2p-form (or p-Riemann tensor). Topological aspects of such generalised gravitational instantons and those of associated (through spin connections) generalised Yang-Mills instantons are briefly mentioned. The possibility of a study of surface deformations at the horizons of our class of ``p-black holes'' leading to Virasoro algebras with a p-dependent hierarchy of central charges is commented on. Remarks in conclusion indicate directions for further study and situate our formalism in a broader context.

hep-th↗
arXiv1999

Nonselfdual solutions for gauge fields in Schwarzschild and deSitter backgrounds for dimension $d\geq 4$

A particularly simple class of nonselfdual solutions are obtained for gauge fields in Schwarzschild and deSitter backgrounds. For Lorentz signature these have finite energy and finite action for Euclidean signature. In each case one obtains either real or a pair of complex conjugate solutions. The actions are easily computed for any dimension d. Numerical values are given for d= 4,6,7,8,9,10. It is explained why d=5 is a very special case. Possible continuations and generalizations of the results obtained are indicated. A particular solution for AdS_4 background is presented in the Appendix.

hep-th↗
arXiv1998

Maps and twists relating $U(sl(2))$ and the nonstandard $U_{h}(sl(2))$: unified construction

A general construction is given for a class of invertible maps between the classical $U(sl(2))$ and the Jordanian $U_{h}(sl(2))$ algebras. Different maps are directly useful in different contexts. Similarity trasformations connecting them, in so far as they can be explicitly constructed, enable us to translate results obtained in terms of one to the other cases. Here the role of the maps is studied in the context of construction of twist operators between the cocommutative and noncocommutative coproducts of the $U(sl(2))$ and $U_{h}(sl(2))$ algebras respectively. It is shown that a particular map called the `minimal twist map' implements the simplest twist given directly by the factorized form of the ${\cal R}_{h}$-matrix of Ballesteros-Herranz. For other maps the twist has an additional factor obtainable in terms of the similarity transformation relating the map in question to the minimal one. The series in powers of $h$ for the operator performing this transformation may be obtained up to some desired order, relatively easily. An explicit example is given for one particularly interesting case. Similarly the classical and the Jordanian antipode maps may be interrelated by a similarity transformation. For the `minimal twist map' the transforming operator is determined in a closed form.

math.QA↗
arXiv1998

Implementation of an iterative map in the construction of (quasi)periodic instantons: chaotic aspects and discontinuous rotation numbers

An iterative map of the unit disc in the complex plane (Appendix) is used to explore certain aspects of selfdual, four dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits. The iterations do not correspond directly to a discretized time evolution of the gauge fields. They are implemented in an indirect fashion. First, (t,r,θ,ϕ) being the standard coordinates, the (r,t) half plane is mapped on the unit disc in an appropriate way. This provides an (r,t) parametrization (Sec.1) of Z_0, the starting point of the iterations and makes the iterates increasingly complex functions of r and t. These are then incorporated as building blocks in the generating function of the fields (Sec.2). We explain (starting in Sec.1 and at different stages) in what sense and to what extent some remarkable features of our map (indicated in the title) are thus carried over into the continuous time development of the fields. Special features for quasiperiodicity are studied (Sec.3). Spinor solutions (Sec.4) and propagators (Sec.5) are discussed from the point of view of the mapping. Several possible generalizations are indicated (Sec.6). Some broader topics are discussd in conclusion (Sec.7).

hep-th↗
arXiv1997

General Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_{q}$-matrices

A class of transformations of $R_q$-matrices is introduced such that the $q\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation matrix is singular itself at $q\to 1$ limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for $sl(2)$ but also for algebras of higher dimensions. Explicit constructions are presented starting with ${\cal U}_q(sl(2))$ and ${\cal U}_q(sl(3))$, while choosing $R_q$ for (fund. rep.)$\otimes$(arbitrary irrep.). The treatment for the general case and various perspectives are indicated. Our method yields nonstandard deformations along with a nonlinear map of the $h$-Borel subalgebra on the corresponding classical Borel subalgebra. For ${\cal U}_h(sl(2))$ this map is extended to the whole algebra and compared with another one proposed by us previously.

q-alg↗
arXiv1996

Combined (q,h)-Deformation as a Nonlinear Map on $U_q(sl(2))$

The generators $(J_{\pm}, J_0)$ of the algebra $U_q(sl(2))$ is our starting point. An invertible nonlinear map involving, apart from q, a second arbitrary complex parameter h, defines a triplet $({\hat X},{\hat Y},{\hat H})$. The latter set forms a closed algebra under commutation relations. The nonlinear algebra $U_{q,h}(sl(2))$, thus generated, has two different limits. For $q \to 1$, the Jordanian h-deformation $U_{h}(sl(2))$ is obtained. For $h \to 0$, the q-deformed algebra $U_{q}(sl(2))$ is reproduced. From the nonlinear map, the irreducible representations of the doubly-deformed algebra $U_{q,h}(sl(2))$ may be directly and explicitly obtained form the known representations of the algebra $U_q(sl(2))$. Here we consider only generic values of q.

q-alg↗
arXiv1996

A Generalization of $U_h(sl(2))$ via Jacobian Elliptic Function

A two-parametric generalization of the Jordanian deformation $U_h (sl(2))$ of $sl(2)$ is presented. This involves Jacobian elliptic functions. In our deformation $U_{(h,k)}(sl(2))$, for $k^2=1$ one gets back $U_h(sl(2))$. The constuction is presented via a nonlinear map on $sl(2)$. This invertible map directly furnishes the highest weight irreducible representations of $U_{(h,k)}(sl(2))$. This map also provides two distinct induced Hopf stuctures, which are exhibited. One is induced by the classical $sl(2)$ and the other by the distinct one of $U_h(sl(2))$. Automorphisms related to the two periods of the elliptic functions involved are constructed. Translations of one generator by half and quarter periods lead to interesting results in this context. Possibilities of applications are discussed briefly.

q-alg↗
arXiv1996

Irreducible Representations of Jordanian Quantum Algebra ${\cal U}_h(sl(2))$ Via a Nonlinear Map

The generators of the Jordanian quantum algebra ${\cal U}_h(sl(2))$ are expressed as nonlinear invertible functions of the classical $sl(2)$ generators. This permits immediate explicit construction of the finite dimensional irreducible representations of the algebra ${\cal U}_h(sl(2))$. Using this construction, new finite dimensional solutions of the Yang-Baxter equation may be obtained.

q-alg↗
arXiv1996

On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations

By solving a set of recursion relations for the matrix elements of the ${\cal U}_h(sl(2))$ generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the ${\cal U}_h(sl(2))$ algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction, yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$ algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.

q-alg↗