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V. D. Ivashchuk - One of the best experts on this subject based on the ideXlab platform.

  • On flux integrals for generalized Melvin solution related to simple Finite-Dimensional Lie Algebra
    The European Physical Journal C, 2017
    Co-Authors: V. D. Ivashchuk
    Abstract:

    A generalized Melvin solution for an arbitrary simple Finite-Dimensional Lie Algebra $$\mathcal G$$ G is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of $$\mathcal G$$ G . It is governed by a set of n moduli functions $$H_s(z)$$ H s ( z ) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earLier that these functions should be polynomials—the so-called fluxbrane polynomials. These polynomials depend upon integration constants $$q_s$$ q s , $$s = 1,\dots ,n$$ s = 1 , ⋯ , n . In the case when the conjecture on the polynomial structure for the Lie Algebra $$\mathcal G$$ G is satisfied, it is proved that 2-form flux integrals $$\Phi ^s$$ Φ s over a proper 2 d submanifold are finite and obey the relations $$q_s \Phi ^s = 4 \pi n_s h_s$$ q s Φ s = 4 π n s h s , where the $$h_s > 0$$ h s > 0 are certain constants (related to dilatonic coupling vectors) and the $$n_s$$ n s are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, $$s = 1,\dots ,n$$ s = 1 , ⋯ , n . The main relations of the paper are valid for a solution corresponding to a Finite-Dimensional semi-simple Lie Algebra $$\mathcal G$$ G . Examples of polynomials and fluxes for the Lie Algebras $$A_1$$ A 1 , $$A_2$$ A 2 , $$A_3$$ A 3 , $$C_2$$ C 2 , $$G_2$$ G 2 and $$A_1 + A_1$$ A 1 + A 1 are presented.

Jakob Palmkvist - One of the best experts on this subject based on the ideXlab platform.

  • Borcherds and Kac-Moody extensions of simple Finite-Dimensional Lie Algebras
    Journal of High Energy Physics, 2012
    Co-Authors: Jakob Palmkvist
    Abstract:

    We study the Borcherds superAlgebra obtained by adding an odd (fermionic) null root to the set of simple roots of a simple Finite-Dimensional Lie Algebra. We compare it to the Kac-Moody Algebra obtained by replacing the odd null root by an ordinary simple root, and then adding more simple roots, such that each node that we add to the Dynkin diagram is connected to the previous one with a single line. This generalizes the situation in maximal supergravity, where the E _ n symmetry Algebra can be extended either to a Borcherds superAlgebra or to the Kac-Moody Algebra E _11, and both extensions can be used to derive the spectrum of p -form potentials in the theory. We show that also in the general case, the Borcherds and Kac-Moody extensions lead to the same ‘ p -form spectrum’ of representations of the simple Finite-Dimensional Lie Algebra.

V. N. Tolstoy - One of the best experts on this subject based on the ideXlab platform.

  • On Some Lie BiAlgebra Structures on Polynomial Algebras and their Quantization
    Communications in Mathematical Physics, 2008
    Co-Authors: S. M. Khoroshkin, I. I. Pop, M. E. Samsonov, A. A. Stolin, V. N. Tolstoy
    Abstract:

    We study classical twists of Lie biAlgebra structures on the polynomial current Algebra $${\mathfrak{g}[u]}$$ , where $${\mathfrak{g}}$$ is a simple complex Finite-Dimensional Lie Algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r -matrices fall into classes labelled by the vertices of the extended Dynkin diagram of $${\mathfrak{g}}$$ . We give the complete classification of quasi-trigonometric r -matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of $${\mathfrak{sl}(n)}$$ .

Ivashchuk V.d. - One of the best experts on this subject based on the ideXlab platform.

  • On flux integrals for generalized Melvin solution related to simple Finite-Dimensional Lie Algebra
    Springer New York LLC, 2020
    Co-Authors: Ivashchuk V.d.
    Abstract:

    A generalized Melvin solution for an arbitrary simple Finite-Dimensional Lie Algebra G is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of G. It is governed by a set of n moduli functions Hs(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earLier that these functions should be polynomials—the so-called fluxbrane polynomials. These polynomials depend upon integration constants qs, s= 1 , ⋯ , n. In the case when the conjecture on the polynomial structure for the Lie Algebra G is satisfied, it is proved that 2-form flux integrals Φ s over a proper 2d submanifold are finite and obey the relations qsΦ s= 4 πnshs, where the hs> 0 are certain constants (related to dilatonic coupling vectors) and the ns are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, s= 1 , ⋯ , n. The main relations of the paper are valid for a solution corresponding to a Finite-Dimensional semi-simple Lie Algebra G. Examples of polynomials and fluxes for the Lie Algebras A1, A2, A3, C2, G2 and A1+ A1 are presented. © 2017, The Author(s)

  • Black brane solutions governed by fluxbrane polynomials
    'Elsevier BV', 2020
    Co-Authors: Ivashchuk V.d.
    Abstract:

    A family of composite black brane solutions in the model with scalar fields and fields of forms is presented. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat "internal" spaces. The solutions are governed by moduli functions Hs (s=1,..., m) obeying non-linear differential equations with certain boundary conditions imposed. These master equations are equivalent to Toda-like equations and depend upon the non-degenerate (m×m) matrix A. It was conjectured earLier that the functions Hs should be polynomials if A is a Cartan matrix for some semisimple Finite-Dimensional Lie Algebra (of rank m). It is shown that the solutions to master equations may be found by using so-called fluxbrane polynomials which can be calculated (in principle) for any semisimple Finite-Dimensional Lie Algebra. Examples of dilatonic charged black hole (0-brane) solutions related to Lie Algebras A1, A2, C2 and G2 are considered. © 2014 Elsevier B.V

  • Multidimensional gravity, flux and black brane solutions governed by polynomials
    2020
    Co-Authors: Ivashchuk V.d., Melnikov V.n.
    Abstract:

    Two famiLies of composite black brane solutions are overviewed, fluxbrane and black brane ones, in a model with scalar fields and fields of forms. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat "internal" spaces. The solutions are governed by moduli functions Hs (for fluxbranes) and Hs (for black branes), obeying nonlinear differential equations with certain boundary conditions. Themaster equations for Hs and Hs are equivalent to Toda-like equations and depend on a nondegenerate matrix A related to brane intersection rules. The functions Hs and Hs, as was conjectured and confirmed (at least partly) earLier, should be polynomials in proper variables if A is a Cartan matrix of some semisimple Finite-Dimensional Lie Algebra. The fluxbrane polynomials Hs were shown to be used for the construction of black brane polynomials Hs. This approach is illustrated by examples of nonextremal electric black p-brane solutions related to Lie Algebras A2, C2, and G2. © 2014 Pleiades Publishing, Ltd

S. M. Khoroshkin - One of the best experts on this subject based on the ideXlab platform.

  • On Some Lie BiAlgebra Structures on Polynomial Algebras and their Quantization
    Communications in Mathematical Physics, 2008
    Co-Authors: S. M. Khoroshkin, I. I. Pop, M. E. Samsonov, A. A. Stolin, V. N. Tolstoy
    Abstract:

    We study classical twists of Lie biAlgebra structures on the polynomial current Algebra $${\mathfrak{g}[u]}$$ , where $${\mathfrak{g}}$$ is a simple complex Finite-Dimensional Lie Algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r -matrices fall into classes labelled by the vertices of the extended Dynkin diagram of $${\mathfrak{g}}$$ . We give the complete classification of quasi-trigonometric r -matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of $${\mathfrak{sl}(n)}$$ .