scholarly journals Exploring exceptional Drinfeld geometries

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Chris D. A. Blair ◽  
Daniel C. Thompson ◽  
Sofia Zhidkova
Keyword(s):  
Field Theory ◽  
Lie Algebra ◽  
Algebraic Structure ◽  
Leibniz Algebra ◽  
Maximal Supergravity ◽  
Structure Constants

Abstract We explore geometries that give rise to a novel algebraic structure, the Exceptional Drinfeld Algebra, which has recently been proposed as an approach to study generalised U-dualities, similar to the non-Abelian and Poisson-Lie generalisations of T-duality. This algebra is generically not a Lie algebra but a Leibniz algebra, and can be realised in exceptional generalised geometry or exceptional field theory through a set of frame fields giving a generalised parallelisation. We provide examples including “three-algebra geometries”, which encode the structure constants for three-algebras and in some cases give novel uplifts for CSO(p, q, r) gaugings of seven-dimensional maximal supergravity. We also discuss the M-theoretic embedding of both non-Abelian and Poisson-Lie T-duality.

scholarly journals Jacobi-Lie T-plurality

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Jose J. Fernandez-Melgarejo ◽  
Yuho Sakatani
Keyword(s):  
Field Theory ◽  
Leibniz Algebra ◽  
Lie Derivative ◽  
Lie Bialgebra ◽  
Drinfel’D Double ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Structure Constants ◽  
One To One ◽  
The Spectator

We propose a Leibniz algebra, to be called DD^++, which is a generalization of the Drinfel’d double. We find that there is a one-to-one correspondence between a DD^++ and a Jacobi–Lie bialgebra, extending the known correspondence between a Lie bialgebra and a Drinfel’d double. We then construct generalized frame fields E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+EAM∈O(D,D)×ℝ+ satisfying the algebra \hat{\pounds}_{E_A}E_B = - X_{AB}{}^C\,E_C£̂EAEB=−XABCEC, where X_{AB}{}^CXABC are the structure constants of the DD^++ and \hat{\pounds}£̂ is the generalized Lie derivative in double field theory. Using the generalized frame fields, we propose the Jacobi–Lie TT-plurality and show that it is a symmetry of double field theory. We present several examples of the Jacobi–Lie TT-plurality with or without Ramond–Ramond fields and the spectator fields.

scholarly journals Gauged double field theory as an L∞ algebra

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Eric Lescano ◽  
Martín Mayo
Keyword(s):  
Field Theory ◽  
Algebraic Structure ◽  
Classical Field ◽  
Field Theories ◽  
Lie Derivative ◽  
Linear Perturbation ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Symmetry Transformations ◽  
Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

Algebraic structure of Dirac–Pauli equation with account to AMM of neutron in the presence of magnetic field with cylindrical symmetry

2021 ◽  
pp. 2150121
Author(s):  
Masoud Seidi
Keyword(s):  
Magnetic Field ◽  
Lie Algebra ◽  
Algebraic Structure ◽  
Anomalous Magnetic Moment ◽  
Casimir Operator ◽  
Cylindrical Symmetry ◽  
Pauli Equation ◽  
Eigenvalues And Eigenfunctions ◽  
Ladder Operators ◽  
Nu Method

The eigenvalues and eigenfunctions of Dirac–Pauli equation have been obtained for a neutron with anomalous magnetic moment (AMM) in the presence of a strong magnetic field with cylindrical symmetry. In our calculations, the Nikiforov and Uvarov (NU) method has been used. Using the eigenfunctions and construction of the ladder operators, we show that these generators satisfy su(2) Lie algebra and computed the second-order Casimir operator of the lie algebra.

scholarly journals THE UNIVERSAL LINK POLYNOMIAL

1992 ◽  
Vol 07 (05) ◽  
pp. 877-945 ◽  
Author(s):  
E. GUADAGNINI
Keyword(s):  
Field Theory ◽  
Closed Form ◽  
Lie Algebra ◽  
Wilson Line ◽  
Expectation Values ◽  
Simple Lie Algebra ◽  
Chern Simons

The solution of the non-Abelian SU (N) quantum Chern–Simons field theory defined in R3 is presented. It is shown how to compute the expectation values of the Wilson line operators, associated with oriented framed links, in closed form. The main properties of the universal link polynomial, defined by these expectation values, are derived in the case of a generic real simple Lie algebra. The resulting polynomials for some simple examples of links are reported.

QUANTUM LIOUVILLE FIELD THEORY ON A MANIFOLD WITH BOUNDARY

1998 ◽  
Vol 13 (25) ◽  
pp. 2057-2063
Author(s):  
S. A. APIKYAN
Keyword(s):  
Field Theory ◽  
Functional Equations ◽  
Ward Identity ◽  
Semiclassical Approximation ◽  
Boundary Structure ◽  
Manifold With Boundary ◽  
Structure Constants ◽  
Liouville Field Theory

This letter studies the quantum Liouville field theory on a manifold with boundary. The boundary conformal Ward identity (CWI) is written and its semiclassical approximation is analyzed. This establishes a method of finding the accessory parameters of the theory with boundary. The boundary structure constants of the theory are defined and the functional equations which determine them are derived.

Some Properties of ID∗-n-Lie-derivations of Leibniz algebras

2021 ◽  
pp. 2250054
Author(s):  
G. R. Biyogmam ◽  
C. Tcheka ◽  
D. A. Kamgam
Keyword(s):  
Lie Algebra ◽  
Leibniz Algebra ◽  
Central Series ◽  
Algebra Structure ◽  
Leibniz Algebras ◽  
Lie Derivations

The concepts of [Formula: see text]-derivations and [Formula: see text]-central derivations have been recently presented in [G. R. Biyogmam and J. M. Casas, [Formula: see text]-central derivations, [Formula: see text]-centroids and [Formula: see text]-stem Leibniz algebras, Publ. Math. Debrecen 97(1–2) (2020) 217–239]. This paper studies the notions of [Formula: see text]-[Formula: see text]-derivation and [Formula: see text]-[Formula: see text]-central derivation on Leibniz algebras as generalizations of these concepts. It is shown that under some conditions, [Formula: see text]-[Formula: see text]-central derivations of a non-Lie-Leibniz algebra [Formula: see text] coincide with [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations, that is, [Formula: see text]-[Formula: see text]-derivations in which the image is contained in the [Formula: see text]th term of the lower [Formula: see text]-central series of [Formula: see text] and vanishes on the upper [Formula: see text]-central series of [Formula: see text] We prove some properties of these [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations. In particular, it is shown that the Lie algebra structure of the set of [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations is preserved under [Formula: see text]-[Formula: see text]-isoclinism.

Summary of Elements of Algebraic Theory

1995 ◽  
Author(s):  
F. Iachello ◽  
R. D. Levine
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Structure ◽  
Binary Operation ◽  
Algebraic Theory ◽  
Quantum Mechanical ◽  
Ordinary Quantum

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras. The binary operation (“multiplication”) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, B] = AB - BA. A set of operators {X} is a Lie algebra when it is closed under commutation.

scholarly journals Pseudo-metric 2-step nilpotent Lie algebras

2018 ◽  
Vol 18 (2) ◽  
pp. 237-263 ◽  
Author(s):  
Christian Autenried ◽  
Kenro Furutani ◽  
Irina Markina ◽  
Alexander Vasiľev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Rational Structure ◽  
Lie Triple System ◽  
Nilpotent Lie Algebras ◽  
Lie Bracket ◽  
Orthogonal Lie Algebra ◽  
Structure Constants ◽  
Scalar Products

Abstract The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that a 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with the Lie bracket. This choice of the standard pseudo-metric form allows us to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo H-type algebras have bases with rational structure constants, which implies that the corresponding pseudo H-type groups admit lattices.

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