The Experts below are selected from a list of 76563 Experts worldwide ranked by ideXlab platform

Piergiulio Tempesta - One of the best experts on this subject based on the ideXlab platform.

Uwe Helmke - One of the best experts on this subject based on the ideXlab platform.

  • Lie Theory for Quantum Control
    GAMM-Mitteilungen, 2008
    Co-Authors: Gunther Dirr, Uwe Helmke
    Abstract:

    One of the main theoretical challenges in quantum computing is the design of explicit schemes that enable one to effectively factorize a given final unitary operator into a product of basic unitary operators. As this is equivalent to a constructive controllability task on a Lie group of special unitary operators, one faces interesting classes of bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In this paper we give a review on recent Lie-theoretical developments in finite-dimensional quantum control that play a key role for solving such factorization problems on a compact Lie group. After a brief introduction to basic terms and concepts from quantum mechanics, we address the fundamental control theoretic issues for bilinear control systems and survey standard techniques fromLie Theory relevant for quantum control. Questions of controllability, accessibility and time optimal control of spin systems are in the center of our interest. Some remarks on computational aspects are included as well. The idea is to enable the potential reader to understand the problems in clear mathematical terms, to assess the current state of the art and get an overview on recent developments in quantum control-an emerging interdisciplinary field between physics, control and computation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Ezra Getzler - One of the best experts on this subject based on the ideXlab platform.

  • Lie Theory for nilpotent l algebras
    Annals of Mathematics, 2009
    Co-Authors: Ezra Getzler
    Abstract:

    The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor γ from L ∞ -algebras concentrated in degree > -n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L ∞ algebras (i.e., chain complexes), the functor γ is the Dold-Kan simplicial set.

  • Lie Theory for nilpotent L-infinity algebras
    Annals of Mathematics, 2009
    Co-Authors: Ezra Getzler
    Abstract:

    The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infinity algebras concentrated in degree >-n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L-infinity algebras (i.e. chain complexes), the functor gamma is the Dold-Kan simplicial set.

Vladislav Kharchenko - One of the best experts on this subject based on the ideXlab platform.

  • Quantum Lie Theory: A Multilinear Approach
    2016
    Co-Authors: Vladislav Kharchenko
    Abstract:

    Elements of noncommutative algebra.- Poincar'e-Birkhoff-Witt basis.- Quantizations of Kac-Moody algebras.- Algebra of skew-primitive elements.- Multilinear operations.- Braided Hopf algebras.- Binary structures.- Algebra of primitive nonassociative polynomials.

  • Quantum Lie Theory - Quantum Lie Theory
    Lecture Notes in Mathematics, 2015
    Co-Authors: Vladislav Kharchenko
    Abstract:

    This is an introduction to the mathematics behind the phrase “quantum Lie algebra”. The numerous attempts over the last 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been widely accepted. In this book, an alternative approach is developed that includes multivariable operations. Among the problems discussed are the following: a PBW-type theorem; quantum deformations of Kac--Moody algebras; generic and symmetric quantum Lie operations; the Nichols algebras; the Gurevich--Manin  Lie algebras;  and Shestakov--Umirbaev  operations for the Lie Theory of nonassociative products.  Opening with an introduction for beginners and continuing as a textbook for graduate students in physics and mathematics, the book can also be used as a reference by more advanced readers. With the exception of the introductory chapter, the content of this monograph has not previously appeared in book form

  • algebra of primitive nonassociative polynomials
    2015
    Co-Authors: Vladislav Kharchenko
    Abstract:

    In this chapter, we consider nonassociative primitive polynomials as operations for nonassociative Lie Theory in a similar manner as how we considered the skew-primitive polynomials as operations for quantum Lie Theory in Chaps. 4 and 5. Many of the well-known generalizations of Lie algebras involve only one or two operations. For instance, Malcev algebras have one binary bracket; Lie triple systems have one ternary bracket; Bol and Lie-Yamaguti algebras have one binary bracket and one ternary bracket; and Akivis algebras have two operations, an antisymmetric binary bracket and a ternary bracket (related to commutator and associator), with only one identity that relates the two operations and generalizes the Jacobi identity. The notion of Akivis algebra initially appears to be a proper analog to Lie algebras for the Theory of nonassociative products. However, the question raised by K.H. Hofmann and K. Strambach of whether the commutator and associator are the only primitive operations in a nonassociative bialgebra was answered negatively. If true, it would have corroborated the fundamental role of Akivis algebras for nonassociative Lie Theory. In 2002, I.P. Shestakov and U.U. Umirbaev discovered infinitely many independent operations, thus proving the theorems considered in this chapter. These results demonstrate that Shestakov-Umirbaev operations together with the commutator form a complete set of nonassociative Lie operations, whereas Theorem 8.3 is a PBW basis theorem for the Lie Theory of nonassociative products.

Eckhard Meinrenken - One of the best experts on this subject based on the ideXlab platform.

  • clifford algebras and Lie Theory
    2013
    Co-Authors: Eckhard Meinrenken
    Abstract:

    Preface.- Conventions.- List of Symbols.- 1 Symmetric bilinear forms.- 2 Clifford algebras.- 3 The spin representation.- 4 Covariant and contravariant spinors.- 5 Enveloping algebras.- 6 Weil algebras.- 7 Quantum Weil algebras.- 8 Applications to reductive Lie algebras.- 9 D(g k) as a geometric Dirac operator.- 10 The Hopf-Koszul-Samelson Theorem.- 11 The Clifford algebra of a reductive Lie algebra.- A Graded and filtered super spaces.- B Reductive Lie algebras.- C Background on Lie groups.- References.- Index.

  • Lie Theory and the chern weil homomorphism
    Annales Scientifiques De L Ecole Normale Superieure, 2005
    Co-Authors: Anton Alekseev, Eckhard Meinrenken
    Abstract:

    Abstract Let P → B be a principal G-bundle. For any connection θ on P, the Chern–Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra W g = S g ∗ ⊗ ∧ g ∗ into the algebra of differential forms A = Ω ( P ) . Invariant polynomials ( S g ∗ ) inv ⊂ W g map to cocycles, and the induced map in cohomology ( S g ∗ ) inv → H ( A basic ) is independent of the choice of θ. The algebra Ω ( P ) is an example of a commutative g -differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern–Weil construction generalizes to all such algebras. In this paper, we introduce a canonical Chern–Weil map W g → A for possibly non-commutative g -differential algebras with connection. Our main observation is that the generalized Chern–Weil map is an algebra homomorphism “up to g -homotopy”. Hence, the induced map ( S g ∗ ) inv → H basic ( A ) is an algebra homomorphism. As in the standard Chern–Weil Theory, this map is independent of the choice of connection. Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouviere's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott–Shulman complex.

  • Lie Theory and the Chern–Weil homomorphism
    Annales Scientifiques de l’École Normale Supérieure, 2005
    Co-Authors: Anton Alekseev, Eckhard Meinrenken
    Abstract:

    Abstract Let P → B be a principal G-bundle. For any connection θ on P, the Chern–Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra W g = S g ∗ ⊗ ∧ g ∗ into the algebra of differential forms A = Ω ( P ) . Invariant polynomials ( S g ∗ ) inv ⊂ W g map to cocycles, and the induced map in cohomology ( S g ∗ ) inv → H ( A basic ) is independent of the choice of θ. The algebra Ω ( P ) is an example of a commutative g -differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern–Weil construction generalizes to all such algebras. In this paper, we introduce a canonical Chern–Weil map W g → A for possibly non-commutative g -differential algebras with connection. Our main observation is that the generalized Chern–Weil map is an algebra homomorphism “up to g -homotopy”. Hence, the induced map ( S g ∗ ) inv → H basic ( A ) is an algebra homomorphism. As in the standard Chern–Weil Theory, this map is independent of the choice of connection. Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouviere's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott–Shulman complex.

  • Lie Theory and the chern weil homomorphism
    arXiv: Representation Theory, 2003
    Co-Authors: Anton Alekseev, Eckhard Meinrenken
    Abstract:

    We introduce a canonical Chern-Weil map for possibly non-commutative g-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism ``up to g-homotopy''. Hence, the induced map from invariant polynomials to the basic cohomology is an algebra homomorphism. As in the standard Chern-Weil Theory, this map is independent of the choice of connection. Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouviere's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.