Bicovariant Quantum Algebras and Quantum Lie Algebras

Abstract

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y\equiv L^{+}S{L}^{-}$ being a special case --- generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for $Y$ in $S{O}_q(N)$.

38 pages

References in corpus (5)

• An introduction to non-commutative differential geometry on quantum groups
• Differential Geometry on Linear Quantum Groups
• Quantum and Braided Linear Algebra
• Constant Solutions of Reflection Equations and Quantum Groups
• The 2-Dimensional Quantum Euclidean Algebra

Cited by in corpus (31)

• Noncommutative Geometry and Gravity
• Generalized Quantum Relativistic Kinematics: a Stability Point of View
• Identical Particles and Quantum Symmetries
• Noncommutative Symmetries and Gravity
• Distance Measurement and $\kappa$-Deformed Propagation of Light and Heavy Probes
• Characteristic Relations for Quantum Matrices
• On second quantization on noncommutative spaces with twisted symmetries
• Spectral extension of the quantum group cotangent bundle
• q-Deformed quantum Lie algebras
• Covariant differential complexes on quantum linear groups
• Unbraiding the braided tensor product
• A quantum algebraic description of D-branes on group manifolds
• On Quantum Lie Algebras and Quantum Root Systems
• Hopf stars, twisted Hopf stars and scalar products on quantum spaces
• On the algebraic structure of differential calculus on quantum groups
• More on quantum groups from the the quantization point of view
• Dirac operator on $\kappa$-Minkowski space bicovariant differential calculus and deformed U(1) gauge theory
• Universal Enveloping Algebra and Differential Calculi on Orthogonal q-groups
• Toward a q-Deformed Standard Model
• Deformed Chern-Simons Theories
• Bicovariant Calculus in Quantum Theory and a Generalization of the Gauss Law
• On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N
• Extended reflection equation algebras, the braid group on a handlebody and associated link polynomials
• $q$-Deformed Chern Characters for Quantum Groups $S{U}_q(N)$
• Braided Hopf Algebras and Differential Calculus
• Twisted Quantum Lax Equations
• On construction of unitary quantum group differential calculus
• Implications of the Hopf algebra properties of noncommutative differential calculi
• q-Deformed Superalgebras
• Gauge transformations and quasitriangularity
• The quantum Cartan algebra associated to a bicovariant differential calculus