De Rham Cohomology and Hodge Decomposition For Quantum Groups

Heckenberger, I.; Schüler, Axel

doi:10.1112/plms/83.3.743

Cited by 18 publications

(9 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. Since ·, · is a homomorphism of the left A-modules Γ ⊗ A Γ ∧ and Γ ∧ , it is easy to see that (15) gives a well-defined action of the A-bimodule Γ on Γ ∧ . Hence this action can be extended to an action ⊲ on Γ ∧ of the algebra Γ ⊗ A .…”

Section: Quantum Clifford Algebrasmentioning

confidence: 99%

Spin geometry on quantum groups via covariant differential calculi

Heckenberger

2003

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

Let A be a cosemisimple Hopf * -algebra with antipode S and let Γ be a left-covariant first order differential * -calculus over A such that Γ is selfdual (see Section 2) and invariant under the Hopf algebra automorphism S 2 . A quantum Clifford algebra Cl(Γ, σ, g) is introduced which acts on Woronowicz' external algebra Γ ∧ . A minimal left ideal of Cl(Γ, σ, g) which is an A-bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on Γ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL q (2) and its bicovariant 4D ± -calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed.

show abstract

Section: Quantum Clifford Algebrasmentioning

confidence: 99%

Spin geometry on quantum groups via covariant differential calculi

Heckenberger

2003

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then it is shown that the inclusion map B → X is a fibration as defined earlier (see Theorem 10.5). For the development of noncommutative homogenous spaces the reader should refer to [11,15]. Again the quantum Hopf fibration ι : A (S 2 q ) ֒→ A (SL q (2)) is an example of this situation and we explicitly prove that it is a differentiable fibration for one of two standard four-dimensional bicovariant calculi on A (SL q (2)).…”

Section: Introductionmentioning

confidence: 88%

The serre spectral sequence of a noncommutative fibration for de Rham cohomology

Beggs

Brzeziński

2005

Acta Math.

View full text Add to dashboard Cite

For differential calculi on noncommutative algebras, we construct a twisted de Rham cohomology using flat connections on modules. This has properties similar, in some respects, to sheaf cohomology on topological spaces. We also discuss generalised mapping properties of these theories, and relations of these properties to corings. Using this, we give conditions for the Serre spectral sequence to hold for a noncommutative fibration. This might be better read as giving the definition of a fibration in noncommutative differential geometry. We also study the multiplicative structure of such spectral sequences. Finally we show that some noncommutative homogeneous spaces satisfy the conditions to be such a fibration, and in the process clarify the differential structure on these homogeneous spaces. We also give two explicit examples of differential fibrations: these are built on the quantum Hopf fibration with two different differential structures. Flat connections and cohomology with twisted coefficientsThe classical Serre spectral sequence uses cohomology with a non-trivial coefficient bundle. In this section we discuss flat connections on modules in noncommutative geometry, and how this can be used to define de Rham cohomology with non-trivial coefficient modules.By a differential calculus on a noncommutative algebra A we mean a differential graded algebra (d, Ω * A) such that Ω 0 A = A. The product in Ω * A (for * ≥ 1) is denoted by the wedge ∧ (although Ω * A is not graded anticommutative in general). The density condition says that Ω n+1 A ⊂ A.dΩ n A, but we will not require this till later.The cohomology of (d, Ω * A) is denoted by H * dR (A) and referred to as a de Rham cohomology of A. Recall that a connection in a left A-module E is a map ∇ : E → Ω 1 A ⊗ A E satisfying the Leibniz rule, for all a ∈ A, e ∈ E, ∇(a.e) = da ⊗ e + a∇e.

show abstract

“…For other results on a Hodge-type decomposition and on de Rham cohomology in the setting of differential calculi over quantum groups, see [3].…”

Section: A Hodge-type Decomposition and The Dirac And Laplace Operatorsmentioning

confidence: 99%

Differential calculi over quantum groups

Murphy

2003

Banach Center Publications

View full text Add to dashboard Cite

An exposition is given of recent work of the author and others on the differential calculi that occur in the setting of compact quantum groups. The principal topics covered are twisted graded traces, an extension of Connes' cyclic cohomology, invariant linear functionals on covariant calculi and the Hodge, Dirac and Laplace operators in this setting. Some new results extending the classical de Rham theorem and Poincaré duality are also discussed.

show abstract

De Rham Cohomology and Hodge Decomposition For Quantum Groups

Abstract: Let Γ = Γ τ,z be one of the N 2 -dimensional bicovariant first order differential calculi for the quantum groups GL q (N ), SL q (N ),

Cited by 18 publications

References 16 publications

Spin geometry on quantum groups via covariant differential calculi

Spin geometry on quantum groups via covariant differential calculi

The serre spectral sequence of a noncommutative fibration for de Rham cohomology

Differential calculi over quantum groups

Contact Info

Product

Resources

About