The variation procedure on a teleparallel manifold is studied. The main problem is the non-commutativity of the variation with the Hodge dual map. We establish certain useful formulas for variations and restate the master formula due to Hehl and his collaborates. Our approach is different and sometimes easier for applications. By introducing the technique of the variational matrix we find neces...

In this paper we will prove an integral inequality of Stampacchia-type for a fourth-order elliptic operator on complete and connected Kähler manifolds. Our inequality implies a Hodge-Kodaira orthogonal decomposition for the Sobolev-type space W(X). In particular we will able to prove, under suitable topological conditions on the manifold X, the existence of an isomorphism between the Aeppli gro...

We construct the BRST cohomology under a positive-definite inner product and obtain the Hodge decomposition theorem at a non-degenerate state vector space V . The harmonic states isomorphic with a BRST cohomology class correspond to the physical Hilbert space with positive norm as long as the completeness of QBRST is satisfied. We explicitly define a “co-BRST” operator and analyze the quartet m...

In [4], Lin and Sjamaar show how to use symplectic Hodge theory to obtain canonical equivariant extensions of closed forms in Hamiltonian actions of compact connected Lie groups on closed symplectic manifolds which have the strong Lefschetz property. In this paper, we show how to do the same using classical Hodge theory. This has the advantage of applying far more generally. Our method makes us...

In this paper we give a thoughtful exposition of the Clifford algebra Cl(HV ) of multivecfors which is naturally associated with a hyperbolic space HV , whose elements are called vecfors. Geometrical interpretation of vecfors and multivecfors are given. Poincaré automorphism (Hodge dual operator) is introduced and several useful formulas derived. The role of a particular ideal in Cl(HV ) whose ...

The machinery of braided geometry introduced previously is used now to construct the ǫ ‘totally antisymmetric tensor’ on a general braided vector space determined by R-matrices. This includes natural q-Euclidean and q-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as ̃ SOq(4) or ̃ SOq(1, 3). The Hodge ∗ operator and differentials are also cons...

The machinery of braided geometry introduced previously is used now to construct thètotally antisymmetric tensor' on a general braided vector space determined by R-matrices. This includes natural q-Euclidean and q-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as g SO q (4) or g SO q (1; 3). The Hodge operator and diierentials are also constru...

We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the ‘quantum groups approach’. We apply the author’s previous classification theorem, implying that calculi in the factorisable case correspond to blocks in the dual, to classify differential calculi on the quantum codouble C(G)◮<CG = D∗(G) of a finite group G. We give D∗(S3) as an example. We ...

Let X be a complex manifold, and Z an irreducible closed analytic subset. We have the polarizable Hodge Module ICZQ H whose underlying perverse sheaf is the intersection complex ICZQ. See [16]. Let (M,F ) be its underlying filtered DX -Module. ThenM is the unique regular holonomic DX -Module which corresponds to ICZC by the Riemann-Hilbert correspondence [9] [13], and it is relatively easy to d...

The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a smoothly bounded domain in R, and on the half space R + . The novelty is that the topology used is not an L 2 topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary value problem belonging to a class of problems first intr...