Exterior Algebra Of Forms Research Articles

The standard complex of quantum enveloping algebras

We consider quantized derivation representations of Hopf algebras in some associative algebras . The algebra of quantized differential operators of bialgebra representations, which are special representations by derivations, are constructed. For the quantum enveloping algebras of Lie algebras associated with the root systems , , and we define a deformation of the exterior algebra of forms, and by applying the above results it is shown that the quantized adjoint representation of induces a bialgebra representation of in . The resulting algebra of differential operators is a deformation of the standard Koszul complex of Lie algebras. It admits an exterior derivative which is, in particular, a -module morphism. Hence cohomology groups of relative to some -module can be constructed.

Tangent lifts of Poisson and related structures

The derivation $d_T$ on the exterior algebra of forms on a manifold $M$ with values in the exterior algebra of forms on the tangent bundle $TM$ is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

The contact ideal

If Y is a fibered manifold over a base manifold X, a differential form ρ, defined on the (finite) τ-jet prolongation J τ Y of Y, is said to be contact, if it vanishes along the τ-jet prolongation J τ γ of every section γ of Y, i.e., (J τγ)∗ρ = 0 for all γ. The contact forms define a subcomplex of the de Rham complex on J τ Y, and an ideal in the exterior algebra of forms on J τ Y, called the contact ideal. The contact ideal is not generated by linear forms. Together with contact forms, we consider a modified notion of a strongly contact form which leads to a modified subcomplex of the de Rham complex. The local structure of all contact, and strongly contact forms is described. Applications to the higher order variational calculus on fibered manifolds are given.

Cuntz deformations of the exterior algebra

We study here possible deformations of the usual exterior algebra of forms in an N-dimensional space X. For consistency reasons, these deformations are Cuntz derivations on the commutative algebra of functions of X and, moreover, they are solutions of the quantum Yang-Baxter equation. Finally, consistent deformations of the exterior algebra in the N = 1 and N = 2 cases are explicitly constructed and the relation of the present approach to the differential calculus on quantum spaces is briefly discussed.

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2)L ⊗ SU(2)R ⊗ U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.