There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

A detailed study is made of the noncommutative geometry of Rq, the quantum space covariant under the quantum group SOq(3). For each of its two SOq(3)-covariant differential calculi we find its metric, the corresponding frame and two torsion-free covariant derivatives that are metric compatible up to a conformal factor and which yield both a vanishing linear curvature. A discussion is given of v...

The phase space of a particle on a group manifold can be split in left and right sectors, in close analogy with the chiral sectors in Wess Zumino Witten models. We perform a classical analysis of the sectors, and the geometric quantization in the case of SU(2). The quadratic relation, classically identifying SU(2) as the sphere S3, is replaced quantum mechanically by a similar condition on non-...

This article is a further contribution to our research [1] into a class of infinite-dimensional Lie algebras L∞(N+, N−) generalizing the standard W∞ algebra, viewed as a tensor operator algebra of SU(1, 1) in a group-theoretic framework. Here we interpret L∞(N+, N−) either as a infinite continuation of pseudo-unitary symmetries U(N+, N−), or as a “higher-U(N+, N−)-spin extension” of the diffeom...

Much of twentieth century physics, whether it be Classical or Quantum, has been based on the concept of spacetime as a differentiable manifold. While this work has culminated in the standard model, it is now generally accepted that in the light of recent experimental results, we have to go beyond the standard model. On the other hand Quantum SuperString Theory and a recent model of Quantized Sp...

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization ...

In [6], employing commutative algebra, we showed that if the number of principal curvatures is 4 and if the multiplicities m1 and m2 of the principal curvatures satisfy m2 ≥ 2m1 − 1, then the isoparametric hypersurface is of the type constructed by Ozeki-Takeuchi and Ferus-Karcher-Münzner [18], [11]. This leaves only four multiplicity pairs (m1, m2) = (3, 4), (4, 5), (6, 9) and (7, 8) unsettled...

Let L be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold (X,ω). We assume that the first homology group H1(L;A) with coefficients in a commutative ring A injects into the group H1(X;A) and that X contains no Maslov zero pseudoholomorphic disc with boundary on L. Then, we prove that for every generic choice of a tame almost-complex structure J on X, every relative ...

We prove that if N is a closed simply connected manifold and j : L →֒ T ∗N is an exact Lagrangian embedding, then H2(N) → H2(L) is injective and the image of π2(L) → π2(N) has finite index. Viterbo proved that there is a transfer map on free loopspaces H∗(L0N) → H∗(L0L) which commutes under the inclusion of constant loops with the ordinary transfer map H∗(N) → H∗(L). This commutative diagram sti...

We construct a cubical CW-complex CK(M3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M3. We construct CK(R3) by attaching cells to CK(R3) for every degenerate 1-singular and 2-singular knot, and we show that π1(CK(R 3)) = 1 and π2(CK(R 3)) = Z. We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R3 we show...