Recently the correlation functions of the so–called Itzykson-Zuber/Harish-Chandra integrals were computed (by one of the authors and collaborators) for all classical groups using an integration formula that relates integrals over compact groups with respect to the Haar measure and Gaussian integrals over a maximal nilpotent Lie subalgebra of their complexification. Since the integration formula...

The main difficulty in the nonrelativistic case is that one can only use a smaller group of commuting vector fields. In particular, since one cannot use the generators of the hyperbolic rotations, due to the different wave speeds, the earlier approach of Klainerman [9] breaks down. This is because the invariant Sobolev inequality that plays a key role in [9] does not hold if one uses a smaller ...

M. O'Sullivan and P. Ekman (2004) claim to have discovered 29 wizards of deception detection. The present commentary offers a statistical critique of the evidence for this claim. Analyses reveal that chance can explain results that the authors attribute to wizardry. Thus, by the usual statistical logic of psychological research, O'Sullivan and Ekman's claims about wizardry are gratuitous. Even ...

Let g be a simple, finite-dimensional Lie superalgebra over C. These have been classified by V. Kac. Unless g is a Lie algebra or a Lie superalgebra of type osp(1, 2n), the category of finite-dimensional representations of g is not semisimple; q.v. [6]. This leads to a classification problem. For example, in [3], the representation theory of sl(m,n) is worked out by showing it is wild when m,n ...

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II1 factor M containing an “exotic” maximal abelian subalgebra A: as an A,A-bimodule, L(M) is neither coarse nor discrete. Thus we show that there exist II1 factors with such property but without Cartan subalgebras. It also follows from Voiculescu’s free entropy results that M is not an inter...

Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac–Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwist...

We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left invariant vector fields. We study the duality between vector fields and 1-forms and generalize the construction to tensor fields. A Lie derivative along any (also non left invariant) vector field is proposed and a puzzling ambiguity in its definition di...

We present a systematic procedure for the determination of a complete set of kth-order (k ≥ 2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that th...

We completely describe the decompositions (into indecomposable submodules) of the tensor products of irreducible sl(2)-modules in characteristic 3. The answer resembles analogous decompositions for the Lie superalgebra sl(1|1).