Nested Archimedean copulas recently gained interest since they generalize the wellknown class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference it is important to have the density. The present work fills this gap. A general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas in arbitrary dimensions if the number of nesting levels is not too large. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.

A Lie version of Turaev's G-Frobenius algebras from 2dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a g-quasi-Frobenius Lie algebra for g a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra (q, β) together with a left g-module structure which acts on q via derivations and for which β is g-invariant.Geometrically, g-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group G which acts via symplectic Lie group automorphisms. In addition to geometry, g-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, g-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in Rep(g). If g is now equipped with a Lie bialgebra structure, then the categorical formulation of G-Frobenius algebras given in [18] suggests that the Lie version of a G-Frobenius algebra is a quasi-Frobenius Lie object in Rep(D(g)), where D(g) is the associated (semiclassical) Drinfeld double. We show that if g is a quasitriangular Lie bialgebra, then every g-quasi-Frobenius Lie algebra has an induced D(g)-action which gives it the structure of a D(g)-quasi-Frobenius Lie algebra.

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists Dβ(k[G]), β ∈ Z3(G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.

In this note, we extend the idea of G-Frobenius algebras (G-FAs) for G a finite group to the case where G is replaced by a finite groupoid. These new structures, which we call groupoid Frobenius algebras, have twists that are entirely analogous to the universal G-FA twists by Z 2 (G, k × ). The usefulness of these new structures comes from recognizing that there is a large collection of G-FAs that can be regarded as non-trivial groupoid Frobenius algebras. As a consequence of this, the twists associated to groupoid Frobenius algebras can be brought to bear on the problem of twisting G-FAs. In particular, we show that for any integer n ≥ 2, there exists a class of G-FAs with twists by Z n (G, k × ).

For a smooth manifold $M$, it was shown in \cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of \cite{BPH} and construct covariant derivatives along MVFs which are not induced by affine connections on $TM$. We call this more general class of covariant derivatives \textit{higher affine connections}. In addition, we also propose a framework which gives rise to non-induced higher connections; this framework is obtained by equipping the full exterior bundle $\wedge^\bullet TM$ with an associative bilinear form $\eta$. Since the latter can be shown to be equivalent to a set of differential forms of various degrees, this framework also provides a link between higher connections and multisymplectic geometry.Comment: 37 pages; main definition and results generalized; substantial changes after section