This is the second part of the paper (the rst part is published in Journal of AMS 9 1135. In the rst part, we deened for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space Hom(L X i X i ; U) is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group U q sl n...

An invariant mean on a group G is a normalized, positive, translation invariant linear functional defined on the space of all bounded complex valued functions on G. Some groups possess an invariant mean (or are said to be amenable), while others do not. In particular, all abelian groups are amenable [2, §17.5]. An invariant mean on a group need not be inversion invariant. It is quite easy to sh...

using a generalized spherical mean operator, we obtain a generalization of titchmarsh&apos;s theorem for the dunkl transform for functions satisfying the (&apos;; p)-dunkl lipschitz condition in the space lp(rd;wl(x)dx), 1 < p 6 2, where wl is a weight function invariant under the action of an associated re ection group.

We describe an algorithm for splitting a permutation representation of a finite group into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection into these subspaces. An important element of the algorithm is the calculation of Gröbner bases of polynomial ideals. A preliminary implementatio...

Abstract. We define a BV-structure on the Hochschild-cohomology of a unital, associative algebra A with a symmetric, invariant and non-degenerate inner product. The induced Gerstenhaber algebra is the one described in Gerstenhaber’s original paper on Hochschild-cohomology. We also prove the corresponding theorem in the homotopy case, namely we define the BV-structure on the Hochschild-cohomolog...

We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad O, generalizing the construction already known for the associative operad. This is done by defining a colored operad b O, which describes modules over O with invariant inner products. We show that b O satisfies Koszulness and identify algebras over a resolution of b O in terms of derivations and module map...

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden’s identity, the expression fo...

An Ad K invariant inner product on the Lie algebra of a compact connected Lie group K extends to a Hermitian inner product on the Lie algebra of the complexified Lie group Kc. The Laplace-Beltrami operator, ∆, on Kc induced by the Hermitian inner product determines, for each number a > 0, a Green’s function ra by means of the identity (a2−∆/4)−1 = ra∗. The Hilbert space of holomorphic functions...

In model-based control design one often has to describe the plant by a linear model. Deriving such a model poses issues of parameterization, estimation, and validation of the model before designing the controller. In this paper, a direct data-driven control method is proposed for designing controllers that can handle constraints without deriving a model of the plant and directly from data. A hi...