Under the assumptions that q is not a root of unity and that the differentials du) of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groups A nu B n ,C n and D n . We prove that apart one dimensional differential calculi and from finitely many values of q, there are precisely In such calculi on the quantum group A n -ι = SL q (n) for n Ξ> 3. All these calculi have the dimension n 2 . For the quantum groups B n , C n and D n we show that except for finitely many q there exist precisely two iV 2 -dimensional bicovariant calculi for N ^ 3, where N = In + 1 for B n and N = 2n for C n ,D n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker ε are determined. In the limit q-*l two of the 2n calculi for A n -\ and one of the two calculi for B n , C n and D n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.

Suppose that q is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groups GL q (N), O q (N) and Sp q (N) for which the differentials άu ι j of the matrix entries u ι j generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi on GL q {N) for N ^ 3. In the limit q -> 1 only two of these calculi give the ordinary differential calculus on GL(N). Our second main theorem states that apart from finitely many q there exist precisely two differential calculi with these properties on O q (N) and Sp q (N) for N ^ 4. This strengthens the corresponding result proved in our previous paper [SS2]. There are four such calculi on O q (3). We introduce two new 4-dimensional bicovariant differential calculi on O q (3).

When humans are about to manipulate an object, our brains use visual cues to recall an internal representation to predict its weight and scale the lifting force accordingly. Such a long-term force profile, formed through repeated experiences with similar objects, has been proposed to improve manipulative performance. Skillful object manipulation is crucial for many animals, particularly those that rely on tools for foraging. However, despite enduring interest in tool use in non-human animals, there has been very little investigation of their ability to form an expectation about an object’s weight. In this study, we tested whether wild chimpanzees use long-term force profiles to anticipate the weight of a nut-cracking hammer from its size. To this end, we conducted a field experiment presenting chimpanzees with natural wooden hammers and artificially hollowed, lighter hammers of the same size and external appearance. We used calibrated videos from camera traps to extract kinematic parameters of lifting movements. We found that, when lacking previous experience, chimpanzees lifted hollowed hammers with a higher acceleration than natural hammers (overshoot effect). After using a hammer to crack open one nut, chimpanzees tuned down the lifting acceleration for the hollowed hammers, but continued lifting natural hammers with the same acceleration. Our results show that chimpanzees anticipate the weight of an object using long-term force profiles and suggest that, similarly to humans, they use internal representations of weight to plan their lifting movements.Electronic supplementary materialThe online version of this article (10.1007/s10071-017-1144-0) contains supplementary material, which is available to authorized users.

Abstract.We study N 2 − 1 dimensional left-covariant differential calculi on the quantum group SLq(N ). In this way we obtain four classes of differential calculi which are algebraically much simpler as the bicovariant calculi. The algebra generated by the left-invariant vector fields has only quadratic-linear relations and posesses a Poincaré-Birkhoff-Witt basis. We use the concept of universal (higher order) differential calculus associated with a given left-covariant first order differential calculus. It turns out that the space of left-invariant k-forms has the dimensionas in the case of the corresponding classical Lie group SL(N ).

Let Γ = Γ τ,z be one of the N 2 -dimensional bicovariant first order differential calculi for the quantum groups GL q (N ), SL q (N ),