Journal of Generalized Lie Theory and Applications

A differential calculus of the first order over multi-braided quantum groups is developed. In analogy with the standard theory, left/right-covariant and bicovariant differential structures are introduced and investigated. Furthermore, antipodally covariant calculi are studied. The concept of the *-structure on a multi-braided quantum group is formulated, and in particular the structure of left-covariant *-covariant calculi is analyzed. These structures naturally incorporate the idea of the quantum Lie algebra associated to a given multibraded quantum group, the space of left-invariant forms corresponding to the dual of the Lie algebra itself. A special attention is given to differential calculi covariant with respect to the action of the associated braid system. In particular it is shown that the left/right braided-covariance appears as a consequence of the left/right-covariance relative to the group action. Braided counterparts of all basic results of the standard theory are found.

We consider Lie algebras of dimension 3 up to isomorphism. We construct a noncommutative ane spectrum of the isomorphism classes as a noncommutative k-algebra M, using noncommutative deformation theory. This k-algebra is an example of a noncommutative structure.

It is explained how the time evolution of the operadic variables may be introduced by using the operadic Lax equation. The operadic Lax representations for the harmonic oscilla- tor are constructed in 3-dimensional binary anti-commutative algebras. As an example, an operadic Lax representation for the harmonic oscillator in the Lie algebra sl(2) is constructed.

A time transformation technique for Nambu–Poisson systems is developed, and its structural properties are examined. The approach is based on extension of the phase space P into P¯ = P×R, where the additional variable controls the time-stretching rate. It is shown that time transformation of a system on P can be realised as an extended system on P¯, with an extended Nambu–Poisson structure. In addition, reversible systems are studied in conjunction with the Nambu–Poisson structure. The application in mind is adaptive numerical integration by splitting of Nambu–Poisson Hamiltonians. As an example, a novel integration method for the rigid body problem is presented and analysed.

It is shown that some operators in quantum mechanics have hidden structures that are unobservable in principle. These structures are based on a supersymmetric decomposition of the momentum operator, and a nonassociative decomposition of the spin operator.