in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

It is proved that several usual methods of reduction for ordinary differential equations, that do not come from the Lie theory, are derived from the existence of C∞ -symmetries. This kind of symmetries is also applied to obtain two successive reductions of an equation that lacks Lie point symmetries but is a reduced equation of another one with a three dimensional Lie algebra of point symmetrie...

Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is...

Let g be a Lie algebra over a field F of characteristic zero, let C be a certain tensor category of representations of g, and C a certain category of duals. In [M 4] we associated to C and C by a Tannaka reconstruction a monoid M with a coordinate ring F [M ] of matrix coefficients, as well as a Lie algebra Lie(M). We interpreted the Tannaka monoid M algebraic geometrically as a weak algebraic ...

using the riemann-liouville fractional differintegral operator, the lie theory is reformulated. the fractional poisson bracket over the fractional phase space as 3n state vector is defined to be the fractional lie derivative. its properties are outlined and proved. a theorem for the canonicity of the transformation using the exponential operator is proved. the conservation of its generator is p...

The purpose of this brief note is to show that the set of states reachable from a given initial state for a finite dimensional quantum system is equal to the orbit under the Lie group generated by the Lie algebra generated by the internal and external Hamiltonians, even if this Lie group is not compact. This result has been explicitly mentioned in several talks by the author. This note is not b...

Let G be a connected real Lie group. The universal representation kernel, Ko, oí G is defined as the intersection of all kernels of continuous finite dimensional representations of G. Evidently, Ko is a closed normal subgroup of G, and it is known from a theorem due to Goto (cf. [l, Theorem 7.1]) that G/Ko has a faithful continuous finite dimensional representation. Thus Kg is the smallest norm...