Discusses certain one-parameter subgroups of the low-rank unitary groups called root subgroups. Unitary groups also have representations of Lie type which means they consist of transformations that act as automorphisms of an underlying Lie algebra, in this case the special linear algebra. Exploring this definition of the unitary groups, we find a correlation, via exponentiation, to the basis elements of Lie algebra.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 1978 / Vita. / Bibliography: leaves 96-97. / Ph. D. / Ph. D. Massachusetts Institute of Technology, Department of Mathematics

In this scientific work, we use two effective methods : Lie groups theory and the finite element method, to explain why the transition from laminar flow to turbulence flow depends on the variation of the Reynolds number. We restrict ourselves to the case of incompressible viscous Newtonian fluid flows. Their governing equations, i.e. the continuity and Navier-Stokes equations are established and investigated. Their solutions are expressed explicitly thanks to Lie's theory. The stability theory, which leads to an eigenvalue problem is used together with the finite element method, showing a way to compute the critical Reynolds number, for which the transition to turbulence occurs. The stationary flow is also studied and a finite element method, the Newton method, is used to prove the stability of its convergence, which is guaranteed for small variations of the Reynolds number. / Mathematical Sciences / M.Sc. (Applied Mathematics)

We present a geometric construction of almost Poisson brackets for nonholonomic mechanical systems whose configuration space is a Lie group G. We study the so-called LL and LR systems where the kinetic energy defines a left invariant metric on G and the constraints are invariant with respect to left (respectively right) translation on G.For LL systems, the equations on the momentum phase space, T*G, can be left translated onto g*, the dual space of the Lie algebra g. We show that the reduced equations on g* can be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the standard Lie-Poisson bracket onto the constraint space.For LR systems, we use ideas of semidirect product reduction to transfer the equations on T*G into the dual Lie algebra, s*, of a semidirect product. This provides a natural Lie algebraic setting for the equations of motion commonly found in the literature. We show that these equations can also be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the Lie-Poisson structure on s* onto a constraint submanifold.In both cases the constraint functions are Casimirs of the bracket and are satisfied automatically. Our construction is a natural generalization of the classical ideas of Lie-Poisson and semidirect product reduction to the nonholonomic case. It also sets a convenient stage for the study of Hamiltonization of certain nonholonomic systems.Our examples include the Suslov and the Veselova problems of constrained motion of a rigid body, and the Chaplygin sleigh.In addition we study the almost Poisson reduction of the Chaplygin sphere. We show that the bracket given byBorisov and Mamaev is obtained by reducing a nonstandard almost Poisson bracket that is obtained by projecting a non-canonical bivector onto the constraint submanifold using the Lagrange-D'Alembert principle.The examples that we treat show that it is possible to cast the reduced equations of motion of certain nonholonomic systems in Hamiltonian form (in the Poisson formulation) either by multiplication by a conformal factor, by the use of nonstandard brackets or simply by reduction methods.

The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.

Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of S∞.

A large part of the structure of the objects in the theory of Dooley and Wildberger [Funktsional. Anal. I Prilozhen. 27 (1993), no. 1, 25-32] and that of Rouviere [Compositio Math. 73 (1990), no. 3, 241-270] can be described by considering a connected, finite-dimentional symmetric space G/H (as defined by Rouviere), with ???exponential map???, Exp, from L G/L H to G/H, an action, ???: K ??? Aut??(G) (where Aut?? (G) is the projection onto G/H of all the automorphisms of G which leave H invariant), of a Lie group, K, on G/H and the corresponding action, ???# , of K on L G/L H defined by g ??? L (???g), along with a quadruple (s, E, j, E#), where s is a ???# - invariant, open neighbourhood of 0 in L G/L H, E is a test-function subspace of C??? (Exp s), j ?? C??? (s), and E# is a test-function subspace of C??? (s) which contains { j.f Exp: f ?? E }. Of interest is the question: Is the function ???: ?? ??? ????, where ??: f ??? j.f Exp, a local associative algebra homomorphism from F# with multiplication defined via convolution with respect to a function e: s x s ??? C, to F, with the usual convolution for its multiplication (where F is the space of all ??? - invariant distributions of E and F# is the space of all ???# - invariant distributions of E#)? For this system of objects, we can show that, to some extent, the choice of the function j is not critical, for it can be ???absorbed??? into the function e. Also, when K is compact, we can show that ??? ker ?? = { f ?? E : ???k f (???g) dg = 0}. These results turn out to be very useful for calculations on s2 ??? G/H, where G = SO(3) and H??? SO(3) with H ??? SO(2) with ??? : h ??? Lh, as we can use these results to show that there is no quadruple (s, E, j, E#) for SO(3)/H with j analytic in some neighbourhood of 0 such that ??? is a local homomorphism from F# to F. Moreover, we can show that there is more than one solution for the case where s, E and E# are as chosen by Rouviere, if e is does not have to satisfy e(??,??) = e(??,??).

Representations of SO(4,2;R) are constructed using 4 x 4 and 2 x 2 matrices with elements in H' ��� C . Using 2 x 2 matrix representations of C and H', the 4 x 4 representation is interpreted in terms of 16 x 16 real matrices. Finally, the known isomorphism between the conformal group and SO(4,2;R) is written explicitly in terms of the 4 x 4 representation. The 4 x 4 construction should generalize to matrices with elements in K' ��� K for K any normed division algebra over the reals and K' any split algebra over the reals. / Graduation date: 2013

No description available.

No description available.