Free motion on the Poisson SU(n) group

SL(N,C) is the phase space of the Poisson SU(N). We calculate explicitly the symplectic structure of SL(N,C), define an analogue of the Hamiltonian of the free motion on SU(N) and solve the corresponding equations of motion. Velocity is related to the momentum by a non-linear Legendre transformation.Comment: LaTeX, 10 page

Phase spaces related to standard classical rr-matrices

Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding phase spaces (symplectic groupoids).Comment: 20 pages, LaTeX, no figure

Planck-scale relativity from quantum κ\kappa-Poincar\'e algebra

Extending the commutator algebra of quantum $\kappa$-Poincar\'e symmetry to the whole of the phase space, and assuming that this algebra is to be covariant under action of deformed Lorentz generators, we derive the transformation properties of positions under the action of deformed boosts. It turns out that these transformations leave invariant the quadratic form in the position space, which is the Minkowski metric and that the boosts saturate. The issues of massless and massive particles motion, as well as time dilatation and length contraction in this new framework are also studied.Comment: 14 pages, LaTeX, no figure

Constant curvature solutions of Grassmannian sigma models: (1) Holomorphic solutions

We present a general formula for the Gaussian curvature of curved holomorphic 2-spheres in Grassmannian manifolds G(m, n). We then show how to construct such solutions with constant curvature. We also make some relevant conjectures for the admissible constant curvatures in G(m, n) and give some explicit expressions, in particular, for G(2, 4) and G(2, 5).Comment: 14 page

Exploring the Process of Being Me as the Central Focus of Our Spiritual Journey

The struggle of being me, or living my true self, has been a longtime wrestling match for humanity that has not always produced a raised-arm victor. Personal identity struggles are as old as human life and can be traced all the way back to Adam and Eve. Yet every human being has that same desire: to know themselves and be known and still be accepted for who they really are. It has raised its head through people living in the shadow of rejection and struggling within their relationships. Yet, I claim that the struggle of being me is designed by God to be humankind\u27s primary spiritual journey and that being me can only be discovered through converging two life roads: God-sensing and people-loving. Other solutions to being me have pervaded popular society. Psychology has focused on the retraining of the mind, sociology the retraining of behavior, science the retraining of personality type, religion the retraining of morals, and spirituality the retraining of the soul. Each has made progress in some part, but has not completely been successful in helping people understand and know themselves as part of a God-intended spiritual journey. I plan to show that being me is humankind\u27s primary spiritual journey because: it is God\u27s 01iginal plan; it is humankind\u27s innate drive; and it is a lifelong spiritual journey. I also plan to show that being me can be discovered through two roads: the road of God-sensing because it begins in sensing God and results in congruent holiness; and the road of people-loving because being me is defined in life relationships and creates freedom to love others. The thesis will be presented in the project of writing a personal book that will target both a Christian and non-Christian audience using a style that will draw from personal experiences

Some Comments on BPS systems

We look at simple BPS systems involving more than one field. We discuss the conditions that have to be imposed on various terms in Lagrangians involving many fields to produce BPS systems and then look in more detail at the simplest of such cases. We analyse in detail BPS systems involving 2 interacting Sine-Gordon like fields, both when one of them has a kink solution and the second one either a kink or an antikink solution. We take their solitonic static solutions and use them as initial conditions for their evolution in Lorentz covariant versions of such models. We send these structures towards themselves and find that when they interact weakly they can pass through each other with a phase shift which is related to the strength of their interaction. When they interact strongly they repel and reflect on each other. We use the method of a modified gradient flow in order to visualize the solutions in the space of fields.Comment: 27 pages, 17 figure