370 research outputs found
Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
We employ new calculational technique and present complete list of classical
-matrices for complex homogeneous orthogonal Lie algebra
, the rotational symmetry of four-dimensional
complex space-time. Further applying reality conditions we obtain the classical
-matrices for all possible real forms of :
Euclidean , Lorentz , Kleinian
and quaternionic Lie algebras.
For we get known four classical Lorentz -matrices,
but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we provide
new results and mention some applications.Comment: 13 pages; typos corrected. v3 matches version published in PL
Quantum deformations of Euclidean, Lorentz, Kleinian and quaternionic symmetries in unified setting -- Addendum
In our previous paper we obtained a full classification of nonequivalent
quasitriangular quantum deformations for the complex Euclidean Lie
symmetry . The result was presented in the form of
a list consisting of three three-parameter, one two-parameter and one
one-parameter nonisomorphic classical -matrices which provide 'directions'
of the nonequivalent quantizations of . Applying
reality conditions to the complex -matrices we
obtained the nonisomorphic classical -matrices for all possible real forms
of : Euclidean , Lorentz
, Kleinian and quaternionic
Lie algebras. In the case of and
real symmetries these -matrices give the full
classifications of the inequivalent quasitriangular quantum deformations,
however for and the
classifications are not full. In this paper we complete these classifications
by adding three new three-parameter -real -matrices and
one new three-parameter -real -matrix. All
nonisomorphic classical -matrices for all real forms of
are presented in the explicite form what is
convenient for providing the quantizations. We will mention also some
applications of our results to the deformations of space-time symmetries and
string -models.Comment: 10 pages. We supplement results of our previous paper by adding new
and -matrices needed for the
complete classification of real classical -matrices for all four real
forms of $\mathfrak{o}(4;\mathbb{C})
Invariant solutions and Noether symmetries in Hybrid Gravity
Symmetries play a crucial role in physics and, in particular, the Noether
symmetries are a useful tool both to select models motivated at a fundamental
level, and to find exact solutions for specific Lagrangians. In this work, we
consider the application of point symmetries in the recently proposed
metric-Palatini Hybrid Gravity in order to select the functional
form and to find analytical solutions for the field equations and for the
related Wheeler-DeWitt (WDW) equation. We show that, in order to find out
integrable models, conformal transformations in the Lagrangians
are extremely useful. In this context, we explore two conformal transformations
of the forms and . For the former conformal
transformation, we found two cases of functions where the field
equations admit Noether symmetries. In the second case, the Lagrangian reduces
to a Brans-Dicke-like theory with a general coupling function. For each case,
it is possible to transform the field equations by using normal coordinates to
simplify the dynamical system and to obtain exact solutions. Furthermore, we
perform quantization and derive the WDW equation for the minisuperspace model.
The Lie point symmetries for the WDW equation are determined and used to find
invariant solutions.Comment: 12 pages, 1 figur
Jordanian Twist Quantization of D=4 Lorentz and Poincare Algebras and D=3 Contraction Limit
We describe in detail two-parameter nonstandard quantum deformation of D=4
Lorentz algebra , linked with Jordanian deformation of
. Using twist quantization technique we obtain
the explicit formulae for the deformed coproducts and antipodes. Further
extending the considered deformation to the D=4 Poincar\'{e} algebra we obtain
a new Hopf-algebraic deformation of four-dimensional relativistic symmetries
with dimensionless deformation parameter. Finally, we interpret
as the D=3 de-Sitter algebra and calculate the contraction
limit ( -- de-Sitter radius) providing explicit Hopf algebra
structure for the quantum deformation of the D=3 Poincar\'{e} algebra (with
masslike deformation parameters), which is the two-parameter light-cone
-deformation of the D=3 Poincar\'{e} symmetry.Comment: 13 pages, no figure
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