14 research outputs found
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})
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
Jordanian Quantum Deformations of D=4 Anti-de-Sitter and Poincare Superalgebras
We consider a superextension of the extended Jordanian twist, describing
nonstandard quantization of anti-de-Sitter () superalgebra in
the form of Hopf superalgebra. The super-Jordanian twisting function and
corresponding basic coproduct formulae for the generators of are
given in explicit form. The nonlinear transformation of the classical
superalgebra basis not modifying the defining algebraic relations but
simplifying coproducts and antipodes is proposed. Our physical application is
to interpret the new super-Jordanian deformation of superalgebra as
deformed D=4 supersymmetries. Subsequently we perform suitable
contraction of quantum Jordanian superalgebra and obtain new
-deformation of D=4 Poincare superalgebra, with the bosonic sector
describing the light cone -deformation of Poincare symmetries.Comment: LaTeX,13 pages, comments and references added, to be published in
Eur.Phys.J.
Basic Twist Quantization of osp(1|2) and kappa-- Deformation of D=1 Superconformal Mechanics
The twisting function describing a nonstandard (super-Jordanian) quantum
deformation of is given in explicite closed form. The quantum
coproducts and universal R-matrix are presented. The non-uniqueness of the
twisting function as well as two real forms of the deformed
superalgebras are considered. One real quantum superalgebra is
interpreted as describing the -deformation of D=1, N=1 superconformal
algebra, which can be applied as a symmetry algebra of N=1 superconformal
mechanics.Comment: 13 pages,LaTeX, v2. References added, typos correcte
Once again about quantum deformations of D=4 Lorentz algebra: twistings of q-deformation
This paper together with the previous one (arXiv:hep-th/0604146) presents the
detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf
algebra in terms of complex and real generators. We describe here in detail two
quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of
the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation
an Abelian twist depending on Cartan generators of o(3,1) is used. The second
example of twisting provides a quantum deformation of Cremmer-Gervais type for
the Lorentz algebra. For completeness we describe also twisting of the Lorentz
algebra by standard Jordanian twist. By twist quantization techniques we obtain
for these deformations new explicit formulae for the deformed coproducts and
antipodes of the o(3,1)-generators.Comment: 17 page
N=1/2 Deformations of Chiral Superspaces from New Quantum Poincare and Euclidean Superalgebras
We present a large class of supersymmetric classical r-matrices, describing
the supertwist deformations of Poincare and Euclidean superalgebras. We
consider in detail new family of four supertwists of N=1 Poincare superalgebra
and provide as well their Euclidean counterpart. The proposed supertwists are
better adjusted to the description of deformed D=4 Euclidean supersymmetries
with independent left-chiral and right-chiral supercharges. They lead to new
quantum superspaces, obtained by the superextension of twist deformations of
spacetime providing Lie-algebraic noncommutativity of space-time coordinates.
In the Hopf-algebraic Euclidean SUSY framework the considered supertwist
deformations provide an alternative to the N=1/2 SUSY Seiberg's star product
deformation scheme.Comment: 17 pages, LaTeX, new (re-worked) revised and extended versio
Deformation of orthosymplectic Lie superalgebra osp(1|2)
Triangular deformation of the orthosymplectic Lie superalgebra osp(1|4) is
defined by chains of twists. Corresponding classical r-matrix is obtained by a
contraction procedure from the trigonometric r-matrix. The carrier space of the
constant r-matrix is the Borel subalgebra.Comment: LaTeX, 8 page