Cayley-Klein Groups Research Articles

Контракции калибровочных групп и спонтанное нарушение симметрии

Contractions of gauge models with orthogonal Cayley-Klein groups SO(2; ϵ), SO(3; ϵ), unitary groups SU(2; ϵ) as gauge groups are studied. In the limit of zero contraction parameters, orthogonal groups are isomorphic to the nonsemisimple Euclidean and Newton groups of the corresponding dimension, and the spaces of matter fields become fibered spaces with a degenerate metric. Particular attention is paid to the coordination of spontaneous symmetry breaking with the group contraction procedure. It is shown that contracted gauge theories describe the same set of fields with the same masses as theories with the original simple groups, if the chosen vacuum in the corresponding limit belonged to the base of the fibered space of matter fields. Lagrangians of the models depending on the contraction parameters are obtained, which makes it possible to trace the order of zeroing of terms in the Lagrangians as the contraction parameters tend to zero.

Thermodynamic properties of quantum models based on the complex unitary Cayley-Klein groups

In this work, the goal is to reach the thermodynamics properties produced by a family of the integrable Hamiltonians of the unitary groups of rank one in the Cayley-Klein space. In fact, this makes it possible through the general commutator relations between their generators of the desired Cayley-Klein group under investigation. On the other hand, by converting the coordinates to the Euler parameterization, the metric obtained from the Casimir is rewritten in terms of these coordinates, which is equal to the same the Lagrangian of the geodesic motion of the corresponding space. By matching coordinates, Lagrangian is written in terms of canonical momenta, and so the Hamiltonian can be obtained in two forms: one for a free particle and another for an involved particle with the integral potential. Hence, by solving the Hamiltonian of each of the two states in a different way(NU and SUSY QM methods), the quantum model for its corresponding system is shown in terms of the contraction parameter (κ1). Finally, with finding eigenvalues for each of these two systems, their thermodynamic properties are discussed with changing two parameters λ and δ, respectively. In our study, one can only see a phase transition in the heat capacity diagram at critical point τ0. In other words, it can be stated that a phase transition of the second type possible occurs for free particle in high temperature. Also, other results are reported in the end.

Method of categorical extension of Cayley-Klein groups

The method of categorical extension of the Cayley-Klein groups is developed. The method uses the Cayley-Klein spaces as objects of the Cayley-Klein category endowed with all possible linear relations or bilinear forms as morphisms.

Quantum Orthogonal Cayley-Klein Groups in Cartesian Basis

The similarity transformations of quantum orthogonal groups are developed and FRT theory is reformulated to the Cartesian basis. The quantum orthogonal Cayley-Klein groups are introduced as the algebra functions over an associative algebra with the nilpotent generators. The quantum orthogonal Cayley-Klein algebras are obtained as the dual objects to the corresponding quantum groups.

The Quantum Symplectic Cayley-Klein Groups

The contraction method applied to the construction of the nonsemisimple quantum symplectic Cayley-Klein groups Fun(Spq(n;j)). This groups has been realised as Hopf algebra of the noncommutative functions over the algebra with nilpotent generators. The dual quantum algebras spq(n;j) are constructed.

The matrix quantum unitary Cayley-Klein groups

The author defines the matrix quantum unitary Cayley-Klein groups of arbitrary dimensions by generalizing the R-matrix theory of the quantum groups to the case of degenerate Hermitic forms. The corresponding quantum groups are obtained from the appropriate classical quantum groups by contractions.

Transitions: Contractions and analytical continuations of the Cayley-Klein Groups

As a foundation for Klein's fundamental idea about the connection of geometry and its motion group and the unified description of all Cayley-Klein geometries, a method of group transitions including contractions as well as analytical continuations of the groups is developed. The generators and Casimir operators of an arbitrary Cayley-Klein group are obtained from those of the classical orthogonal group. The classification of all possible transitions between the Cayley-Klein groups is given. The physically important case of the kinematic groups is discussed.

The Jordan–Schwinger representations of Cayley–Klein groups. I. The orthogonal groups

The Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the orthogonal groups. The Jordan–Schwinger representations of Cayley–Klein groups are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent and Fock states.

The Jordan–Schwinger representations of Cayley–Klein groups. III. The symplectic groups

The symplectic Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the classical symplectic groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

The Jordan–Schwinger representations of Cayley–Klein groups. II. The unitary groups

The unitary Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.