Representations Of Uq Research Articles

Smirnov's integrals and the quantum Knizhnik-Zamolodchikov equation of level 0

We study the quantum Knizhnik-Zamolodchikov equation (1984) of level 0 associated with the spin 1/2 representation of Uq(SL2). We find an integral formula for solutions in the case of an arbitrary total spin and mod q mod <1. In the formula, different solutions can be obtained by taking different integral kernels with the cycle of integration being fixed.

Unitary highest weight representation of Uq(su(1,1)) when q is a root of unity

Unitary representation of Uq(su(1,1)) with q being roots of unity are studied. The authors construct unitary irreducible highest weight modules and find that the representations are discrete series owing to the unitarity. Moreover, it is revealed that each unitary irreducible highest weight module is equivalent to the tensor product of two modules. They show that one of them is just the unitary irreducible highest weight module of su(1,1) and the other is the same as that of Uq(su(2)). It is also shown that the number of the latter modules is finite in their representation.

Quantum and classical mechanics of q-deformed systems

Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of Uq(m) in a space of functions on a phase space spanned by commutative coordinates is constructed.

Irreducible highest weight representations of quantum groupsU q (gl(n, C))

Explicit recurrence formulas of canonical realization (boson representation) for quantum enveloping algebrasUq(gl(n, C)) are given. Using them, irreducible highest weight representations ofUq(gl(n, C)) are obtained as restriction of representation of Fock space to invariant subspace generated by vacuum as a cyclic vector.

The q-deformed boson realization of representation of quantum universal enveloping algebras for q a root of unity: II. The subalgebra chain of Uq(Cl)

For pt.I see ibid., vol.24, p.3265, (1991). Applying the method of the q-deformed boson realization for the quantum group to the case where q is a root of unity the authors establish a set of standard basis for the representation space of the quantum universal enveloping algebra Uq(Cl) associated with a typical subalgebra chain of Uq(Cl). On this basis they systematically obtain irreducible and indecomposable representations of Uq(Cl) and its subalgebras. They discuss Uq(C2) in detail.

Braid group action on theq-Weyl algebra

We provide a braid group action on theq-deformed Weyl algebraW q (n). The restriction of this action to the representations ofU q (A n−1 ) andU q (C n ) inW q (n) is seen to agree with the braid group action introduced by Lusztig on these quantum algebras.