428,644 research outputs found
Dual canonical bases for the quantum general linear supergroup
Dual canonical bases of the quantum general linear supergroup are constructed
which are invariant under the multiplication of the quantum Berezinian. By
setting the quantum Berezinian to identity, we obtain dual canonical bases of
the quantum special linear supergroup {\s O}_q(SL_{m\mid n}). We apply the
canonical bases to study invariant subalgebras of the quantum supergroups under
left and right translations. In the case , it is shown that each invariant
subalgebra is spanned by a part of the dual canonical bases. This in turn leads
to dual canonical bases for any Kac module constructed by using an analogue of
Borel-Weil theorem.Comment: 32 page
Interatomic collisions in two-dimensional and quasi-two-dimensional confinements with spin-orbit coupling
We investigate the low-energy scattering and bound states of two
two-component fermionic atoms in pure two-dimensional (2D) and quasi-2D
confinements with Rashba spin-orbit coupling (SOC). We find that the SOC
qualitatively changes the behavior of the 2D scattering amplitude in the
low-energy limit. For quasi-2D systems we obtain the analytic expression for
the effective-2D scattering amplitude and the algebraic equations for the
two-atom bound state energy. Based on these results, we further derive the
effective 2D interaction potential between two ultracold atoms in the quasi-2D
confinement with Rashba SOC. These results are crucial for the control of the
2D effective physics in quasi-2D geometry via the confinement intensity and the
atomic three-dimensional scattering length.Comment: 13pages, 5 figure
Minimal Uncertainty States For Quantum Groups
The problem of how to obtain quasi-classical states for quantum groups is
examined. A measure of quantum indeterminacy is proposed, which involves
expectation values of some natural quantum group operators. It is shown that
within any finite dimensional irreducible representation, the highest weight
vector and those unitarily related to it are the quasi-classical states.Comment: 4 pages, late
Quantum superalgebra representations on cohomology groups of non-commutative bundles
Quantum homogeneous supervector bundles arising from the quantum general
linear supergoup are studied. The space of holomorphic sections is promoted to
a left exact covariant functor from a category of modules over a quantum
parabolic sub-supergroup to the category of locally finite modules of the
quantum general linear supergroup. The right derived functors of this functor
provides a form of Dolbeault cohomology for quantum homogeneous supervector
bundles. We explicitly compute the cohomology groups, which are given in terms
of well understood modules over the quantized universal enveloping algebra of
the general linear superalgebra.Comment: 24 page
Quantum supergroups and topological invariants of three - manifolds
The Reshetikhin - Turaeve approach to topological invariants of three -
manifolds is generalized to quantum supergroups. A general method for
constructing three - manifold invariants is developed, which requires only the
study of the eigenvalues of certain central elements of the quantum supergroup
in irreducible representations. To illustrate how the method works,
at odd roots of unity is studied in detail, and the
corresponding topological invariants are obtained.Comment: 22 page
Topological Invariants For Lens Spaces And Exceptional Quantum Groups
The Reshetikhin - Turaev invariants arising from the quantum groups
associated with the exceptional Lie algebras , and at odd
roots of unity are constructed and explicitly computed for all the lens spaces.Comment: LaTeX 10 page
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