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 $n=1$, 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

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, $U_q(gl(2|1))$ 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 $G_2$, $F_4$ and $E_8$ at odd roots of unity are constructed and explicitly computed for all the lens spaces.Comment: LaTeX 10 page

Serre presentations of Lie superalgebras

An analogue of Serre's theorem is established for finite dimensional simple Lie superalgebras, which describes presentations in terms of Chevalley generators and Serre type relations relative to all possible choices of Borel subalgebras. The proof of the theorem is conceptually transparent; it also provides an alternative approach to Serre's theorem for ordinary Lie algebras.Comment: 45 page

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