15 research outputs found
On piecewise trivial Hopf—Galois extensions
We discuss a noncommutative generalization of compact principal bundles that can be trivialized relative to the finite covering by closed sets. In this setting we present bundle reconstruction and reduction
A class of quadratic deformations of Lie superalgebras
We study certain Z_2-graded, finite-dimensional polynomial algebras of degree
2 which are a special class of deformations of Lie superalgebras, which we call
quadratic Lie superalgebras. Starting from the formal definition, we discuss
the generalised Jacobi relations in the context of the Koszul property, and
give a proof of the PBW basis theorem. We give several concrete examples of
quadratic Lie superalgebras for low dimensional cases, and discuss aspects of
their structure constants for the `type I' class. We derive the equivalent of
the Kac module construction for typical and atypical modules, and a related
direct construction of irreducible modules due to Gould. We investigate in
detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie
superalgebra sl(n/1). We formulate the general atypicality conditions at level
1, and present an analysis of zero-and one-step atypical modules for a certain
family of Kac modules.Comment: 26pp, LaTeX. Original title: "Finite dimensional quadratic Lie
superalgebras"; abstract re-worded; text clarified; 3 references added;
rearrangement of minor appendices into text; new subsection 4.
Quantum Bundle Description of the Quantum Projective Spaces
We realise Heckenberger and Kolb's canonical calculus on quantum projective
(n-1)-space as the restriction of a distinguished quotient of the standard
bicovariant calculus for Cq[SUn]. We introduce a calculus on the quantum
(2n-1)-sphere in the same way. With respect to these choices of calculi, we
present quantum projective (N-1)-space as the base space of two different
quantum principal bundles, one with total space Cq[SUn], and the other with
total space Cq[S^(2n-1)]. We go on to give Cq[CP^n] the structure of a quantum
framed manifold. More specifically, we describe the module of one-forms of
Heckenberger and Kolb's calculus as an associated vector bundle to the
principal bundle with total space Cq[SUn]. Finally, we construct strong
connections for both bundles.Comment: 33 pages; minor changes, to appear in Comm. Math. Phy
Dirac Operators on Quantum Projective Spaces
We construct a family of self-adjoint operators D_N which have compact
resolvent and bounded commutators with the coordinate algebra of the quantum
projective space CP_q(l), for any l>1 and 0<q<1. They provide 0^+ dimensional
equivariant even spectral triples. If l is odd and N=(l+1)/2, the spectral
triple is real with KO-dimension 2l mod 8.Comment: 54 pages, no figures, dcpic, pdflate
On the Hochschild (co)homology of quantum homogeneous spaces
The recent result of Brown and Zhang establishing Poincaré duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podleś quantum 2-sphere