3,725 research outputs found
Generalized Cluster States Based on Finite Groups
We define generalized cluster states based on finite group algebras in
analogy to the generalization of the toric code to the Kitaev quantum double
models. We do this by showing a general correspondence between systems with CSS
structure and finite group algebras, and applying this to the cluster states to
derive their generalization. We then investigate properties of these states
including their PEPS representations, global symmetries, and relationship to
the Kitaev quantum double models. We also discuss possible applications of
these states.Comment: 23 pages, 4 figure
Toric Varieties with NC Toric Actions: NC Type IIA Geometry
Extending the usual actions of toric manifolds by
allowing asymmetries between the various factors, we build
a class of non commutative (NC) toric varieties . We
construct NC complex dimension Calabi-Yau manifolds embedded in
by using the algebraic geometry method. Realizations
of NC toric group are given in presence and absence of
quantum symmetries and for both cases of discrete or continuous spectrums. We
also derive the constraint eqs for NC Calabi-Yau backgrounds
embedded in and work out their
solutions. The latters depend on the Calabi-Yau condition , being the charges of % ;
but also on the toric data of the polygons associated to . Moreover,
we study fractional branes at singularities and show that, due to the
complete reducibility property of group representations,
there is an infinite number of fractional branes. We also give the
generalized Berenstein and Leigh quiver diagrams for discrete and continuous
representation spectrums. An illustrating example is
presented.Comment: 25 pages, no figure
Root systems from Toric Calabi-Yau Geometry. Towards new algebraic structures and symmetries in physics?
The algebraic approach to the construction of the reflexive polyhedra that
yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres
reveals graphs that include and generalize the Dynkin diagrams associated with
gauge symmetries. In this work we continue to study the structure of graphs
obtained from reflexive polyhedra. We show how some particularly defined
integral matrices can be assigned to these diagrams. This family of matrices
and its associated graphs may be obtained by relaxing the restrictions on the
individual entries of the generalized Cartan matrices associated with the
Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras.
These graphs keep however the affine structure, as it was in Kac-Moody Dynkin
diagrams. We presented a possible root structure for some simple cases. We
conjecture that these generalized graphs and associated link matrices may
characterize generalizations of these algebras.Comment: 24 pages, 6 figure
Extensions of some classical local moves on knot diagrams
In the present paper, we consider local moves on classical and welded
diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation,
Delta, fused, band-pass and welded band-pass moves. Interrelationship between
these moves is discussed and, for each of these move, we provide an algebraic
classification. We address the question of relevant welded extensions for
classical moves in the sense that the classical quotient of classical object
embeds into the welded quotient of welded objects. As a by-product, we obtain
that all of the above local moves are unknotting operations for welded (long)
knots. We also mention some topological interpretations for these combinatorial
quotients.Comment: 18 pages; this paper is an entirely new version of "On forbidden
moves and the Delta move": the exposition has been totally revised, and
several new results have been added; to appear in Michigan Math.
Cyclic Homology and Quantum Orbits
A natural isomorphism between the cyclic object computing the relative cyclic
homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic
object computing the cyclic homology of a Galois coalgebra with SAYD
coefficients is presented. The isomorphism can be viewed as the
cyclic-homological counterpart of the Takeuchi-Galois correspondence between
the left coideal subalgebras and the quotient right module coalgebras of a Hopf
algebra. A spectral sequence generalizing the classical computation of
Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous
quotient-coalgebra-Galois extensions is constructed. A Pontryagin type
self-duality of the Takeuchi-Galois correspondence is combined with the cyclic
duality of Connes in order to obtain dual results on the invariant cyclic
homology, with SAYD coefficients, of algebras of invariants in homogeneous
quotient-coalgebra-Galois extensions. The relation of this dual result with the
Chern character, Frobenius reciprocity, and inertia phenomena in the local
Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the
Clifford theory is discussed
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