513 research outputs found
Spin networks, quantum automata and link invariants
The spin network simulator model represents a bridge between (generalized)
circuit schemes for standard quantum computation and approaches based on
notions from Topological Quantum Field Theories (TQFT). More precisely, when
working with purely discrete unitary gates, the simulator is naturally modelled
as families of quantum automata which in turn represent discrete versions of
topological quantum computation models. Such a quantum combinatorial scheme,
which essentially encodes SU(2) Racah--Wigner algebra and its braided
counterpart, is particularly suitable to address problems in topology and group
theory and we discuss here a finite states--quantum automaton able to accept
the language of braid group in view of applications to the problem of
estimating link polynomials in Chern--Simons field theory.Comment: LateX,19 pages; to appear in the Proc. of "Constrained Dynamics and
Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 200
Knot invariants and higher representation theory
We construct knot invariants categorifying the quantum knot variants for all
representations of quantum groups. We show that these invariants coincide with
previous invariants defined by Khovanov for sl_2 and sl_3 and by
Mazorchuk-Stroppel and Sussan for sl_n.
Our technique is to study 2-representations of 2-quantum groups (in the sense
of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible
representations. These are the representation categories of certain finite
dimensional algebras with an explicit diagrammatic presentation, generalizing
the cyclotomic quotient of the KLR algebra. When the Lie algebra under
consideration is , we show that these categories agree with
certain subcategories of parabolic category O for gl_k.
We also investigate the finer structure of these categories: they are
standardly stratified and satisfy a double centralizer property with respect to
their self-dual modules. The standard modules of the stratification play an
important role as test objects for functors, as Vermas do in more classical
representation theory.
The existence of these representations has consequences for the structure of
previously studied categorifications. It allows us to prove the non-degeneracy
of Khovanov and Lauda's 2-category (that its Hom spaces have the expected
dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke
algebras are symmetric Frobenius.
In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps
between representations of quantum groups are used to define polynomial knot
invariants. We show that the categorifications of tensor products are related
by functors categorifying these maps, which allow the construction of bigraded
knot homologies whose graded Euler characteristics are the original polynomial
knot invariants.Comment: 99 pages. This is a significantly rewritten version of
arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been
significantly improved. These earlier papers have been left up mainly in the
interest of preserving references. v3: final version, to appear in Memoirs of
the AMS. Proof of nondegeneracy moved to separate erratu
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