280 research outputs found
Generating loop graphs via Hopf algebra in quantum field theory
We use the Hopf algebra structure of the time-ordered algebra of field
operators to generate all connected weighted Feynman graphs in a recursive and
efficient manner. The algebraic representation of the graphs is such that they
can be evaluated directly as contributions to the connected n-point functions.
The recursion proceeds by loop order and vertex number.Comment: 22 pages, LaTeX + AMS + eepic; new section with alternative recursion
formula added, further minor changes and correction
Structure of semisimple Hopf algebras of dimension
Let be prime numbers with , and an algebraically closed
field of characteristic 0. We show that semisimple Hopf algebras of dimension
can be constructed either from group algebras and their duals by means
of extensions, or from Radford biproduct R#kG, where is the group
algebra of group of order , is a semisimple Yetter-Drinfeld Hopf
algebra in of dimension . As an application,
the special case that the structure of semisimple Hopf algebras of dimension
is given.Comment: 11pages, to appear in Communications in Algebr
Weak Hopf algebras corresponding to Cartan matrices
We replace the group of group-like elements of the quantized enveloping
algebra of a finite dimensional semisimple Lie algebra
by some regular monoid and get the weak Hopf algebra
. It is a new subclass of weak Hopf algebras
but not Hopf algebras. Then we devote to constructing a basis of
and determine the group of weak Hopf algebra
automorphisms of when is not a root of
unity.Comment: 21 page
Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
We derive and study a quantum group g(p,q) that is Kazhdan--Lusztig-dual to
the W-algebra W(p,q) of the logarithmic (p,q) conformal field theory model. The
algebra W(p,q) is generated by two currents and of dimension
(2p-1)(2q-1), and the energy--momentum tensor T(z). The two currents generate a
vertex-operator ideal with the property that the quotient W(p,q)/R is the
vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2 p q)
of irreducible g(p,q)-representations is the same as the number of irreducible
W(p,q)-representations on which acts nontrivially. We find the center of
g(p,q) and show that the modular group representation on it is equivalent to
the modular group representation on the W(p,q) characters and
``pseudocharacters.'' The factorization of the g(p,q) ribbon element leads to a
factorization of the modular group representation on the center. We also find
the g(p,q) Grothendieck ring, which is presumably the ``logarithmic'' fusion of
the (p,q) model.Comment: 52pp., AMSLaTeX++. half a dozen minor inaccuracies (cross-refs etc)
correcte
k-deformed Poincare algebras and quantum Clifford-Hopf algebras
The Minkowski spacetime quantum Clifford algebra structure associated with
the conformal group and the Clifford-Hopf alternative k-deformed quantum
Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem
context. The resulting algebra is equivalent to the deformed anti-de Sitter
algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into
account, together with the associated quantum Clifford algebra and a (not
braided) deformation of the periodicity Atiyah-Bott-Shapiro theorem.Comment: 10 pages, RevTeX, one Section and references added, improved content
Products, coproducts and singular value decomposition
Products and coproducts may be recognized as morphisms in a monoidal tensor
category of vector spaces. To gain invariant data of these morphisms, we can
use singular value decomposition which attaches singular values, ie generalized
eigenvalues, to these maps. We show, for the case of Grassmann and Clifford
products, that twist maps significantly alter these data reducing degeneracies.
Since non group like coproducts give rise to non classical behavior of the
algebra of functions, ie make them noncommutative, we hope to be able to learn
more about such geometries. Remarkably the coproduct for positive singular
values of eigenvectors in yields directly corresponding eigenvectors in
A\otimes A.Comment: 17 pages, three eps-figure
R-Matrix Formulation of the Quantum Inhomogeneous Groups Iso_qr(N) and Isp_qr(N)
The quantum commutations and the orthogonal (symplectic) conditions
for the inhomogeneous multiparametric -groups of the type are
found in terms of the -matrix of . A consistent
Hopf structure on these inhomogeneous -groups is constructed by means of a
projection from . Real forms are discussed: in
particular we obtain the -groups , including the quantum
Poincar\'e group.Comment: 14 pages, latex, no figure
Twisting algebras using non-commutative torsors
Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can
be used to twist comodule algebras. After surveying and extending the
literature on the subject, we prove a theorem that affords a presentation by
generators and relations for the algebras obtained by such twisting. We give a
number of examples, including new constructions of the quantum affine spaces
and the quantum tori.Comment: 27 pages. Masuoka is a new coauthor. Introduction was revised.
Sections 1 and 2 were thoroughly restructured. The presentation theorem in
Section 3 is now put in a more general framework and has a more general
formulation. Section 4 was shortened. All examples (quantum affine spaces and
tori, twisting of SL(2), twisting of the enveloping algebra of sl(2)) are
left unchange
On quantum group SL_q(2)
We start with the observation that the quantum group SL_q(2), described in
terms of its algebra of functions has a quantum subgroup, which is just a usual
Cartan group.
Based on this observation we develop a general method of constructing quantum
groups with similar property. We also describe this method in the language of
quantized universal enveloping algebras, which is another common method of
studying quantum groups.
We carry our method in detail for root systems of type SL(2); as a byproduct
we find a new series of quantum groups - metaplectic groups of SL(2)-type.
Representations of these groups can provide interesting examples of bimodule
categories over monoidal category of representations of SL_q(2).Comment: plain TeX, 19 pages, no figure
Quantum Lie algebras associated to and
Quantum Lie algebras \qlie{g} are non-associative algebras which are
embedded into the quantized enveloping algebras of Drinfeld and Jimbo
in the same way as ordinary Lie algebras are embedded into their enveloping
algebras. The quantum Lie product on \qlie{g} is induced by the quantum
adjoint action of . We construct the quantum Lie algebras associated to
and . We determine the structure constants and the
quantum root systems, which are now functions of the quantum parameter .
They exhibit an interesting duality symmetry under .Comment: Latex 9 page
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