449 research outputs found
On Generators and Congenerators
The question of the existence of generators and cogenerators
i n a category is of i n t e r e s t i n view of the special
adjoint functor theorem. ISBELL has given an example (unpublished)
which shows t h a t the existence of a cogenerator
i s a necessary part of the hypothesis of the special
adjoint functor theorem. This example also shows t h a t the
category of groups has no cogenerator. (Clearly the f r e e
group on one element i s a generator i n the category of
groups.) It is well known t h a t there e x i s t generators and
cogenerators i n the categories of commutative groups,
Comrnutative Lie algebras (over a f i e l d ) and commutative
r e s t r i c t e d Lie algebras, because a l l of these categories
are module categories. By ISBELL1s r e s u l t when one drops
the condition of cornmutativity for the category of commut
a t i v e groups there i s no longer a cogenerator. We have
Proved similar r e s u l t s for the categories of commutative
Lie algebras and commutative r e s t r i c t e d Lie algebras. The
r e s u l t s are summarized i n the l i s t below where we have
included some r e l a t e d categories
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
Algebra Structures on Hom(C,L)
We consider the space of linear maps from a coassociative coalgebra C into a
Lie algebra L. Unless C has a cocommutative coproduct, the usual symmetry
properties of the induced bracket on Hom(C,L) fail to hold. We define the
concept of twisted domain (TD) algebras in order to recover the symmetries and
also construct a modified Chevalley-Eilenberg complex in order to define the
cohomology of such algebras
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
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
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
Cotensor Coalgebras in Monoidal Categories
We introduce the concept of cotensor coalgebra for a given bicomodule over a
coalgebra in an abelian monoidal category. Under some further conditions we
show that such a cotensor coalgebra exists and satisfies a meaningful universal
property. We prove that this coalgebra is formally smooth whenever the comodule
is relative injective and the coalgebra itself is formally smooth
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
Noncommutative fields and actions of twisted Poincare algebra
Within the context of the twisted Poincar\'e algebra, there exists no
noncommutative analogue of the Minkowski space interpreted as the homogeneous
space of the Poincar\'e group quotiented by the Lorentz group. The usual
definition of commutative classical fields as sections of associated vector
bundles on the homogeneous space does not generalise to the noncommutative
setting, and the twisted Poincar\'e algebra does not act on noncommutative
fields in a canonical way. We make a tentative proposal for the definition of
noncommutative classical fields of any spin over the Moyal space, which has the
desired representation theoretical properties. We also suggest a way to search
for noncommutative Minkowski spaces suitable for studying noncommutative field
theory with deformed Poincar\'e symmetries.Comment: 20 page
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