387 research outputs found
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
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
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
On Iterated Twisted Tensor Products of Algebras
We introduce and study the definition, main properties and applications of
iterated twisted tensor products of algebras, motivated by the problem of
defining a suitable representative for the product of spaces in noncommutative
geometry. We find conditions for constructing an iterated product of three
factors, and prove that they are enough for building an iterated product of any
number of factors. As an example of the geometrical aspects of our
construction, we show how to construct differential forms and involutions on
iterated products starting from the corresponding structures on the factors,
and give some examples of algebras that can be described within our theory. We
prove a certain result (called ``invariance under twisting'') for a twisted
tensor product of two algebras, stating that the twisted tensor product does
not change when we apply certain kind of deformation. Under certain conditions,
this invariance can be iterated, containing as particular cases a number of
independent and previously unrelated results from Hopf algebra theory.Comment: 44 pages, 21 figures. More minor typos corrections, one more example
and some references adde
Quantifying structural damage from self-irradiation in a plutonium superconductor
The 18.5 K superconductor PuCoGa5 has many unusual properties, including
those due to damage induced by self-irradiation. The superconducting transition
temperature decreases sharply with time, suggesting a radiation-induced Frenkel
defect concentration much larger than predicted by current radiation damage
theories. Extended x-ray absorption fine-structure measurements demonstrate
that while the local crystal structure in fresh material is well ordered, aged
material is disordered much more strongly than expected from simple defects,
consistent with strong disorder throughout the damage cascade region. These
data highlight the potential impact of local lattice distortions relative to
defects on the properties of irradiated materials and underscore the need for
more atomic-resolution structural comparisons between radiation damage
experiments and theory.Comment: 7 pages, 5 figures, to be published in PR
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
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
Noncommutative Symmetries and Gravity
Spacetime geometry is twisted (deformed) into noncommutative spacetime
geometry, where functions and tensors are now star-multiplied. Consistently,
spacetime diffeomorhisms are twisted into noncommutative diffeomorphisms. Their
deformed Lie algebra structure and that of infinitesimal Poincare'
transformations is defined and explicitly constructed.
This allows to construct a noncommutative theory of gravity.Comment: 26 pages. Lectures given at the workshop `Noncommutative Geometry in
Field and String Theories', Corfu Summer Institute on EPP, September 2005,
Corfu, Greece. Version 2: Marie Curie European Reintegration Grant
MERG-CT-2004-006374 acknowledge
The Hopf modules category and the Hopf equation
We study the Hopf equation which is equivalent to the pentagonal equation,
from operator algebras. A FRT type theorem is given and new types of quantum
groups are constructed. The key role is played now by the classical Hopf
modules category. As an application, a five dimensional noncommutative
noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres
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