248 research outputs found
The Weyl group of the fine grading of associated with tensor product of generalized Pauli matrices
We consider the fine grading of sl(n,\mb C) induced by tensor product of
generalized Pauli matrices in the paper. Based on the classification of maximal
diagonalizable subgroups of PGL(n,\mb C) by Havlicek, Patera and Pelantova,
we prove that any finite maximal diagonalizable subgroup of PGL(n,\mb C)
is a symplectic abelian group and its Weyl group, which describes the symmetry
of the fine grading induced by the action of , is just the isometry group of
the symplectic abelian group . For a finite symplectic abelian group, it is
also proved that its isometry group is always generated by the transvections
contained in it
Sign refinement for combinatorial link Floer homology
Link Floer homology is an invariant for links which has recently been
described entirely in a combinatorial way. Originally constructed with mod 2
coefficients, it was generalized to integer coefficients thanks to a sign
refinement. In this paper, thanks to the spin extension of the permutation
group we give an alternative construction of the combinatorial link Floer chain
complex associated to a grid diagram with integer coefficients. We prove that
the filtered homology of this complex is an invariant for the link and that it
gives the previous sign refinement by means of a 2-cohomological class
corresponding to the spin extension of the permutation group.Comment: 17 pages, 10 figures. correction of the Alexander grading and of the
formula of lemma 5.2 of the sign refinemen
Twisted K-Theory for the Orbifold [*/G]
We study the relationship between the twisted Orbifold K-theories
{^{\alpha}}K_{orb}(\textsl{X}) and {^{\alpha'}}K_{orb}(\textsl{Y}) for two
different twists and of the
Orbifolds \textsl{X}=[*/G] and \textsl{Y}=[*/G'] respectively, for and
finite groups. We prove that under suitable hypothesis over the twisting
and the group we obtain an isomorphism between these twisted
K-theories.Comment: version accepted in Pacific Journal of Mathematic
A note on the Schur multiplier of a nilpotent Lie algebra
For a nilpotent Lie algebra of dimension and dim, we find
the upper bound dim, where denotes the
Schur multiplier of . In case the equality holds if and only if
, where is an abelian Lie algebra of dimension
and H(1) is the Heisenberg algebra of dimension 3.Comment: Paper in press in Comm. Algebra with small revision
Detection of Symmetry Protected Topological Phases in 1D
A topological phase is a phase of matter which cannot be characterized by a
local order parameter. It has been shown that gapped phases in 1D systems can
be completely characterized using tools related to projective representations
of the symmetry groups. We show how to determine the matrices of these
representations in a simple way in order to distinguish between different
phases directly. From these matrices we also point out how to derive several
different types of non-local order parameters for time reversal, inversion
symmetry and symmetry, as well as some more general cases
(some of which have been obtained before by other methods). Using these
concepts, the ordinary string order for the Haldane phase can be related to a
selection rule that changes at the critical point. We furthermore point out an
example of a more complicated internal symmetry for which the ordinary string
order cannot be applied.Comment: 12 pages, 9 Figure
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
The twisted Drinfeld double (or quasi-quantum double) of a finite group with
a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid
is the loop (or inertia) groupoid of the original group and the twisting is
shown geometrically to be the loop transgression of the 3-cocycle. The twisted
representation theory of finite groupoids is developed and used to derive
properties of the Drinfeld double, such as representations being classified by
their characters.
This is all motivated by gerbes and 3-dimensional topological quantum field
theory. In particular the representation category of the twisted Drinfeld
double is viewed as the `space of sections' associated to a transgressed gerbe
over the loop groupoid.Comment: 25 pages, 10 picture
Coinvariant algebras and fake degrees for spin Weyl groups of classical type
The coinvariant algebra of a Weyl group plays a fundamental role in several
areas of mathematics. The fake degrees are the graded multiplicities of the
irreducible modules of a Weyl group in its coinvariant algebra, and they were
computed by Steinberg, Lusztig and Beynon-Lusztig. In this paper we formulate a
notion of spin coinvariant algebra for every Weyl group. Then we compute all
the spin fake degrees for each classical Weyl group, which are by definition
the graded multiplicities of the simple modules of a spin Weyl group in the
spin coinvariant algebra. The spin fake degrees for the exceptional Weyl groups
are given in a sequel.Comment: v2, 39 pages, title modified (with "of classical type" added), the
original version was split into two parts following editor's suggestion; this
v2 is the part one (to appear in Math. Proc. Cambridge Philos. Soc.), with a
sequel dealing with the exceptional typ
Dijkgraaf-Witten invariants of surfaces and projective representations of groups
We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective
representations of groups. As an application we prove that the complex
Dijkgraaf-Witten invariants of surfaces of positive genus are positive
integers.Comment: second version: a mistake corrected in the non-orientable case and a
few improvements adde
The Drinfel'd Double and Twisting in Stringy Orbifold Theory
This paper exposes the fundamental role that the Drinfel'd double \dkg of
the group ring of a finite group and its twists \dbkg, \beta \in
Z^3(G,\uk) as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold
theories and their twistings.
The results pertain to three different aspects of the theory. First, we show
that --Frobenius algebras arising in global orbifold cohomology or K-theory
are most naturally defined as elements in the braided category of
\dkg--modules. Secondly, we obtain a geometric realization of the Drinfel'd
double as the global orbifold --theory of global quotient given by the
inertia variety of a point with a action on the one hand and more
stunningly a geometric realization of its representation ring in the braided
category sense as the full --theory of the stack . Finally, we show
how one can use the co-cycles above to twist a) the global orbifold
--theory of the inertia of a global quotient and more importantly b) the
stacky --theory of a global quotient . This corresponds to twistings
with a special type of 2--gerbe.Comment: 35 pages, no figure
On satellites in semi-abelian categories: Homology without projectives
Working in a semi-abelian context, we use Janelidze's theory of generalised
satellites to study universal properties of the Everaert long exact homology
sequence. This results in a new definition of homology which does not depend on
the existence of projective objects. We explore the relations with other
notions of homology, and thus prove a version of the higher Hopf formulae. We
also work out some examples.Comment: 29 pages; major changes in Example 4.15, minor changes throughout the
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