461 research outputs found
The adjoint representation of group algebras and enveloping algebras
In this paper we study the Hopf adjoint action of group algebras and enveloping algebras. We are particularly concerned with determining when these representations are faithful. Delta methods allow us to reclute the problem to certain better behaved subalgebras. Nevertheless, the problem remains open in the finite group and finite-dimensional Lie algebra cases
Invariant IdeaIs of Abelian Group AIgebras Under the Action of SimpIe Linear Groups
Invariant IdeaIs of Abelian Group AIgebras Under the Action of SimpIe Linear Groups
Irreducible actions and compressible modules
Any finite set of linear operators on an algebra yields an operator
algebra and a module structure on A, whose endomorphism ring is isomorphic
to a subring of certain invariant elements of . We show that if is
a critically compressible left -module, then the dimension of its
self-injective hull over the ring of fractions of is bounded by the
uniform dimension of and the number of linear operators generating .
This extends a known result on irreducible Hopf actions and applies in
particular to weak Hopf action. Furthermore we prove necessary and sufficient
conditions for an algebra A to be critically compressible in the case of group
actions, group gradings and Lie actions
Reversible skew laurent polynomial rings and deformations of poisson automorphisms
A skew Laurent polynomial ring S = R[x(+/- 1); alpha] is reversible if it has a reversing automorphism, that is, an automorphism theta of period 2 that transposes x and x(-1) and restricts to an automorphism gamma of R with gamma = gamma(-1). We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of the two most familiar examples of simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus, both of which are deformations of Poisson algebras over the base field F. Their reversing automorphisms are deformations of Poisson automorphisms of those Poisson algebras. In each case, the ring of invariants of the Poisson automorphism is the coordinate ring B of a surface in F-3 and the ring of invariants S-theta of the reversing automorphism is a deformation of B and is a factor of a deformation of F[x(1), x(2), x(3)] for a Poisson bracket determined by the appropriate surface
Homology and Derived Series of Groups II: Dwyer's Theorem
We give new information about the relationship between the low-dimensional
homology of a group and its derived series. This yields information about how
the low-dimensional homology of a topological space constrains its fundamental
group. Applications are given to detecting when a set of elements of a group
generates a subgroup ``large enough'' to map onto a non-abelian free solvable
group, and to concordance and grope cobordism of links. We also greatly
generalize several key homological results employed in recent work of
Cochran-Orr-Teichner, in the context of classical knot concordance.
In 1963 J. Stallings established a strong relationship between the
low-dimensional homology of a group and its lower central series quotients. In
1975 W. Dwyer extended Stallings' theorem by weakening the hypothesis on the
second homology groups. The naive analogues of these theorems for the derived
series are false. In 2003 the second author introduced a new characteristic
series, associated to the derived series, called the torsion-free derived
series. The authors previously established a precise analogue, for the
torsion-free derived series, of Stallings' theorem. Here our main result is the
analogue of Dwyer's theorem for the torsion-free derived series. We also prove
a version of Dwyer's theorem for the rational lower central series. We apply
these to give new results on the Cochran-Orr-Teichner filtration of the
classical link concordance group.Comment: 26 pages. In this version, we have included a new proof of part of
the main theorem. The new proof is somewhat simpler and stays entirely in the
world of group homology and homological algebra rather than using
Eilenberg-Mac Lane spaces. Other minor corrections. This is the final version
to appear in Geometry & Topolog
Airborne Wind Shear Detection and Warning Systems. Fourth Combined Manufacturers' and Technologists' Conference, part 2
The Fourth Combined Manufacturers' and Technologists' Conference was hosted jointly by NASA Langley Research Center (LaRC) and the Federal Aviation Administration (FAA) in Williamsburg, Virginia, on April 14-16, 1992. The meeting was co-chaired by Dr. Roland Bowles of LaRC and Bob Passman of the FAA. The purpose of the meeting was to transfer significant ongoing results of the NASA/FAA Joint Airborne Wind Shear Program to the technical industry and to pose problems of current concern to the combined group. It also provided a forum for manufacturers to review forward-look technology concepts and for technologists to gain an understanding of the problems encountered by the manufacturers during the development of airborne equipment and the FAA certification requirements. The present document has been compiled to record the essence of the technology updates and discussions which follow each
Noncommutative knot theory
The classical abelian invariants of a knot are the Alexander module, which is
the first homology group of the the unique infinite cyclic covering space of
S^3-K, considered as a module over the (commutative) Laurent polynomial ring,
and the Blanchfield linking pairing defined on this module. From the
perspective of the knot group, G, these invariants reflect the structure of
G^(1)/G^(2) as a module over G/G^(1) (here G^(n) is the n-th term of the
derived series of G). Hence any phenomenon associated to G^(2) is invisible to
abelian invariants. This paper begins the systematic study of invariants
associated to solvable covering spaces of knot exteriors, in particular the
study of what we call the n-th higher-order Alexander module, G^(n+1)/G^(n+2),
considered as a Z[G/G^(n+1)$-module. We show that these modules share almost
all of the properties of the classical Alexander module. They are torsion
modules with higher-order Alexander polynomials whose degrees give lower bounds
for the knot genus. The modules have presentation matrices derived either from
a group presentation or from a Seifert surface. They admit higher-order linking
forms exhibiting self-duality. There are applications to estimating knot genus
and to detecting fibered, prime and alternating knots. There are also
surprising applications to detecting symplectic structures on 4-manifolds.
These modules are similar to but different from those considered by the author,
Kent Orr and Peter Teichner and are special cases of the modules considered
subsequently by Shelly Harvey for arbitrary 3-manifolds.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-19.abs.htm
Simple algebras of Weyl type
Over a field of any characteristic, for a commutative associative algebra
with an identity element and for the polynomial algebra of a
commutative derivation subalgebra of , the associative and the Lie
algebras of Weyl type on the same vector space are
defined. It is proved that , as a Lie algebra (modular its center) or as
an associative algebra, is simple if and only if is -simple and
acts faithfully on . Thus a lot of simple algebras are obtained.Comment: 9 pages, Late
Lie bialgebras of generalized Witt type
In a paper by Michaelis a class of infinite-dimensional Lie bialgebras
containing the Virasoro algebra was presented. This type of Lie bialgebras was
classified by Ng and Taft. In this paper, all Lie bialgebra structures on the
Lie algebras of generalized Witt type are classified. It is proved that, for
any Lie algebra of generalized Witt type, all Lie bialgebras on are
coboundary triangular Lie bialgebras. As a by-product, it is also proved that
the first cohomology group is trivial.Comment: 14 page
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