333 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 $A$ yields an operator
algebra $B$ and a module structure on A, whose endomorphism ring is isomorphic
to a subring $A^B$ of certain invariant elements of $A$. We show that if $A$ is
a critically compressible left $B$-module, then the dimension of its
self-injective hull $A$ over the ring of fractions of $A^B$ is bounded by the
uniform dimension of $A$ and the number of linear operators generating $B$.
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

### 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 $W$ of generalized Witt type, all Lie bialgebras on $W$ are
coboundary triangular Lie bialgebras. As a by-product, it is also proved that
the first cohomology group $H^1(W,W \otimes W)$ is trivial.Comment: 14 page

### Branch Rings, Thinned Rings, Tree Enveloping Rings

We develop the theory of ``branch algebras'', which are infinite-dimensional
associative algebras that are isomorphic, up to taking subrings of finite
codimension, to a matrix ring over themselves. The main examples come from
groups acting on trees.
In particular, for every field k we construct a k-algebra K which (1) is
finitely generated and infinite-dimensional, but has only finite-dimensional
quotients;
(2) has a subalgebra of finite codimension, isomorphic to $M_2(K)$;
(3) is prime;
(4) has quadratic growth, and therefore Gelfand-Kirillov dimension 2;
(5) is recursively presented;
(6) satisfies no identity;
(7) contains a transcendental, invertible element;
(8) is semiprimitive if k has characteristic $\neq2$;
(9) is graded if k has characteristic 2;
(10) is primitive if k is a non-algebraic extension of GF(2);
(11) is graded nil and Jacobson radical if k is an algebraic extension of
GF(2).Comment: 35 pages; small changes wrt previous versio

### The Ideal Intersection Property for Groupoid Graded Rings

We show that if a groupoid graded ring has a certain nonzero ideal property,
then the commutant of the center of the principal component of the ring has the
ideal intersection property, that is it intersects nontrivially every nonzero
ideal of the ring. Furthermore, we show that for skew groupoid algebras with
commutative principal component, the principal component is maximal commutative
if and only if it has the ideal intersection property

### High-performance diamond-based single-photon sources for quantum communication

Quantum communication places stringent requirements on single-photon sources.
Here we report a theoretical study of the cavity Purcell enhancement of two
diamond point defects, the nickel-nitrogen (NE8) and silicon-vacancy (SiV)
centers, for high-performance, near on-demand single-photon generation. By
coupling the centers strongly to high-finesse optical photonic-bandgap cavities
with modest quality factor Q = O(10^4) and small mode volume V = O(\lambda^3),
these system can deliver picosecond single-photon pulses at their zero-phonon
lines with probabilities of 0.954 (NE8) and 0.812 (SiV) under a realistic
optical excitation scheme. The undesirable blinking effect due to transitions
via metastable states can also be suppressed with O(10^{-4}) blinking
probability. We analyze the application of these enhanced centers, including
the previously-studied cavity-enhanced nitrogen-vacancy (NV) center, to
long-distance BB84 quantum key distribution (QKD) in fiber-based, open-air
terrestrial and satellite-ground setups. In this comparative study, we show
that they can deliver performance comparable with decoy state implementation
with weak coherent sources, and are most suitable for open-air communication.Comment: 12 pages, 6 figures, 3 tables, revisions to excitation parameter

### Noncommutative Geometry of Finite Groups

A finite set can be supplied with a group structure which can then be used to
select (classes of) differential calculi on it via the notions of left-, right-
and bicovariance. A corresponding framework has been developed by Woronowicz,
more generally for Hopf algebras including quantum groups. A differential
calculus is regarded as the most basic structure needed for the introduction of
further geometric notions like linear connections and, moreover, for the
formulation of field theories and dynamics on finite sets. Associated with each
bicovariant first order differential calculus on a finite group is a braid
operator which plays an important role for the construction of distinguished
geometric structures. For a covariant calculus, there are notions of invariance
for linear connections and tensors. All these concepts are explored for finite
groups and illustrated with examples. Some results are formulated more
generally for arbitrary associative (Hopf) algebras. In particular, the problem
of extension of a connection on a bimodule (over an associative algebra) to
tensor products is investigated, leading to the class of `extensible
connections'. It is shown that invariance properties of an extensible
connection on a bimodule over a Hopf algebra are carried over to the extension.
Furthermore, an invariance property of a connection is also shared by a `dual
connection' which exists on the dual bimodule (as defined in this work).Comment: 34 pages, Late

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