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

Simple Lie Color Algebras of Weyl Type

For an $(\epsilon,G)$-color-commutative associative algebra $A$ with an identity element over a field $F$ of characteristic not 2, and for a color-commutative subalgebra $D$ of color-derivations of $A$, denote by $A[D]$ the associative subalgebra of ${\rm End}(A)$ generated by $A$ (regarding as operators on $A$ via left multiplication) and $D$. It is easily proved that, as an associative algebra, $A[D]$ is $G$-graded simple if and only if $A$ is \G-graded $D$-simple. Suppose $A$ is \G-graded $D$-simple. Then, (a) $A[D]$ is a free left $A$-module; (b) as a Lie color algebra, the subquotient $[A[D],A[D]]/Z(A[D])\cap[A[D],A[D]]$ is simple (except one minor case), where $Z(A[D])$ is the color center of $A[D]$. The structure of this subquotient is explicitly described.Comment: 10 pages, latex; to appear in Israel J. Mat

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

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

Knot Concordance and Higher-Order Blanchfield Duality

In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} of the classical knot concordance group C. The filtration is important because of its strong connection to the classification of topological 4-manifolds. Here we introduce new techniques for studying C and use them to prove that, for each natural number n, the abelian group F_n/F_{n.5} has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson-Gordon and Gilmer, contain slice knots.Comment: Corrected Figure in Example 8.4, Added Remark 5.11 pointing out an important strengthening of Theorem 5.9 that is needed in a subsequent pape

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 $F$ of any characteristic, for a commutative associative algebra $A$ with an identity element and for the polynomial algebra $F[D]$ of a commutative derivation subalgebra $D$ of $A$, the associative and the Lie algebras of Weyl type on the same vector space $A[D]=A\otimes F[D]$ are defined. It is proved that $A[D]$, as a Lie algebra (modular its center) or as an associative algebra, is simple if and only if $A$ is $D$-simple and $A[D]$ acts faithfully on $A$. Thus a lot of simple algebras are obtained.Comment: 9 pages, Late

Crowdfunding: Geography, Social Networks, and the Timing of Investment Decisions

We examine a crowdfunding platform that connects artists with funders. Although the Internet reduces many distance-related frictions, local and distant funders exhibit different funding patterns. Local funders appear less responsive to information about the cumulative funds raised by an artist. However, this distance effect appears to proxy for a social effect: it is largely explained by funders who likely have an offline social relationship with the artist (“friends and family”). Yet, this social effect does not persist past the first investment, suggesting that it may be driven by an activity like search but not monitoring. Thus, although the platform seems to diminish many distance-sensitive costs, it does not eliminate all of them. These findings provide a deeper understanding of the abilities and limitations of online markets to facilitate transactions and convey information between buyers and sellers with varying degrees of social connectedness

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