The Experts below are selected from a list of 56712 Experts worldwide ranked by ideXlab platform
Stefano Rossi - One of the best experts on this subject based on the ideXlab platform.
-
Connected components of compact matrix quantum groups and finiteness Conditions
Journal of Functional Analysis, 2014Co-Authors: Lucio S. Cirio, Alessandro D'andrea, Claudia Pinzari, Stefano RossiAbstract:Abstract We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1. We give necessary and sufficient Conditions for normality of the identity component, in the sense of Wang, and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending Chain Condition on the representation ring, called Lie property, which characterizes Lie groups in the classical case and reduces to group Noetherianity of the discrete group dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A u ( F ) is not of Lie type. We discuss an example arising from the compact real form of U q ( sl 2 ) for q 0 .
-
Connected components of compact matrix quantum groups and finiteness Conditions
arXiv: Quantum Algebra, 2012Co-Authors: Lucio S. Cirio, Alessandro D'andrea, Claudia Pinzari, Stefano RossiAbstract:We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1. We give necessary and sufficient Conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending Chain Condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an example arising from the compact real form of U_q(sl_2) for q
Susan J. Sierra - One of the best experts on this subject based on the ideXlab platform.
-
A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras.
arXiv: Rings and Algebras, 2020Co-Authors: Omar Leon Sanchez, Susan J. SierraAbstract:We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending Chain Condition (ACC) on Poisson ideals. We define a combinatorial Condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $\mathbb N^k$. Our main result is: Theorem. If $\mathfrak g$ is a Dicksonian graded Lie algebra over a field of characteristic zero, then the symmetric algebra $S(\mathfrak g)$ satisfies the ACC on radical Poisson ideals. As an application, we establish this ACC for the symmetric algebra of any graded simple Lie algebra of polynomial growth, and for the symmetric algebra of the Virasoro algebra. We also derive some consequences connected to the Poisson primitive spectrum of finitely Poisson-generated algebras.
-
ideals in the enveloping algebra of the positive witt algebra
Algebras and Representation Theory, 2020Co-Authors: Alexey V Petukhov, Susan J. SierraAbstract:Let W+ be the positive Witt algebra, which has a $\mathcal {C}$ -basis $\{e_{n}: n \in \mathcal {Z}_{\geq 1}\}$ , with Lie bracket [ei,ej] = (j − i)ei+j. We study the two-sided ideal structure of the universal enveloping algebra U(W+) of W+. We show that if I is a (two-sided) ideal of U(W+) generated by quadratic expressions in the ei, then U(W+)/I has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending Chain Condition. We conjecture that analogous facts hold for arbitrary ideals of U(W+), and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra S(W+).
Steve Hofmann - One of the best experts on this subject based on the ideXlab platform.
-
bmo solvability and absolute continuity of harmonic measure
Journal of Geometric Analysis, 2018Co-Authors: Steve HofmannAbstract:We show that for a uniformly elliptic divergence form operator L, defined in an open set \(\Omega \) with Ahlfors–David regular boundary, BMO solvability implies scale-invariant quantitative absolute continuity (the weak-\(A_\infty \) property) of elliptic-harmonic measure with respect to surface measure on \(\partial \Omega \). We do not impose any connectivity hypothesis, qualitative, or quantitative; in particular, we do not assume the Harnack Chain Condition, even within individual connected components of \(\Omega \). In this generality, our results are new even for the Laplacian. Moreover, we obtain a partial converse, assuming in addition that \(\Omega \) satisfies an interior Corkscrew Condition, in the special case that L is the Laplacian.
-
A NEW CHARACTERIZATION OF CHORD-ARC DOMAINS
2015Co-Authors: Jonas Azzam, Steve Hofmann, Jose ́ María Martell, Tatiana ToroAbstract:Abstract. We show that if Ω ⊂ Rn+1, n ≥ 1, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Har-nack Chain Conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior Corkscrew points at all scales, so that in fact, Ω is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew Conditions, and an interior Harnack Chain Condition. We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems
-
Contents
2014Co-Authors: Steve Hofmann, María MartellAbstract:Abstract. We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [RR]. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain Ω ⊂ R n+1, n ≥ 2, with a uniformly rectifiable boundary, which satisfies the Harnack Chain Condition plus an interior (but not exterior) corkscrew Condition. In a companion paper to this one [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant L q bounds, with q> 1, o
-
uniform rectifiability and harmonic measure ii poisson kernels in lp imply uniform rectifiability
Duke Mathematical Journal, 2014Co-Authors: Steve Hofmann, Jose Maria Martell, Ignacio UriartetueroAbstract:We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n≥2, for an Ahlfors–David regular domain Ω⊂Rn+1 which satisfies the Harnack Chain Condition plus an interior (but not exterior) corkscrew Condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of ∂Ω.
-
uniform rectifiability and harmonic measure iii riesz transform bounds imply uniform rectifiability of boundaries of 1 sided nta domains
arXiv: Classical Analysis and ODEs, 2012Co-Authors: Steve Hofmann, Jose Maria Martell, Svitlana MayborodaAbstract:Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the "Riesz Transform bound" $$\sup_{\varepsilon>0}\int_E\left|\int_{\{y\in E:|x-y|>\varepsilon\}}\frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)\right|^2 dH^n(x) \leq C \int_E|f|^2 dH^n .$$ Assume further that $E$ is the boundary of a domain $\Omega\subset \mathbb{R}^{n+1}$ satisfying the Harnack Chain Condition plus an interior (but not exterior) Corkscrew Condition. Then $E$ is uniformly rectifiable.
Sierra, Susan J. - One of the best experts on this subject based on the ideXlab platform.
-
A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras
2020Co-Authors: Sanchez, Omar Leon, Sierra, Susan J.Abstract:We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending Chain Condition (ACC) on Poisson ideals. We define a combinatorial Condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $\mathbb N^k$. Our main result is: Theorem. If $\mathfrak g$ is a Dicksonian graded Lie algebra over a field of characteristic zero, then the symmetric algebra $S(\mathfrak g)$ satisfies the ACC on radical Poisson ideals. As an application, we establish this ACC for the symmetric algebra of any graded simple Lie algebra of polynomial growth, and for the symmetric algebra of the Virasoro algebra. We also derive some consequences connected to the Poisson primitive spectrum of finitely Poisson-generated algebras.Comment: 29 pages; comments welcome; v2 minor changes to introduction, submitte
-
Ideals in the enveloping algebra of the positive Witt algebra
2019Co-Authors: Petukhov Alexey, Sierra, Susan J.Abstract:Let $W_+$ be the positive Witt algebra, which has a $C$-basis $\{e_n: n \in Z_{\geq 1}\}$, with Lie bracket $[ e_i, e_j] = (j-i) e_{i+j}$. We study the two-sided ideal structure of the universal enveloping algebra $U(W_+)$ of $W_+$. We show that if $I$ is a (two-sided) ideal of $U(W_+)$ generated by quadratic expressions in the $e_i$, then $U(W_+)/I$ has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending Chain Condition. We conjecture that analogous facts hold for arbitrary ideals of $U(W_+)$, and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra $S(W_+)$.Comment: 22 pages; v2 extensive revisions to Section 4 to improve readability of proofs. To appear in Algebras and Representation Theor
Lucio S. Cirio - One of the best experts on this subject based on the ideXlab platform.
-
Connected components of compact matrix quantum groups and finiteness Conditions
Journal of Functional Analysis, 2014Co-Authors: Lucio S. Cirio, Alessandro D'andrea, Claudia Pinzari, Stefano RossiAbstract:Abstract We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1. We give necessary and sufficient Conditions for normality of the identity component, in the sense of Wang, and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending Chain Condition on the representation ring, called Lie property, which characterizes Lie groups in the classical case and reduces to group Noetherianity of the discrete group dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A u ( F ) is not of Lie type. We discuss an example arising from the compact real form of U q ( sl 2 ) for q 0 .
-
Connected components of compact matrix quantum groups and finiteness Conditions
arXiv: Quantum Algebra, 2012Co-Authors: Lucio S. Cirio, Alessandro D'andrea, Claudia Pinzari, Stefano RossiAbstract:We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1. We give necessary and sufficient Conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending Chain Condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an example arising from the compact real form of U_q(sl_2) for q