Recent zbMATH articles in MSC 22https://zbmath.org/atom/cc/222024-02-15T19:53:11.284213ZUnknown authorWerkzeugReal logarithms of semi-simple matriceshttps://zbmath.org/1526.150072024-02-15T19:53:11.284213Z"Pertici, Donato"https://zbmath.org/authors/?q=ai:pertici.donatoSummary: We study the differential and topological structures of the set of real logarithms of any semi-simple non-singular matrix and of the set of real skew-symmetric logarithms of any special orthogonal matrix.The structure of algebraic Baer \(\ast \)-algebrashttps://zbmath.org/1526.160332024-02-15T19:53:11.284213Z"Szűcs, Zsolt"https://zbmath.org/authors/?q=ai:szucs.zsolt"Takács, Balázs"https://zbmath.org/authors/?q=ai:takacs.balazsThe authors study properties of complex \(\ast\)-algebras in which every element is algebraic, that is, the root of a polynomial with complex coefficients. In particular, this class includes all finite-dimensional complex \(\ast\)-algebras.
The first main result is a characterization of when such a \(\ast\)-algebra \(A\) admits a \(C^\ast\)-seminorm. Among the equivalent conditions are the following:
\begin{itemize}
\item If \(a\in A\) with \(a^\ast a=0\), then \(a=0\).
\item \(A\) is semi-simple and every self-adjoint element has spectrum contained in \(\mathbb R\).
\item Every self-adjoint element is a linear combination of orthogonal projections.
\end{itemize}
The second main result is a structure theorem for algebraic Baer \(\ast\)-algebras. Following \textit{S. K. Berberian} [Baer \(^*\)-rings. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0242.16008)], a \(\ast\)-algebra is called a Baer \(\ast\)-algebra if every right annihilator is the principal right ideal generated by a projection. The finite-dimensional Baer \(\ast\)-algebras are exactly the (underlying \(\ast\)-algebras of) finite-dimensional \(C^\ast\)-algebras.
The authors show that a \(\ast\)-algebra is a Baer \(\ast\)-algebra in which every element is algebraic if and only if it is the direct sum of a finite-dimensional Baer \(\ast\)-algebra and a commutative Baer \(\ast\)-algebra in which every element is algebraic.
As a corollary it is shown that the group algebra \(\mathbb C[G]\) is a Baer \(\ast\)-algebra in which every element is algebraic if and only if \(G\) is finite.
Reviewer: Melchior Wirth (Klosterneuburg)BGG category \(\mathscr{O}\) and \(\mathbb{Z}\)-graded representation theoryhttps://zbmath.org/1526.170132024-02-15T19:53:11.284213Z"Hu, Jun"https://zbmath.org/authors/?q=ai:hu.jun.1This paper is a survey on the \(\mathbb{Z}\)-graded version of the Bernstein-Gelfand-Gelfand (BGG for short) category \(\mathcal{O}\) of representations for complex semisimple Lie algebras. For such a Lie algebra \(\mathfrak g\), the BGG category \(\mathcal{O}\) is the full subcategory of all finitely generated \(U(\mathfrak{g})\)-modules that are locally finite under the action of the upper-triangular part \(U(\mathfrak{n})\) and have weight space decomposition for the Cartan subalgebra \(\mathfrak{h}\).
This category admits the block decomposition, with each block \(\mathcal{O}_\lambda\) being equivalent to the category of finite dimensional modules over a finite dimensional Koszul \(\mathbb{C}\)-algebra \(A_\lambda\). The main object of study in this paper is the category \(\mathcal{O}_\lambda^{\mathbb{Z}}=A_\lambda-\mathrm{gmod}\) of \(\mathbb{Z}\)-graded \(A_\lambda\)-modules.
The paper starts with recalling main definitions and properties of the BGG category \(\mathcal{O}\), the coinvariant algebra and Soergel's combinatorial \(\mathbb{V}\)-functor, that plays a key role in the study of representation theory of \(\mathcal{O}^{\mathbb{Z}}_\lambda\). The author formulates the Koszul duality statement for endomorphism algebras of parabolic modules and provides an introduction to the theory of projective functors, as well as a categorification result for a Hecke algebra using indecomposable projective functors. The second section is devoted to the Kazhdan-Lusztig conjecture (now a theorem, proved by \textit{A. Beilinson} and \textit{J. Bernstein} [C. R. Acad. Sci., Paris, Sér. I 292, 15--18 (1981; Zbl 0476.14019)] and by \textit{J. L. Brylinski} and \textit{M. Kashiwara} [Invent. Math. 64, 387--410 (1981; Zbl 0473.22009)]) and to generalization of it for the \(\mathbb{Z}\)-graded case, due to the author and \textit{W. Xiao} [``Jantzen coefficients and radical filtrations for generalized Verma modules'', Preprint, \url{arXiv:2004.08758}]. The third section deals with the study of the coinvariant algebra \(C\) and the subalgebra of its parabolic invariants \(C^I\), i.e. invariants under the action of a parabolic subgroup \(W^I\) in the Weyl group of \(\mathfrak{g}\). The author shows that \(C\) and \(C^I\) are graded cellular algebras with a homogeneous symmetrizing form. It allows to construct a degree 0 anti-involution in \(A_\lambda\) and a \(\mathbb{Z}\)-graded duality functor in \(\mathcal{O}^{\mathbb Z}_\lambda\). In the final part, \(\mathbb Z\)-graded lifts of translation functors are studied. In the semiregular case, they were constructed in [\textit{C. Stroppel}, J. Algebra 268, No. 1, 301--326 (2003; Zbl 1040.17002)]; here the author presents a general construction and studies the action of these functors on graded Verma modules, graded projective modules and graded simple modules.
The paper is well-written and provides a comprehensive survey of the results on the \(\mathbb{Z}\)-graded generalization of the BGG category \(\mathcal{O}\), including the most recent ones.
For the entire collection see [Zbl 1508.20002].
Reviewer: Evgeny Smirnov (Moskva)Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of \({\mathcal{S}}(\mathfrak{g})\)https://zbmath.org/1526.170402024-02-15T19:53:11.284213Z"Panyushev, Dmitri I."https://zbmath.org/authors/?q=ai:panyushev.dmitri-i"Yakimova, Oksana S."https://zbmath.org/authors/?q=ai:yakimova.oksana-sSummary: Let \(\mathfrak{g}\) be a semisimple Lie algebra, \( \vartheta \in \Aut(\mathfrak{g})\) a finite order automorphism, and \(\mathfrak{g}_0\) the subalgebra of fixed points of \(\vartheta \). Recently, we noticed that using \(\vartheta\) one can construct a pencil of compatible Poisson brackets on \({\mathcal{S}}(\mathfrak{g})\), and thereby a `large' Poisson-commutative subalgebra \({\mathcal{Z}}(\mathfrak{g},\vartheta )\) of \(\mathcal{S}(\mathfrak{g})^{\mathfrak{g}_0} \). In this article, we study invariant-theoretic properties of \((\mathfrak{g},\vartheta )\) that ensure good properties of \({\mathcal{Z}}(\mathfrak{g},\vartheta )\). Associated with \(\vartheta\) one has a natural Lie algebra contraction \(\mathfrak{g}_{(0)}\) of \(\mathfrak{g}\) and the notion of a \textit{good generating system} (=g.g.s.) in \({\mathcal{S}}(\mathfrak{g})^\mathfrak{g} \). We prove that in many cases the equality \(\text{ind\,}\mathfrak{g}_{(0)}=\text{ind\,}\mathfrak{g}\) holds and \({\mathcal{S}}(\mathfrak{g})^\mathfrak{g}\) has a g.g.s. According to V. G. Kac's classification of finite order automorphisms (1969), \( \vartheta\) can be represented by a Kac diagram, \( \mathcal{K}(\vartheta )\), and our results often use this presentation. The most surprising observation is that \(\mathfrak{g}_{(0)}\) depends only on the set of nodes in \(\mathcal{K}(\vartheta )\) with nonzero labels, and that if \(\vartheta\) is inner and a certain label is nonzero, then \(\mathfrak{g}_{(0)}\) is isomorphic to a parabolic contraction of \(\mathfrak{g}\).Wavefront sets and descent method for finite unitary groupshttps://zbmath.org/1526.200182024-02-15T19:53:11.284213Z"Peng, Zhifeng"https://zbmath.org/authors/?q=ai:peng.zhifeng"Wang, Zhicheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.2Summary: Let \(G\) be a connected reductive algebraic group defined over a finite field \(\mathbb{F}_q\), and let \(\mathfrak{g}\) be the Lie algebra of \(G\). In the 1980s, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group \(G^F\) in the case where \(q\) is a power of a good prime for \(G^F\), and defined (Kawanaka) wavefront sets of the irreducible representations \(\pi\) of \(G^F\) by GGGRs. In [Adv. Math. 94, No. 2, 139--179 (1992; Zbl 0789.20042), Theorem 11.2], \textit{G. Lusztig} showed that if a nilpotent element \(X \in \mathfrak{g}^F\) is ``large'' for an irreducible representation \(\pi\), then the representation \(\pi\) appears with ``small'' multiplicity in the GGGR associated to \(X\). In this paper, we prove that for unitary groups, if \(X\) is the wavefront of \(\pi\), the multiplicity equals one, which generalizes the multiplicity one result of usual Gelfand-Graev representations. Moreover, we give an algorithm to decompose GGGRs for \(\mathrm{U}_n(\mathbb{F}_q)\) and calculate the \(\mathrm{U}_4(\mathbb{F}_q)\) case by this algorithm.Weights for \(\ell\)-local compact groupshttps://zbmath.org/1526.200192024-02-15T19:53:11.284213Z"Semeraro, Jason"https://zbmath.org/authors/?q=ai:semeraro.jasonAn \(\ell\)-compact group is the \(\ell\)-local homotopy theoretic analogue of a compact Lie group with a ``Weyl group'' \(W\) which is a \(\mathbb{Z}_{\ell}\)-reflection group.
In the paper under review, the author studies the \(\mathcal{F}\)-weights for an \(\ell\)-local compact group \(\mathcal{F}\) over a discrete \(\ell\)-toral group \(S\) with discrete torus \(T\). Motivated by Alperin's weight conjecture for simple groups of Lie type, he conjectures that when \(T\) is the unique maximal abelian subgroup of \(S\) up to \(\mathcal{F}\)-conjugacy and every element of \(S\) is \(\mathcal{F}\)-fused into \(T\), the number of weights of \(\mathcal{F}\) is bounded above by the number of ordinary irreducible characters of its Weyl group.
The main result of this paper is the proof of the previous conjecture in the case when \(\mathcal{F}\) is simple and \(|S : T|=\ell\).
Reviewer: Enrico Jabara (Venezia)Norm rigidity for arithmetic and profinite groupshttps://zbmath.org/1526.200422024-02-15T19:53:11.284213Z"Polterovich, Leonid"https://zbmath.org/authors/?q=ai:polterovich.leonid"Shalom, Yehuda"https://zbmath.org/authors/?q=ai:shalom.yehuda"Shem-Tov, Zvi"https://zbmath.org/authors/?q=ai:shemtov.zviFor a result of \textit{A. Ostrowski} [Acta Math. 41, 271--284 (1917; JFM 46.0169.02)], every absolute value on the field of rational numbers \(\mathbb{Q}\) is equivalent to either the standard absolute value, or a \(p\)-adic absolute value for which the closure of \(\mathbb{Z}\) is compact. The main purpose of this paper is a non-abelian analog of this result for \(\mathrm{SL}_{n}(\mathbb{Z})\) with \(n \geq 3\) and related groups of arithmetic type.
Let \(A\) be a commutative ring such that every non-trivial ideal of \(A\) has finite index. The authors prove the following dichotomy property: if \(\mathrm{SL}_{n}(A)\) has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. Then they relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the normal subgroup theorem of \textit{G. A. Margulis} [Discrete subgroups of semisimple Lie groups. Berlin etc.: Springer-Verlag (1991; Zbl 0732.22008)] and the work of \textit{N. Nikolov} and \textit{D. Segal} [Invent. Math. 190, No. 3, 513--602 (2012; Zbl 1268.20031)]. As a consequence they derive constraints to the possible approximations of certain non-residually finite central extensions of arithmetic groups.
Reviewer: Egle Bettio (Venezia)Anosov representations over closed subflowshttps://zbmath.org/1526.200692024-02-15T19:53:11.284213Z"Wang, Tianqi"https://zbmath.org/authors/?q=ai:wang.tianqiLet \(\Gamma\) be a hyperbolic group with a finite symmetric generating set. If \(\Gamma\) acts properly discontinuously and co-compactly on a topological space \(X\) and commutes with a flow \(\Phi\), then every linear representation of \(\Gamma\) gives rise to a flat vector bundle over \(X/\Gamma\) with a linear lift of \(\Phi\). This construction makes it possible to define a dominated splitting condition. Anosov representations correspond to the case when \(X\) is the geodesic flow of \(\Gamma\). In the paper under review, the author considers the more general case when \(X\) is a closed, \(\Gamma\)-invariant subset of the geodesic flow. In this setting the author gives several equivalent characterizations of this type of representations and proves some properties analogous to the classical Anosov representations, such as stability, the Cartan property and regularity properties of the limit maps.
Reviewer: Miguel Paternain (Montevideo)The pro-\(p\)-Iwahori invariants of the universal quotient representation for \(\mathrm{GL}_2(F)\)https://zbmath.org/1526.200702024-02-15T19:53:11.284213Z"Jana, Arindam"https://zbmath.org/authors/?q=ai:jana.arindamSummary: Let \(F\) be a finite extension of \(\mathbb{Q}_p\). Every supersingular mod \(p\) representation of \(\mathrm{GL}_2(F)\) is a quotient of a certain universal module, say \(\pi_r\). The space of pro-\(p\)-Iwahori invariants of \(\pi_r\) is instrumental in classifying the supersingular mod \(p\) representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\). For \(F\neq \mathbb{Q}_p\), the space of pro-\(p\)-Iwahori invariants of \(\pi_r\) has been determined by Schein and Hendel using the spherical Hecke algebra. An alternative way to study mod \(p\) representations of \(\mathrm{GL}_2(F)\) is introduced by Anandavardhanan and Borisagar using the Iwahori-Hecke algebra. In this paper, we give a canonical isomorphism between the universal modules for \(\mathrm{GL}_2(F)\) obtained from the spherical approach and the Iwahori-Hecke approach. We also compute an explicit basis of the pro-\(p\)-Iwahori invariant space of \(\pi_r\) for \(\mathrm{GL}_2(F)\) using the Iwahori-Hecke approach. This method allows us to extend Hendel's computation slightly in the range of \(p\)-adic digits from \(2< r_j<p-3\) to \(1< r_j<p-1\).On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic casehttps://zbmath.org/1526.200712024-02-15T19:53:11.284213Z"Kim, Dongkwan"https://zbmath.org/authors/?q=ai:kim.dongkwanThis article is concerned with the Springer theory of Weyl groups of type \(BC_n\). The two main theorems provide formulas for the restriction of total Springer representations to the maximal parabolic subgroup of type \(BC_{n-1}\), for the Springer theory arising from the nilpotent cone of a symplectic Lie algebra in characteristic \(2\) and for the Springer theory arising from the exotic nilpotent cone of [\textit{S. Kato}, Duke Math. J. 148, No. 2, 305--371 (2009; Zbl 1183.20002)]. The author also explains that the two formulas become equivalent via the comparison of the two aforementioned Springer theories in [\textit{S. Kato}, Am. J. Math. 133, No. 2, 519--553 (2011; Zbl 1242.20056)] and he demonstrates how they can be used to examine the existence of affine pavings of Springer fibers for the symplectic Lie algebra in characteristic \(2\).
Reviewer: Jonathan Gruber (Lausanne)Classification of involutive commutative two-valued groupshttps://zbmath.org/1526.200852024-02-15T19:53:11.284213Z"Buchstaber, Victor M."https://zbmath.org/authors/?q=ai:bukhshtaber.viktor-matveevich"Veselov, Alexander P."https://zbmath.org/authors/?q=ai:veselov.alexander-p"Gaifullin, Alexander A."https://zbmath.org/authors/?q=ai:gaifullin.alexander-aThis nice paper deals with the notion of two-valued group. By a two-valued group we mean a set \(X\) with a two-valued multiplication \(* : X\times X \to \mathrm{Sym}^2(X)\) (where \(\mathrm{Sym}^2(X)\) is the set of all two-element multisets with elements in \(X\)), an identity element \(e\in X\) and an operation of taking an inverse, \(x\mapsto x^{-1}\) with the properties \textit{associativity}, \textit{strong identity} and \textit{existence and uniqueness of an inverse}. A typical example: Let \(G\) be an ordinary (single-valued) group and \(\iota : G \to G\) be an automorphism with \(\iota^2 = \mathrm{id}_G\) (that is, involutive). Then the quotient set \(X = G/\iota\) is endowed with the structure \(\pi(g)* \pi(h) =[\pi (gh),\pi (g\iota(h))],\) \(e_X= e_G,\) and \(\pi(g)^{-1} =\pi(g^{-1})\) is a two-valued group, where \(g, h \in G\), \(e_G\) is the identity element of \(G\), and \(\pi : G\to X\) is the natural projection. In the paper, finitely generated involutive commutative two-valued groups are classified. It is proved every finitely generated involutive commutative two-valued group is isomorphic to a two-valued group in one of three series of such two-valued groups: a primary series, a unipotent series, and a special series (see Theorem 1.10). For both the Hausdorff and locally compact instances, several classification results for topological involutive commutative two-valued groups are presented (see Theorem 5.8). Finally, at the end of the paper, the case of algebraic two-valued groups is also discussed.
Reviewer: Ali Madanshekaf (Semnan)Flexible Hilbert-Schmidt stability versus hyperlinearity for property (T) groupshttps://zbmath.org/1526.200862024-02-15T19:53:11.284213Z"Dogon, Alon"https://zbmath.org/authors/?q=ai:dogon.alonSummary: We prove a statement concerning hyperlinearity for central extensions of property (T) groups in the presence of flexible HS-stability, and more generally, weak ucp-stability. Notably, this result is applied to show that if \(\mathrm{Sp}_{2g}(\mathbb{Z})\) is flexibly HS-stable, then there exists a non-hyperlinear group. Further, the same phenomenon is shown to hold generically for random groups sampled in Gromov's density model, as well as all infinitely presented property (T) groups. This gives new directions for the possible existence of a non-hyperlinear group. Our results yield Hilbert-Schmidt analogues for \textit{L. Bowen} and \textit{P. Burton}'s work relating flexible P-stability of \(\mathrm{PSL}_n(\mathbb{Z})\) and the existence of non-sofic groups [Trans. Am. Math. Soc. 373, No. 6, 4469--4481 (2020; Zbl 1476.20046)].Karl Heinrich Hofmann and the structure of compact groups and pro-Lie groupshttps://zbmath.org/1526.220012024-02-15T19:53:11.284213Z"Morris, Sidney A."https://zbmath.org/authors/?q=ai:morris.sidney-aSummary: This article is dedicated to Karl Heinrich Hofmann on his 90th birthday. The first part of the article records some biographical facts about him. The second part focuses on the research papers and books he published with the author of this article over the last 45 years. These results concern the structure of compact groups and pro-Lie groups.A corrigendum to: ``A note on compact-like semitopological groups''https://zbmath.org/1526.220022024-02-15T19:53:11.284213Z"Peng, Liang-Xue"https://zbmath.org/authors/?q=ai:peng.liangxueCorrigendum of Lemma 3 of [\textit{A. Ravsky}, Carpathian Math. Publ. 11, No. 2, 442--452 (2019; Zbl 1482.22004)].Editorial remarks on Y. Zelenyuk's proof of the nonexistence of infinite increasing chains of principal left ideals in \(\beta \mathbb{Z}\)https://zbmath.org/1526.220032024-02-15T19:53:11.284213ZThis article offers a different version of Yevhen Zelenyuk's proof that there is no strictly increasing chain of principal left ideals of the semigroup \((\beta\mathbb{Z},+)\) in the paper `` \textit{Increasing sequences of principal left ideals of} \(\beta \mathbb{Z}\) \textit{are finite}'', published in the same issue (see Zbl 1503.22002).
Reviewer: María Vicenta Ferrer González (Castelló)Topological \(S\)-act congruencehttps://zbmath.org/1526.220042024-02-15T19:53:11.284213Z"Maity, Sunil Kumar"https://zbmath.org/authors/?q=ai:maity.sunil-kumar"Paul, Monika"https://zbmath.org/authors/?q=ai:paul.monikaSummary: In this paper, we establish the necessary and sufficient condition for an equivalence relation \(\rho\) on an \(S\)-act \(A\) endowed with a topology such that \(A/\rho\) becomes a Hausdorff topological \(S\)-act. Also, we show that if \(A_1\) and \(A_2\) be two topological \(S\)-acts, then for any homomorphism \(\varphi:A_1 \rightarrow A_2\), \(A_1/\operatorname{ker}\varphi\) is a topological \(S\)-act if and only if \(\varphi\) is \(\varphi\)-saturated continuous. Moreover, we establish for any two congruences \(\theta_1\) and \(\theta_2\) on an \(S\)-act \(A\) endowed with a topology, \(\theta_1\cap \theta_2\) is a topological \(S\)-act congruence on \(A\) if and only if the mapping \(\varphi :A \rightarrow A/ \theta_1\times A/\theta_2\), defined by \(\varphi (a) = (a \theta_1, a\theta_2)\), for all \(a\in A\), is \(\varphi\)-saturated continuous, where \(S\) is a topological semigroup.Topological dynamics of groupoid actionshttps://zbmath.org/1526.220052024-02-15T19:53:11.284213Z"Flores, Felipe"https://zbmath.org/authors/?q=ai:flores.felipe"Măntoiu, Marius"https://zbmath.org/authors/?q=ai:mantoiu.mariusSummary: Some basic notions and results in topological dynamics are extended to continuous groupoid actions in topological spaces. We focus mainly on recurrence properties. Besides results that are analogous to the classical case of group actions, but which have to be put in the right setting, there are also new phenomena. Mostly for groupoids whose source map is not open (and there are many), some properties which were equivalent for group actions become distinct in this general framework; we illustrate this with various counterexamples.Spectral multipliers for functions of fixed \(K\)-type on \(L^p(\mathrm{SL}(2,\mathbb{R})\)https://zbmath.org/1526.220062024-02-15T19:53:11.284213Z"Ricci, Fulvio"https://zbmath.org/authors/?q=ai:ricci.fulvio.1"Wróbel, Błażej"https://zbmath.org/authors/?q=ai:wrobel.blazej-janSummary: We prove an \(L^p\) spectral multiplier theorem for functions of the \(K\)-invariant sublaplacian \(L\) acting on the space of functions of fixed \(K\)-type on the group \(SL(2,\mathbb{R})\). As an application we compute the joint \(L^p(SL(2,\mathbb{R}))\) spectrum of \(L\) and the derivative along \(K\).Arithmeticity of some hypergeometric groupshttps://zbmath.org/1526.220072024-02-15T19:53:11.284213Z"Bajpai, Jitendra"https://zbmath.org/authors/?q=ai:bajpai.jitendra"Singh, Sandip"https://zbmath.org/authors/?q=ai:singh.sandip"Singh, Shashank Vikram"https://zbmath.org/authors/?q=ai:singh.shashank-vikramGiven two monic polynomials of the same degree \(n\) one can write using coefficients of one polynomial a matrix
\[
A=\begin{pmatrix}0 & 0&...& 0& -a_0\\
0 & 1&...& 0& -a_1\\
&&\dots\\
0 &0&...&1 &-a_n \end{pmatrix}
\]
and a similar matrix \(B\) for the second polynomial. One can generate a subgroup \(\Gamma\subset GL_n(\mathbb{C})\) by these matrices.
\textit{J. Bajpai} and \textit{S. Singh} [Trans. Am. Math. Soc. 372, No. 11, 7541--7572 (2019; Zbl 1427.22012)] provide a list of 77 (up to scalar shifts) possible pairs of degree five polynomials that are products of cyclotomic polynomials and satisfying the above conditions of \textit{F. Beukers} and \textit{G. Heckman} [Invent. Math. 95, No. 2, 325--354 (1989; Zbl 0663.30044)] so that the Zariski closures of the corresponding hypergeometric groups are either finite or the orthogonal groups of non-degenerate quadratic forms of signature \((p,q)\) with \(p,q\geq 1\).
The arithmeticity of all but 20 of these groups was investigated before. In the paper under review it is shown that one of the remaining 20 pairs correspond to an arithmetic hypergeometric group.
Reviewer: Dmitry Artamonov (Moskva)A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groupshttps://zbmath.org/1526.220082024-02-15T19:53:11.284213Z"Labriet, Quentin"https://zbmath.org/authors/?q=ai:labriet.quentinSummary: This paper is devoted to branching problems for holomorphic discrete series representations of a conformal group \(G\) of a tube domain \(T_{\Omega}\) over a symmetric cone \(\Omega\) restricted to the conformal group \(G'\) of a tube domain \(T_{\Omega'}\) holomorphically embedded in \(T_{\Omega}\). The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. We answer this program with the introduction of another functional model for holomorphic discrete series representations. This model leads to a geometrical interpretation and a close relation to the theory of orthogonal polynomials for such branching problems. This program is illustrated by three cases. First, we consider the \(n\)-fold tensor product of holomorphic discrete series of the universal covering of \(\mathrm{SL}_2(\mathbb{R})\). Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group \(SO_0(2,n)\) to the subgroup \(\mathrm{SO}_0(2,n-p)\), and finally to the subgroup \(\mathrm{SO}_0(2,n-p)\times \mathrm{SO}(p)\).The Langlands-Shahidi method for pairs via types and covershttps://zbmath.org/1526.220092024-02-15T19:53:11.284213Z"Jo, Yeongseong"https://zbmath.org/authors/?q=ai:jo.yeong-seong"Krishnamurthy, M."https://zbmath.org/authors/?q=ai:krishnamurthy.muthuIn this paper under review, the authors compute the local coefficients attached to a pair \((\pi_1, \pi_2)\) of complex supercuspidal representations of the general linear group, using the theory of types and covers à la Bushnell-Kutzko [\textit{C. J. Bushnell} and \textit{P. C. Kutzko}, The admissible dual of \(\text{GL}(N)\) via compact open subgroups. Princeton, NJ: Princeton University Press (1993; Zbl 0787.22016)]. In this process, the authors obtain another proof of a well-known formula of \textit{F. Shahidi} [Am. J. Math. 106, 67--111 (1984; Zbl 0567.22008)] for the corresponding Plancherel constant. The authors remark that their approach can be adapted to other situations of arithmetic interest within the context of the Langlands-Shahidi method, particularly to that of a Siegel Levi subgroup inside a classical group.
Reviewer: Yan Pan (Tucson)Some results on reducibility of parabolic induction for classical groupshttps://zbmath.org/1526.220102024-02-15T19:53:11.284213Z"Lapid, Erez"https://zbmath.org/authors/?q=ai:lapid.erez-moshe"Tadić, Marko"https://zbmath.org/authors/?q=ai:tadic.markoLet \(F\) be a \(p\)-adic field and \(G\) be a classical group over \(F\). Let \(\sigma\) be a supercuspidal irreducible representation of \(G(F)\) and let \(\pi\) be an irreducible representation of \(\mathrm{GL}_n\). The first main result of the paper is: if there exits \(\rho\) in the cuspidal support of \(\pi\) such that \(\rho\rtimes \sigma\) is reducible, then \(\pi\rtimes \sigma\) is also reducible. If there does not exist \(\rho\) in the cuspidal support of \(\pi\) such that \(\rho\rtimes \sigma\) is reducible, it becomes more complicate. The authors gave irreducibility criteriafor \(\pi\rtimes \sigma\) under some restrictions on \(\pi\). The authors then conjectured that if there is no \(\rho\) in the cuspidal support of \(\pi\) such that \(\rho\rtimes \sigma\) is irreducible, then the irreducibility of \(\pi\rtimes \sigma\) depends only on \(\pi\) and not on \(\sigma\). As the authors noted, this conjecture was proved by Ciubotaru-Heiermann using Hecke algebra techniques.
Reviewer: Qing Zhang (Wuhan)Duality for \(K\)-analytic group cohomology of \(p\)-adic Lie groupshttps://zbmath.org/1526.220112024-02-15T19:53:11.284213Z"Thomas, Oliver"https://zbmath.org/authors/?q=ai:thomas.oliverSummary: We prove a duality result for the analytic cohomology of Lie groups over non-archimedean fields acting on locally convex vector spaces by combining Tamme's non-archimedean van Est comparison morphism with Hazewinkel's duality result for Lie algebra cohomology.Mackey-type identity for invariant functions on Lie algebras of finite unitary groups and an applicationhttps://zbmath.org/1526.220122024-02-15T19:53:11.284213Z"Cuenca, Cesar"https://zbmath.org/authors/?q=ai:cuenca.cesar"Olshanski, Grigori"https://zbmath.org/authors/?q=ai:olshanskii.grigorii-iosifovichLet \(G(\infty)\) be the inductive limit of a series of finite groups \(\{G(n)\}\) of Lie type. Let \(\overline{G}\) be some topological completion of \(G(\infty)\) and \(\overline{\mathfrak{g}}\) be the Lie algebra of \(\overline{G}\). The problem that the article tried to address is the classification of ergodic \(\overline{G}\)-invariant Radon measures on \(\overline{\mathfrak{g}}^\ast\), the dual space of \(\overline{\mathfrak{g}}\). The authors show that these measures correspond to the extreme rays of convex cone of positive harmonic functions on some disconnected branching graph or equivalently the boundary of the branching graph.
The authors give a partial solution, Theorem 9.2, to the problem above for the unitary case. The solution shows how to generate an infinite-parameter family of positive harmonic functions in \(\mathrm{Harm}_{>0}(\Gamma^{\mathbb{UB}})\), but does not provide a complete description of the structure of \(\mathrm{Harm}_{>0}(\Gamma^{\mathbb{UB}})\). Theorem 9.2 is inspired by Kerov's mixing construction which relies on the Mackey's theorem that decomposes the composition of restriction and induction operators. Working with invariant functions on Lie algebra, the authors prove a version of Mackey's theorem on the parabolic restriction and parabolic induction in Theorem 5.4.
Reviewer: Rongqing Ye (Louisville)Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groupshttps://zbmath.org/1526.220132024-02-15T19:53:11.284213Z"Gloeckner, Helge"https://zbmath.org/authors/?q=ai:glockner.helge"Tárrega, Luis"https://zbmath.org/authors/?q=ai:tarrega.luisSummary: Let \(M\) be a compact smooth manifold of dimension \(m\) (without boundary) and \(G\) be a finite-dimensional Lie group, with Lie algebra \(\mathfrak{g}\). Let \(H^{>\frac{m}{2}} (M, G)\) be the group of all
mappings \(\gamma : M \to G\) which are \(H^s\) for some \(s >\frac{m}{2}\). We show that \(H^{>\frac{m}{2}} (M, G)\) can be made a regular Lie group in Milnor's sense, modelled on the Silva space \(H^{>\frac{m}{2}} (M, \mathfrak{g}) := \varinjlim_{s>\frac{m}{2}} H^s(M, \mathfrak{g})\), such that
\[
H^{>\frac{m}{2}} (M, G) =\varinjlim{}_{s>\frac{m}{2}} H^s(M, G)
\]
as a Lie group (where \(H^s(M, G)\) is the Hilbert-Lie group of all \(G\)-valued \(H^s\)-mappings on \(M\)). We also explain how the (known) Lie group structure on \(H^s(M, G)\) can be obtained as a special case of a general construction of Lie groups \(\mathcal{F}(M, G)\) whenever function spaces \(\mathcal{F}(U, \mathbb{R})\) on open subsets \(U \subseteq \mathbb{R}^m\) are given, subject to simple axioms.The application of Euler-Rodrigues formula over hyper-dual matriceshttps://zbmath.org/1526.220142024-02-15T19:53:11.284213Z"Ramis, Çağla"https://zbmath.org/authors/?q=ai:ramis.cagla"Yaylı, Yusuf"https://zbmath.org/authors/?q=ai:yayli.yusuf"Zengin, İrem"https://zbmath.org/authors/?q=ai:zengin.irem(no abstract)Graded symmetry groups: plane and simplehttps://zbmath.org/1526.220152024-02-15T19:53:11.284213Z"Roelfs, Martin"https://zbmath.org/authors/?q=ai:roelfs.martin"De Keninck, Steven"https://zbmath.org/authors/?q=ai:de-keninck.stevenPin groups were introduced by \textit{M. F. Atiyah}, \textit{R. Bott} and \textit{A. Shapiro} in [Topology 3, Suppl. 1, 3--38 (1964; Zbl 0146.19001)] in order to describe the symmetries that are the result of combining a finite number of discrete reflections in (hyper)planes.
The paper under review is dedicated to generalizing a conjecture of \textit{M. Riesz} [Fundamental Theories of Physics. 54. Dordrecht: Kluwer Academic Publishers. (1993; Zbl 0823.15028)] stating that a bivector of an \(n\)-dimensional geometric algebra \(\mathbb{R}_{pq}\) can always be decomposed into at most \(\lfloor \frac{n}{2} \rfloor\) simple commuting orthogonal bivectors, where a simple bivector is the outer product of two vectors. The authors extend this conjecture to the wider class of algebras \(\mathbb{R}_{pqr}\), which includes \(r\) null basis vectors, and consider the group of all reflections therein, \(\mathrm{Pin}(p,q,r)\). The main result is the following invariant decomposition (Theorem 1): A product \(U = u_{1}u_{2} \ldots u_{k}\) of \(k\) reflections \(u_{i}\) can be decomposed into exactly \(\lceil \frac{k}{2} \rceil\) simple commuting factors. These are \(\lfloor \frac{k}{2} \rfloor\) products of two reflections, and, for odd \(k\), one extra reflection.
Reviewer: Egle Bettio (Venezia)Polish topologies on groups of non-singular transformationshttps://zbmath.org/1526.220162024-02-15T19:53:11.284213Z"Le Maître, François"https://zbmath.org/authors/?q=ai:le-maitre.francoisSummary: In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure carries no Polish group topology. We then characterize the Borel \(\sigma\)-finite measures \(\lambda\) on a standard Borel space for which the group of \(\lambda\)-preserving transformations has the automatic continuity property. We finally show that the natural Polish topology on the group of all non-singular transformations is actually its only Polish group topology.Topological dynamics of enveloping semigroupshttps://zbmath.org/1526.370012024-02-15T19:53:11.284213Z"Nagar, Anima"https://zbmath.org/authors/?q=ai:nagar.anima"Singh, Manpreet"https://zbmath.org/authors/?q=ai:singh.manpreet.1By topological dynamics we mean the study of a topological group \(T\) (or sometimes, merely a semigroup) acting continuously on a compact Hausdorff space \(X\). Such data \((X,T)\) is called a flow and associated to them,
\textit{R. Ellis} [Trans. Am. Math. Soc. 94, 272--281 (1960; Zbl 0094.17402)]
attached the closure \(E(X,T)\) of the image of \(T\) in the product space \(X^X\) (seen as the space of applications from \(X\) to \(X\)). With the product topology on \(X^X\), this closure is a compact space and endowed with the composition law this is a right topological semigroup. It is called the enveloping semigroup of \((X,T)\).
Whereas this enveloping semigroup is quite abstract and difficult to compute in general, it proved to be very useful in topological dynamics.
If \((X,T)\) is a flow then \(T\) also acts continuously on \(2^X\), the space of closed subspaces of \(X\) and thus one get a new flow \((2^X,T)\) (called the induced system) that is never minimal since \(X\) appears as a subflow.
This monograph focuses on the enveloping semigroup \(E(2^X,T)\). It starts from the basic definitions in topological dynamics and then moves to induced systems, Ellis semigroups and Ellis semigroups for induced systems.
Elementary results are proved but more involved propositions and theorems are not. References to literature are given.
Chapter 5 on the enveloping semigroup \(E(2^X,T)\) is the more original one and contains new results.
For readers interested in an introduction to topological dynamics and enveloping semigroups, the references given at the end of the first chapter can be really helpful.
Reviewer: Bruno Duchesne (Paris)Mixing and double recurrence in probability groupshttps://zbmath.org/1526.370042024-02-15T19:53:11.284213Z"Tserunyan, Anush"https://zbmath.org/authors/?q=ai:tserunyan.anushThe author defines a class probability groups, namely a class of groups equipped with an invariant probability measure. This class also contains the ultraproducts of all locally compact unimodular amenable groups. The basics of the theory of probability/measure-preserving actions for probability groups, including a natural notion of mixing, are developed. A number of connections between mixing and double recurrence in this setting is obtained.
Reviewer: Michael L. Blank (Moskva)Strong orbit equivalence in Cantor dynamics and simple locally finite groupshttps://zbmath.org/1526.370062024-02-15T19:53:11.284213Z"Robert, Simon"https://zbmath.org/authors/?q=ai:robert.simonMany notions in dynamics, notably those related to orbit structure, find their roots in operator algebraic considerations. Strong orbit equivalence is one of these notions, and is defined as follows.
First, if two minimal homeomorphisms \(\varphi,\psi\) of the Cantor space \(X\) have the same orbits, one can uniquely define a cocycle \(c:X\to \mathbb Z\) from \(\psi\) to \(\varphi\) by the equation
\[
\psi(x)=\varphi^{c(x)}(x).
\]
Now, given two minimal homeomorphisms of the Cantor space, we say that they are \emph{strongly orbit equivalent} if they can be conjugated so that they share the same orbits, and both cocycles from the second to the first and from the first to the second have only one discontinuity point.
On the face of it, it is not clear why strong orbit equivalence should be transitive, or why one should consider it at all. A fundamental result of \textit{T. Giordano} et al. [J. Reine Angew. Math. 469, 51--111 (1995; Zbl 0834.46053)] answers both questions. It states that \(\varphi\) and \(\psi\) are strongly orbit equivalent iff the associated crossed products \(C^*\) algebras \(C^*(X,\varphi)\) and \(C^*(X,\psi)\) are isomorphic.
The present paper is motivated by a third characterization of strong orbit equivalence discovered in [\textit{T. Giordano} et al., Isr. J. Math. 111, 285--320 (1999; Zbl 0942.46040)], in terms of a locally finite group \(\Gamma_x^\varphi\). This group is defined as follows: first, the \textit{full group} of a minimal homeomorphism \(\varphi\) is the group of all homeomorphisms whose orbits are contained in those of \(\varphi\). To any \(\psi\) of the full group of \(\varphi\) we can associate a cocycle just as before, and the \textit{topological full group} of \(\varphi\) is the group of all \(\psi\) in the full group of \(\varphi\) whose cocyle is \textit{continuous}. Next, for a given \(x\in X\), the \(\varphi\)-orbit of \(x\) carries a natural linear order given by \(y\leq_{\varphi}z\) iff \(z=\varphi^n(y)\) for some \(n\geq 0\). We can finally define \(\Gamma_x^\varphi\) as the setwise stabilizer of the forward orbit of \(x\): it is the group of all \(\psi\) in the topological full group of \(\varphi\) such that for all \(y\geq_\varphi x\) we have \(\psi(y)\geq_\varphi x\).
Two minimal homeomorphisms \(\varphi\) and \(\psi\) of the Cantor space \(X\) are strongly orbit equivalent if and only if for all \(x,y\in X\), \(\Gamma^\varphi_x\) is isomorphic to \(\Gamma^\psi_y\), as proved in [T. Giordano et al., loc. cit.]. Their proof however relies crucially on deep operator algebraic techniques.
The main result of the present paper is a purely dynamical and very elegant proof of the aforementioned characterization of strong orbit equivalence. It relies on results on ample groups of homeomorphisms found in [\textit{W. Krieger}, Math. Ann. 252, 87--95 (1980; Zbl 0472.54028)].
As a byproduct, using the fact that the derived subgroup of \(\Gamma^\varphi_x\) still resembles strong orbit equivalence and is a simple locally finite group, and the \(S_\infty\)-universality of strong orbit equivalence (see [\textit{J. Melleray}, Isr. J. Math. 236, No. 1, 317--344 (2020; Zbl 1479.03020)]), the author shows that isomorphism of locally finite simple countable groups is a \(S_\infty\)-universal equivalence relation for Borel reducibility.
Reviewer: François Le Maître (Paris)Geodesic planes in the convex core of an acylindrical 3-manifoldhttps://zbmath.org/1526.370402024-02-15T19:53:11.284213Z"McMullen, Curtis T."https://zbmath.org/authors/?q=ai:mcmullen.curtis-t"Mohammadi, Amir"https://zbmath.org/authors/?q=ai:mohammadi.amir"Oh, Hee"https://zbmath.org/authors/?q=ai:oh.heeSummary: Let \(M\) be a convex cocompact, acylindrical hyperbolic 3-manifold of infinite volume, and let \(M^*\) denote the interior of the convex core of \(M\). In this paper we show that any geodesic plane in \(M^*\) is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theorems for planes in convex cocompact 3-manifolds of infinite volume that depend only on the topology of \(M\).Spherical maximal operators on Heisenberg groups: restricted dilation setshttps://zbmath.org/1526.420332024-02-15T19:53:11.284213Z"Roos, Joris"https://zbmath.org/authors/?q=ai:roos.joris"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreas"Srivastava, Rajula"https://zbmath.org/authors/?q=ai:srivastava.rajulaSummary: Consider spherical means on the Heisenberg group with a codimension 2 incidence relation, and associated spherical local maximal functions \(M_E f\) where the dilations are restricted to a set \(E\). We prove \(L^p \to L^q\) estimates for these maximal operators; the results depend on various notions of dimension of \(E\).Uncertainty inequalities for certain connected Lie groupshttps://zbmath.org/1526.430042024-02-15T19:53:11.284213Z"Bansal, Piyush"https://zbmath.org/authors/?q=ai:bansal.piyush"Kumar, Ajay"https://zbmath.org/authors/?q=ai:kumar.ajay"Bansal, Ashish"https://zbmath.org/authors/?q=ai:bansal.ashishSummary: Pitt's inequality for exponential solvable Lie groups with non-trivial center, connected nilpotent Lie groups with non-compact center, Heisenberg motion group and diamond Lie groups has been proved. These inequalities have been used to establish logarithmic uncertainty inequality and Heisenberg uncertainty inequality for the above classes of groups.Analogues of theorems of Chernoff and Ingham on the Heisenberg grouphttps://zbmath.org/1526.430062024-02-15T19:53:11.284213Z"Ganguly, Pritam"https://zbmath.org/authors/?q=ai:ganguly.pritam"Thangavelu, Sundaram"https://zbmath.org/authors/?q=ai:thangavelu.sundaramThe following theorem proved by \textit{P. R. Chernoff} [Bull. Am. Math. Soc. 81, 637--646 (1975; Zbl 0304.47032)] is a higher dimensional analogue of the Denjoy-Carleman theorem on \(\mathbb{R}^n\) which characterizes quasi-analytic functions:
Let \(f\) be a smooth function on \(\mathbb{R}^n\). Assume that \(\Delta^m f \in L^2(\mathbb{R}^n)\) for all \(m \in\mathbb{N}\) and \(\sum_{m=1}^{\infty}\|\Delta^m f\|_{2}^{-\frac{1}{2m}} =\infty\). If \(f\) and all its partial derivatives vanish at a point \(a \in \mathbb{R}^n\), then \(f\) is identically zero.
The authors prove an analogue of Chernoff's theorem for the Laplacian \(\Delta_H\) on the Heisenberg group \(H_n\). Also they apply this theorem to prove Ingham type theorems for the group Fourier transform on \(H_n\) and also for the spectral projections associated to the sublaplacian.
Reviewer: Sanjiv Gupta (Masqaṭ)Polynomially growing harmonic functions on connected groupshttps://zbmath.org/1526.430072024-02-15T19:53:11.284213Z"Perl, Idan"https://zbmath.org/authors/?q=ai:perl.idan"Yadin, Ariel"https://zbmath.org/authors/?q=ai:yadin.arielSummary: We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties.
Our main result shows that (for sufficiently ``nice'' random walk measures) a connected, compactly generated, locally compact group has polynomial volume growth if and only if the space of linear growth harmonic functions has finite dimension.
This characterization is interesting in light of the fact that Gromov's theorem regarding finitely generated groups of polynomial growth does not have an analog in the connected case. That is, there are examples of connected groups of polynomial growth that are not nilpotent by compact. Also, the analogous result for the discrete case has only been established for solvable groups and is still open for general finitely generated groups.The resonances of the Capelli operators for small split orthosymplectic dual pairshttps://zbmath.org/1526.430082024-02-15T19:53:11.284213Z"Bramati, Roberto"https://zbmath.org/authors/?q=ai:bramati.roberto"Pasquale, Angela"https://zbmath.org/authors/?q=ai:pasquale.angela"Przebinda, Tomasz"https://zbmath.org/authors/?q=ai:przebinda.tomaszSummary: Let \((G, G')\) be a reductive dual pair in \(\mathrm{Sp}(\mathsf{W})\) with \(\operatorname{rank} G \leq \operatorname{rank} G'\) and \(G'\) semisimple. The image of the Casimir element of the universal enveloping algebra of \(G'\) under the Weil representation \(\omega\) is a Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of \(\omega\). We compute the resonances of a natural multiple of a translation of this operator for small split orthosymplectic dual pairs. The corresponding resonance representations turn out to be GG$'$-modules in Howe's correspondence. We determine them explicitly.Metamorphism as a covariant transform for the SSR grouphttps://zbmath.org/1526.440032024-02-15T19:53:11.284213Z"Alqurashi, Taghreed"https://zbmath.org/authors/?q=ai:alqurashi.taghreed"Kisil, Vladimir V."https://zbmath.org/authors/?q=ai:kisil.vladimir-vSummary: Metamorphism is a recently introduced integral transform, which is useful in solving partial differential equations. Basic properties of metamorphism can be verified by direct calculations. In this paper, we present metamorphism as a sort of covariant transform and derive its most important features in this way. Our main result is a characterisation of metamorphism's image space. Reading this paper does not require advanced knowledge of group representations or theory of covariant transform.Some applications of group-theoretic Rips constructions to the classification of von Neumann algebrashttps://zbmath.org/1526.460392024-02-15T19:53:11.284213Z"Chifan, Ionuţ"https://zbmath.org/authors/?q=ai:chifan.ionut"Das, Sayan"https://zbmath.org/authors/?q=ai:das.sayan-kumar"Khan, Krishnendu"https://zbmath.org/authors/?q=ai:khan.krishnenduThe left group von Neumann algebra \(L(G)\) of a discrete group \(G\) is the bicommutant of the range of the left regular representation of \(G\) inside the algebra of all bounded linear operators on \(\ell^2(G)\). This is known to be a II\(_1\)-factor precisely when \(G\) is icc (i.e., has infinite nontrivial conjugacy classes). A central question in the classification theory of von Neumann algebras is when \(G\) can be completely reconstructed from \(L(G)\). Such groups are called \(W^*\)-superrigid. A conjecture of Alain Connes predicts that all icc property \((T)\) (Kazhdan property) groups are \(W^*\)-superrigid; note that currently no examples of such (i.e., $W^*$-superrigid property~$(T)$) groups are known yet, and more significantly, there are not so many algebraic features of property \((T)\) groups recognizable at the von Neumann algebraic level.
This is the main focus of the present paper. Some progress is made for property \((T)\) groups that appear as certain fiber products of Belegradek-Osin Rips-type constructions
[\textit{I.~Belegradek} and \textit{D.~Osin}, Groups Geom. Dyn. 2, No.~1, 1--12 (2008; Zbl 1152.20039)].
Using \textit{S.~Popa}'s deformation theory
[\textit{S. Popa}, in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume~I: Plenary lectures and ceremonies. Zürich: European Mathematical Society (EMS). 445--477 (2007; Zbl 1132.46038)],
the authors show that certain algebraic features of property \((T)\) groups are recognizable via their left von Neumann algebra. In particular, as an evidence for the Connes rigidity conjecture, they obtain infinite families of pairwise non-isomorphic property \((T)\) group factors. They use the Rips construction to build examples of property \((T)\) II\(_1\)-factors with masa's without property \((T)\), answering a question of
\textit{Y.-L. Jiang} and \textit{A.~Skalski} [Groups Geom. Dyn. 15, No.~3, 849--892 (2021; Zbl 1489.46066)].
Reviewer: Massoud Amini (Tehran)The \(\kappa\)-nullity of Riemannian manifolds and their splitting tensorshttps://zbmath.org/1526.530162024-02-15T19:53:11.284213Z"Gorodski, Claudio"https://zbmath.org/authors/?q=ai:gorodski.claudio"Guimarães, Felippe"https://zbmath.org/authors/?q=ai:guimaraes.felippe-soaresRecall that for a Riemannian manifold \((M,g)\) the Riemann curvature tensor \(R\) is given by \(R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z\) for smooth vector fields \(X\), \(Y\), \(Z\), where \(\nabla\) is the Levi-Civita connection of \(g\). According to \textit{S.-s. Chern} and \textit{N. H. Kuiper} [Ann. Math. (2) 56, 422--430 (1952; Zbl 0052.17601)] the nullity space at \(p \in M\) is defined by
\begin{align*}
\mathcal{N}|_p = \{z \in T_p M | R_p(x,y)z = 0 \text{ for all }x,y \in T_p M\},
\end{align*}
and the index of nullity at \(p\) is given by \(\nu(p) = \mathrm{dim}\,\mathcal{N}|_p\).
Let \(A_p(x,y)z \equiv g_p(y,z)x-g_p(x,z)y\) and \(B_p(x,y)z \equiv R_p(x,y)z-\kappa A_p(x,y)z\).
For \(\kappa \in \mathbb{R}\), following \textit{T. Ôtsuki} [J. Math. Soc. Japan 6, 221--234 (1954; Zbl 0058.37702)], we similarly define the \(\kappa\)-nullity distribution at \(p\) by
\begin{align*}
\mathcal{N}_\kappa|_p = \{z \in T_p M | B_p(x,y)z = 0 \text{ for all }x,y \in T_p M\},
\end{align*}
and the index of \(\kappa\)-nullity at \(p\) by \(\nu_\kappa(p) = \mathrm{dim}\,\mathcal{N}_\kappa|_p\) to yield a distribution \(\mathcal{N}_\kappa\) and an upper semi-continuous function \(\nu_\kappa\) on \(M\) respectively. The orthogonal distribution \(\mathcal{N}^\perp_\kappa\) is said to be the \(\kappa\)-conullity distribution of \(M\) and its dimension at \(p\) defines a lower semi-continuous function on \(M\).
Since the Levi-Civita connections for \(g\) and \(|\kappa|g\) coincide, the \(\kappa\)-nullity distribution at \(p\) for \(g\) coincides with the \(\kappa/|\kappa|\)-nullity distribution at \(p\) for \(|\kappa|g\) when \(\kappa \neq 0\). So, with no loss of generality, it suffices to restrict our attention to the cases where \(\kappa=-1\), \(\kappa=0\), and \(\kappa=1\).
In this article, the authors consider manifolds of low \(\kappa\)-conullity or \(\kappa\)-nullity where \(\kappa=-1\), \(\kappa=0\), and \(\kappa=1\). They obtain new results for manifolds with low \(\kappa\)-conullity.
In particular, the authors show that a Riemannian \(n\)-manifold with \(n\geq3\) with maximal \((-1)\)-conullity equal to 2,
(i) Has that the rigid motions of the Minkowski plane \(E(1,1) \equiv \mathrm{SO}_0(1,1) \ltimes \mathbb{R}^2\) is locally isometric to an open set \(U\) of points with \((-1)\)-conullity 2 if \(M\) is of constant scalar curvature and the \((-1)\)-conullity distribution is integrable over \(U\);
(ii) Has that the universal cover of \(M\) is homogeneous if \(M\) is complete with finite volume, 3-dimensional or has scalar curvature bounded away from \(-n(n-1)\);
(iii) Is isometric to \(E(1,1)\) or the universal cover of \(\mathrm{SL}(2,\mathbb{R})\) with a left-invariant metric if \(M\) is homogeneous and simply connected.
The authors prove that a simply connected complete \(n\)-Riemannian manifold \(M\) with maximal 0-conullity equal to 2 and positive scalar curvature bounded away from 0 splits as a Riemannian product \(\mathbb{R}^{n-1} \times \Sigma\), where \(\Sigma\) is diffeomorphic to a 2-sphere. Moreover, the authors show that the only simply connected complete Riemannian manifolds with constant 1-conullity equal to 2 and constant scalar curvature are the 3-dimensional Sasakian space forms, namely isometric to 3-dimensional Lie groups with certain left-invariant metrics: the Berger sphere \(\mathrm{SU}(2)\), the universal cover of \(\mathrm{SL}(2,\mathbb{R})\), or the Heisenburg group, \(\mathrm{Nil}^3\).
The proofs concerning Riemannian manifolds with maximal conullity 2 are laid out in Section 3 of the article and all heavily rely on \textit{A. Rosenthal}'s idea of the splitting tensor \(C_T\) [Mich. Math. J. 15, 433--440 (1968; Zbl 0177.24501)]. In the context of scalar curvature that is constant (or bounded away from \(\kappa\)) the authors show in Section 2 that the splitting tensor \(C_T\) for \(T \in \mathcal{N}_\kappa\) and \(||T||=1\) has that \(\mathrm{tr}\, C_T=0\) and \(\det C_T = \kappa\) along the geodesic \(\gamma\) where \(\dot{\gamma}=T\) for cases \(\kappa = -1\), \(\kappa = 0\), and \(\kappa = 1\). The results follow from the heavy restrictions placed on the matrix representation of \(C_T\).
Reviewer: Owen Dearricott (Melbourne)Cauchy's surface area formula in the Heisenberg groupshttps://zbmath.org/1526.530252024-02-15T19:53:11.284213Z"Huang, Yen-Chang"https://zbmath.org/authors/?q=ai:huang.yen-changSummary: We show an analogue of Cauchy's surface area formula for the Heisenberg groups \(\mathbb{H}_n\) for \(n \geq 1\), which states that the p-area of any compact hypersurface \(\Sigma\) in \(\mathbb{H}_n\) with its p-normal vector defined almost everywhere on \(\Sigma\) is the average of its projected p-areas onto the orthogonal complements of all p-normal vectors of the Pansu spheres (up to a constant). The formula provides a geometric interpretation of the p-areas defined by
\textit{J.-F. Hwang} et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 1, 129--177 (2005; Zbl 1158.53306)] in \(H_1\) and \textit{J.-H. Cheng} et al. [Math. Ann. 337, No. 2, 253--293 (2007; Zbl 1109.35009)]
in \(\mathbb{H}_n\) for \(n \geq 2\). We also characterize the projected areas for rotationally symmetric domains in \(\mathbb{H}_n\); namely, for any rotationally symmetric domain with boundary in \(\mathbb{H}_n\), its projected p-area onto the orthogonal complement of any normal vector of the Pansu spheres is a constant, independent of the choice of the projected directions.Left-invariant symplectic structures on diagonal almost abelian Lie groupshttps://zbmath.org/1526.530742024-02-15T19:53:11.284213Z"Castellanos Moscoso, Luis Pedro"https://zbmath.org/authors/?q=ai:castellanos-moscoso.luis-pedroThe classification of left-invariant symplectic structures on Lie groups in low dimensions is known, see [\textit{B.-Y. Chu}, Trans. Am. Math. Soc. 197, 145--159 (1974; Zbl 0261.53039); \textit{M. Goze} and \textit{A. Bouyakoub}, Rend. Semin. Fac. Sci. Univ. Cagliari 57, No. 1, 85--97 (1987; Zbl 0679.17002); \textit{J. R. Gómez} et al., J. Pure Appl. Algebra 156, No. 1, 15--31 (2001; Zbl 0966.17006); \textit{G. Ovando}, Beitr. Algebra Geom. 47, No. 2, 419--434 (2006; Zbl 1155.53042)]. Classifications in geometry are important to study whether a given manifold admits some nice geometric structures and, in the setting of Lie groups, it is natural to ask about the existence of left-invariant structures. A symplectic Lie group is a Lie group $G$ endowed with a left-invariant symplectic form $\omega$ (that is, a nondegenerate closed 2-form). Although there are many interesting results on the structure of symplectic Lie groups and some classifications in low dimensions, the general picture is far from complete. The authors use the method of \textit{T. Hashinaga} et al. [J. Math. Soc. Japan 68, No. 2, 669--684 (2016; Zbl 1353.53058)] to find left-invariant Riemannian metrics to establish a new approach to classify (up to automorphism and scale) left-invariant symplectic structures on Lie groups. Their approach is based on the moduli space of left invariant nondegenerate 2-forms. They apply their method for two particular Lie groups of dimension \(2n\) and give classifications of left-invariant symplectic structures on them.
Reviewer: Manouchehr Misaghian (Prairie View)Full Laplace spectrum of distance spheres in symmetric spaces of rank onehttps://zbmath.org/1526.580082024-02-15T19:53:11.284213Z"Bettiol, Renato G."https://zbmath.org/authors/?q=ai:bettiol.renato-g"Lauret, Emilio A."https://zbmath.org/authors/?q=ai:lauret.emilio-a"Piccione, Paolo"https://zbmath.org/authors/?q=ai:piccione.paoloIt is rare that one can explicitly compute the Laplace spectrum of a closed Riemannian manifold. Even in the homogeneous setting, where techniques from representation theory can be brought to bear on the situation, explicit computation can be elusive. For instance, each sphere carries a metric of constant sectional curvature \(K >0\) and the spectrum of this metric can be computed explicitly. While, these constant curvature metrics account for the homogeneous metrics on an even-dimensional sphere, odd-dimensional spheres admit homogeneous metrics with non-constant sectional curvature, the spectra of which are unknown, in general.
Given an odd-dimensional sphere, the authors provide explicit expressions for the spectra of a two-parameter family of homogeneous metrics determined by an appropriate Hopf bundle (see Theorem~A). This is achieved by implementing a three-step representation-theoretic algorithm (developed in Sec.~2.2) for computing the spectra of the two-parameter family of homogeneous metrics in the canonical variation of a Riemannian submersion of compact normal homogeneous spaces:
\[
K/H \hookrightarrow G/H \stackrel{\pi}{\to} G/K,
\]
where \(H < K <G\) are compact Lie groups. The second step of this algorithm, which requires explicit knowledge of certain branching laws for the spherical representations determined by the pair \((G,H)\), is the main difficulty in computing the spectrum of such metrics. Fortunately, in the cases considered by the authors, the requisite branching laws are known. And, in the event \(K = HL\simeq H \times L\) for some compact Lie group \(L\), the computations can be simplified (see Cor.~2.5).
The metrics considered in Theorem~A do not account for all odd-dimensional homogeneous spheres; however, taken together with the even-dimensional spheres of constant sectional curvature, these homogeneous spheres are an exhaustive list of the distance spheres that can occur in rank-one symmetric spaces. With this in mind, the authors avail themselves of the relationship between the Laplace operator and the Jacobi operator associated to CMC hypersurfaces (see Eq.~7.2) in order to \((1)\) classify the resonant radii for distance spheres in the compact rank-one symmetric spaces \(\mathbb{C}P^{n+1}\), \(\mathbb{H}P^{n+1}\), and \(\textrm{Ca}P^{2}\), for \(n \geq 1\), and \((2)\) demonstrate that distance spheres in the non-compact rank-one symmetric spaces \(\mathbb{C} H^{n +1 }\), \(\mathbb{H}H^{n +1}\) and \(\textrm{Ca}H^{2}\), for \(n \geq 1\), are stable and locally rigid (see Theorem~B).
Reviewer: Craig Sutton (Hanover)Research on key technologies of controlled bidirectional quantum teleportationhttps://zbmath.org/1526.810162024-02-15T19:53:11.284213Z"Yang, Xiaolong"https://zbmath.org/authors/?q=ai:yang.xiaolong"Li, Dongfen"https://zbmath.org/authors/?q=ai:li.dongfen"Zhou, Jie"https://zbmath.org/authors/?q=ai:zhou.jie|zhou.jie.7|zhou.jie.1|zhou.jie.2"Tan, Yuqiao"https://zbmath.org/authors/?q=ai:tan.yuqiao"Zheng, Yundan"https://zbmath.org/authors/?q=ai:zheng.yundan"Liu, Xiaofang"https://zbmath.org/authors/?q=ai:liu.xiaofangSummary: Quantum teleportation is a hot topic of extensive research at present, and bidirectional quantum teleportation is one of the important research directions. In this paper, we propose a scheme to realize controlled bidirectional quantum teleportation by using a six-particle as the quantum channel. In this scheme, the communication parties Alice and Bob need to cooperate to send and receive unknown arbitrary particle states, which can't achieve the purpose alone. During the transmission, Alice and Bob only need to perform simple Bell state measurement and single-particle measurement, and the measurement steps are few. Finally, the purpose can be achieved by performing some appropriate unitary transformations. We validate the scheme on the IBM Quantum Experiment platform and the feasibility by data. In terms of security, we discuss the method of ensuring communication security when this scheme is subjected to external and internal attacks. Finally, we compare it with other bidirectional quantum teleportation schemes, our scheme consumes fewer quantum resources and classical resources, operates fewer times, and has higher transmission efficiency, which fully highlights the advantages of this scheme.Telling compositeness at a distance with outer automorphisms and CPhttps://zbmath.org/1526.810282024-02-15T19:53:11.284213Z"Bischer, Ingolf"https://zbmath.org/authors/?q=ai:bischer.ingolf"Döring, Christian"https://zbmath.org/authors/?q=ai:doring.christian"Trautner, Andreas"https://zbmath.org/authors/?q=ai:trautner.andreasSummary: We investigate charge-parity (CP) and non-CP outer automorphism of groups and the transformation behavior of group representations under them. We identify situations where composite and elementary states that transform in exactly the same representation of the group, transform differently under outer automorphisms. This can be instrumental in discriminating composite from elementary states solely by their quantum numbers with respect to the outer automorphism, i.e. without the need for explicit short distance scattering experiments. We discuss under what conditions such a distinction is unequivocally possible. We cleanly separate the case of symmetry constrained (representation) spaces from the case of multiple copies of identical representations in flavor space, and identify conditions under which non-trivial transformation in flavor space can be enforced for composite states. Next to composite product states, we also discuss composite states in non-product representations. Comprehensive examples are given based on the finite groups \(\Sigma(72)\) and \(D_8\). The discussion also applies to \(\mathrm{SU}(N)\) and we scrutinize recent claims in the literature that \(\mathrm{SU}(2N)\) outer automorphism with antisymmetric matrices correspond to distinct outer automorphisms. We show that outer automorphism transformations with antisymmetric matrices are related by an inner automorphism to the standard \(\mathbb{Z}\) outer automorphism of \(\mathrm{SU}(N)\). As a direct implication, no non-trivial transformation behavior can arise for composite product states under the outer automorphism of \(\mathrm{SU}(N)\).A counterexample to the CFT convexity conjecturehttps://zbmath.org/1526.810362024-02-15T19:53:11.284213Z"Sharon, Adar"https://zbmath.org/authors/?q=ai:sharon.adar"Watanabe, Masataka"https://zbmath.org/authors/?q=ai:watanabe.masatakaSummary: Motivated by the weak gravity conjecture, [\textit{O. Aharony} and \textit{E. Palti}, Phys. Rev. D (3) 104, No. 12, Article ID 126005, 14 p. (2021; \url{doi:10.1103/PhysRevD.104.126005})] conjectured that in any CFT, the minimal operator dimension at fixed charge is a convex function of the charge. In this letter we construct a counterexample to this convexity conjecture, which is a clockwork-like model with some modifications to make it a weakly-coupled CFT. We also discuss further possible applications of this model and some modified versions of the conjecture which are not ruled out by the counterexample.Non-parallel currents and the Newton's third law of motionhttps://zbmath.org/1526.810402024-02-15T19:53:11.284213Z"Epp, V."https://zbmath.org/authors/?q=ai:epp.v-ya|epp.vladimir"Veselkova, A. V."https://zbmath.org/authors/?q=ai:veselkova.a-v(no abstract)Second comment on: ``Maxwell's equations and Lorentz transformations''https://zbmath.org/1526.830012024-02-15T19:53:11.284213Z"Beléndez, Augusto"https://zbmath.org/authors/?q=ai:belendez.augusto"Sirvent-Verdú, Joan Josep"https://zbmath.org/authors/?q=ai:sirvent-verdu.joan-josep"Gallego, Sergi"https://zbmath.org/authors/?q=ai:gallego.sergiSummary: \textit{J. M. Aguirregabiria} et al. [Eur. J. Phys. 43, No. 3, Article ID 035603, 9 p. (2022; Zbl 1521.83002)] obtained the Lorentz transformations by assuming the invariance of Maxwell's equations in vacuum under inertial transformations. However, they do not initially assume that these transformations must be linear. Later, \textit{D. V. Redžić} [Eur. J. Phys. 43, No. 6, Article ID 068002, 5 p. (2022; Zbl 1520.83005)] indicates that it is necessary to explicitly assume the linearity of these transformations to avoid a remarkable and surprising result. Now it is shown that it is not necessary to make the assumption that the transformation equations between inertial reference frames must be linear, but that this linearity is a consequence of the homogeneity of space-time.General form of axially symmetric stationary metric: exact solutions and conservation laws in vacuum fieldshttps://zbmath.org/1526.830032024-02-15T19:53:11.284213Z"Jyoti, Divya"https://zbmath.org/authors/?q=ai:jyoti.divya"Kumar, Sachin"https://zbmath.org/authors/?q=ai:kumar.sachinSummary: The invariant non-static solutions of Einstein's vacuum field equations, corresponding to the most general form of axially symmetric stationary line element that represents a non conformally flat semi-Riemannian spacetime in cylindrical coordinates, are investigated. Lie symmetry method is used for symmetry reduction as well as for obtaining exact solutions in terms of arbitrary functions. The conservation laws are obtained for vacuum equations in axially symmetric gravitational fields. The solutions of Lewis metric and Chandrasekhar metric, are derived from the obtained solutions. By considering the possibilities of arbitrary functions, the graphical behaviour of the solutions is also shown.Lorentz symmetry breaking and entropy correction of Kerr-Newman-AdS black holehttps://zbmath.org/1526.830082024-02-15T19:53:11.284213Z"Li, Ran"https://zbmath.org/authors/?q=ai:li.ran"Yu, Zi-Han"https://zbmath.org/authors/?q=ai:yu.zihan"Yang, Shu-Zheng"https://zbmath.org/authors/?q=ai:yang.shuzhengSummary: In this paper, in the curved space-time of Kerr-Newman-Ads black hole, the scalar particle dynamics equation is modified in consideration of Lorentz symmetry breaking. On this basis, the quantum tunneling radiation of the black hole is precisely modified, and the modified expressions of Hawking temperature and entropy of the black hole are obtained; In order to further obtain the correction effect of the Planck scale, this paper considers a more accurate correction method beyond the semi-classical theory, and further obtains a new expression of the black hole temperature and entropy. The physical significance of a series of new results obtained is also discussed.