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European Mathematical Society Publishing House
2024-03-28 14:19:07
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=PRIMS&vol=53&iss=1&update_since=2024-03-28
Publications of the Research Institute for Mathematical Sciences
Publ. Res. Inst. Math. Sci.
PRIMS
0034-5318
1663-4926
General
10.4171/PRIMS
http://www.ems-ph.org/doi/10.4171/PRIMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Research Institute for Mathematical Sciences, Kyoto University
53
2017
1
Finite $W$-Superalgebras and Dimensional Lower Bounds for the Representations of Basic Lie Superalgebras
Yang
Zeng
Nanjing Audit University, NANJING, CHINA
Bin
Shu
East China Normal University, SHANGHAI, CHINA
Finite W-(super)algebras, basic (classical) Lie superalgebras, modular repre- sentations of Lie (super)algebras, Kac–Weisfeiler conjecture (property) for modular Lie (super)algebras
In this paper we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable, under an assumption on the minimal dimensions of representations of the finite $W$-superalgebra $U(\mathfrak{g}_\mathbb C,e)$ over the field of complex numbers. The afore-mentioned lower bounds for modular representations, as a super-version of the Kac-Weisfeiler conjecture [26], were formulated and proved by Wang-Zhao in [35] for basic Lie superalgebras over an algebraically closed field $\mathbb k$ of positive characteristic $p$. We further conjecture that the assumption is actually satisfied (see Conjecture 1.2). This is to say, the complex finite $W$-superalgebra $U(\mathfrak g_\mathbb C,e)$ affords either one-dimensional or two-dimensional representations, according to the parity of the discriminant number (the difference of dimensions between the odd part of $\mathfrak g_\mathbb C$ and its subspace centralized by $e$). We demonstrate the positivity of the conjecture with examples including all the cases of type $A$, and finally reduce the investigation of the conjecture to the case of rigid nilpotent elements as the situation happens for the ordinary finite $W$-algebras (cf. [29]).
Nonassociative rings and algebras
1
63
10.4171/PRIMS/53-1-1
http://www.ems-ph.org/doi/10.4171/PRIMS/53-1-1
The Hadamard Product and the Karcher Mean of Positive Invertible Operators
Yuki
Seo
Osaka Kyoiku University, OSAKA, JAPAN
Karcher mean, Hadamard product, geometric operator mean, Fiedler inequality
In this paper, we show several operator inequalities involving the Hadamard product and the Karcher mean of $n (≥ 3)$ positive invertible operators on a separable Hilbert space, which are regarded as an $n$-variable operator version of results due to Ando and Aujla–Vasudeva. As applications, we show estimates from above for an $n$-variable version of the Fiedler-type theorem due to Fujii. Moreover, we show an $n$-variable version of the majorization relation due to Ando for the Hadamard product via the Karcher mean of $n (≥ 3)$ positive-defi nite matrices.
Operator theory
Linear and multilinear algebra; matrix theory
65
78
10.4171/PRIMS/53-1-2
http://www.ems-ph.org/doi/10.4171/PRIMS/53-1-2
Kato's Inequality for Magnetic Relativistic Schrödinger Operators
Fumio
Hiroshima
Kyushu University, FUKUOKA, JAPAN
Takashi
Ichinose
Kanazawa University, KANAZAWA, JAPAN
József
Lőrinczi
Loughborough University, LOUGHBOROUGH, UNITED KINGDOM
Kato's inequality, relativistic Schrödinger operator, magnetic relativistic Schrödinger operator
Kato's inequality is shown for the magnetic relativistic Schrödinger operator $H_{A,m}$ defined as the operator theoretical square root of the selfadjoint, magnetic nonrelativistic Schrödinger operator $(-i\nabla-A(x))^2+m^2$ with an $L^{2}_{\mathrm {loc}}$ vector potential $A(x)$.
Operator theory
Quantum theory
79
117
10.4171/PRIMS/53-1-3
http://www.ems-ph.org/doi/10.4171/PRIMS/53-1-3
Two-Weight Norm, Poincaré, Sobolev and Stein–Weiss Inequalities on Morrey Spaces
Kwok-Pun
Ho
The Education University of Hong Kong, HONG KONG, CHINA
Two-weight norm inequality, Poincaré inequality, Sobolev inequality, Stein{ Weiss inequality, Hardy inequality, Rellich inequality, Morrey space, singular integral operator, fractional integral operator
We establish two-weight norm inequalities for singular integral operators and fractional integral operators on Morrey spaces. As a consequence of these inequalities, we obtain two-weight Poincar é and Sobolev inequalities on Morrey spaces. Moreover, we also establish the Stein–Weiss inequality, the Hardy inequality and the Rellich inequality on Morrey spaces.
Real functions
Fourier analysis
Functional analysis
119
139
10.4171/PRIMS/53-1-4
http://www.ems-ph.org/doi/10.4171/PRIMS/53-1-4
On Calderón’s Problem for a System of Elliptic Equations
Oleg
Imanuvilov
Colorado State University, FORT COLLINS, UNITED STATES
Masahiro
Yamamoto
University of Tokyo, TOKYO, JAPAN
Dirichlet-to-Neumann map, uniqueness, gauge equivalence, system of elliptic equations, two dimensions
We consider Calder ón's problem in the case of a partial Dirichlet-to-Neumann map for systems of elliptic equations in a bounded two-dimensional domain. The main result of the paper is as follows: If two systems of elliptic equations generate the same partial Dirichlet-to-Neumann map on some subboundary, then the coeffi cients can be uniquely determined up to gauge equivalence.
Partial differential equations
141
186
10.4171/PRIMS/53-1-5
http://www.ems-ph.org/doi/10.4171/PRIMS/53-1-5
Infinite-Dimensional Manifolds as Ringed Spaces
Michel
Egeileh
Université Saint-Joseph, BEIRUT, LEBANON
Tilman
Wurzbacher
Université de Lorraine, METZ CEDEX 1, FRANCE
Infinite-dimensional manifolds, ringed spaces, smoothness of maps
We analyze the possibility of de fining infi nite-dimensional manifolds as ringed spaces. More precisely, we consider three defi nitions of manifolds modeled on locally convex spaces: in terms of charts and atlases, in terms of ringed spaces and in terms of functored spaces, as introduced by Douady in his thesis. It is shown that for large classes of locally convex model spaces (containing Fr echet spaces and duals of Fr echet–Schwartz spaces), the three defi nitions are actually equivalent. The equivalence of the defi nition via charts with the de finition via ringed spaces is based on the fact that for the classes of model spaces under consideration, smoothness of maps turns out to be equivalent to their scalarwise smoothness (i.e., the smoothness of their composition with smooth real-valued functions).
Global analysis, analysis on manifolds
Manifolds and cell complexes
187
209
10.4171/PRIMS/53-1-6
http://www.ems-ph.org/doi/10.4171/PRIMS/53-1-6
Yano's Conjecture for Two-Puiseux-Pair Irreducible Plane Curve Singularities
Enrique
Artal Bartolo
Universidad de Zaragoza, ZARAGOZA, SPAIN
Pierrette
Cassou-Noguès
Université Bordeaux I, Talence Cedex, FRANCE
Ignacio
Luengo
Universidad Complutense de Madrid, MADRID, SPAIN
Alejandro
Melle-Hernández
Universidad Complutense de Madrid, MADRID, SPAIN
Bernstein polynomial, $b$-exponents, improper integrals
In 1982, Tamaki Yano proposed a conjecture predicting the $b$-exponents of an irreducible plane curve singularity germ that is generic in its equisingularity class. In this article, we prove the conjecture for the case in which the irreducible germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. This hypothesis on the monodromy implies that the $b$-exponents coincide with the opposite of the roots of the Bernstein polynomial, and we compute the roots of the Bernstein polynomial.
Algebraic geometry
Several complex variables and analytic spaces
211
239
10.4171/PRIMS/53-1-7
http://www.ems-ph.org/doi/10.4171/PRIMS/53-1-7