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Zheng Jiqiang - One of the best experts on this subject based on the ideXlab platform.
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Strichartz estimates and wave equation in a conic singular space
'Springer Science and Business Media LLC', 2020Co-Authors: Zhang Junyong, Zheng JiqiangAbstract:Consider the metric cone X=C(Y)=(0,∞)r×Y with metric g=dr2+r2h where the cross section Y is a compact (n−1)-dimensional Riemannian manifold (Y, h). Let Δg be the positive Friedrichs extension Laplacian on X and let Δh be the positive Laplacian on Y, and consider the operator LV=Δg+V0r−2 where V0∈C∞(Y) such that Δh+V0+(n−2)2/4 is a strictly positive operator on L2(Y). In this paper, we prove global-in-time Strichartz estimates without loss regularity for the wave equation associated with the operator LV. It verifies a conjecture in Wang (Remark 2.4 in Ann Inst Fourier 56:1903–1945, 2006) for wave equation. The range of the Admissible Pair is sharp and the range is influenced by the smallest eigenvalue of Δh+V0+(n−2)2/4. To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension n≥3
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Strichartz estimates and wave equation in a conic singular space
'Springer Science and Business Media LLC', 2019Co-Authors: Zhang Junyong, Zheng JiqiangAbstract:Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $\Delta_g$ be the Friedrich extension positive Laplacian on $X$ and let $\Delta_h$ be the positive Laplacian on $Y$, and consider the operator $\LL_V=\Delta_g+V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ such that $\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. In this paper, we prove the global-in-time Strichartz estimates without loss for the wave equation associated with the operator $\LL_V$ which verifies\cite[Remark 2.4]{wang} Wang's conjecture for wave equation. The range of the Admissible Pair is sharp and is influenced by the smallest eigenvalue of $\Delta_h+V_0+(n-2)^2/4$. To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension $n\geq3$.Comment: Comments are welcome! To appear in Mathematische Annale
E. Zelmanov - One of the best experts on this subject based on the ideXlab platform.
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KAC-MOODY LIE ALGEBRAS GRADED BY KAC-MOODY ROOT SYSTEMS
2015Co-Authors: Hechmi Ben Messaoud, Guy Rousseau, G. Benkart, E. ZelmanovAbstract:Abstract. We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C−Admissible Pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If g is a Kac-Moody Lie algebra (with Dynkin diagram indexed by I) and (I, J) is such a C−Admissible Pair, we construct a C−Admissible subalgebra gJ, which is a Kac-Moody Lie algebra of the same type as g, and whose root system Σ grades finitely the Lie algebra g. For an Admissible quotient ρ: I → I we build also a Kac-Moody subalgebra gρ which grades finitely the Lie algebra g. If g is affine or hyperbolic, we prove that the classification of the gradations of g is equivalent to those of the C−Admissible Pairs and of the Admissible quotients. For general Kac-Moody Lie algebras of indefinite type, the situation may be more complicated; it is (less precisely) described by the concept of generalized C−Admissible Pairs. 2000 Mathematics Subject Classification. 17B67. Key words and phrases. Kac-Moody algebra, C−Admissible Pair, gradation. Introduction. The notion of gradation of a Lie algebra g by a finite root system Σ was introduced by S. Berman and R. Moody [7] and further studied b
Rousseau Guy - One of the best experts on this subject based on the ideXlab platform.
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Kac-Moody Lie algebras graded by Kac-Moody root systems
2012Co-Authors: Messaoud, Hechmi Ben, Rousseau GuyAbstract:We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C-Admissible Pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If g is a Kac-Moody Lie algebra (with Dynkin diagram indexed by I) and (I,J) is such a C-Admissible Pair, we construct a C-Admissible subalgebra g^J, which is a Kac-Moody Lie algebra of the same type as g, and whose root system \Sigma grades finitely the Lie algebra g. For an Admissible quotient \rho : I \rightarrow I we build also a Kac-Moody subalgebra g^\rho which grades finitely the Lie algebra g. If g is affine or hyperbolic, we prove that the classification of the gradations of g is equivalent to those of the C-Admissible Pairs and of the Admissible quotients. For general Kac-Moody Lie algebras of indefinite type, the situation may be more complicated; it is (less precisely) described by the concept of generalized C-Admissible Pairs.Comment: 24 page
Tanmay Deshpande - One of the best experts on this subject based on the ideXlab platform.
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minimal idempotents on solvable groups
Selecta Mathematica-new Series, 2016Co-Authors: Tanmay DeshpandeAbstract:In this paper, we begin to develop a theory of character sheaves on an affine algebraic group G defined over an algebraically closed field \(\mathtt {k}\) of characteristic \(p>0\) using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let l be a prime different from p. Following Boyarchenko and Drinfeld (Sel Math, 2008. doi:10.1007/s00029-013-0133-7, arXiv:0810.0794v1), we define the notion of an Admissible Pair on G and the corresponding idempotent in the \(\overline{\mathbb {Q}}_l\)-linear triangulated braided monoidal category \(\mathscr {D}_G(G)\) of conjugation equivariant \(\overline{\mathbb {Q}}_l\)-complexes (under convolution with compact support) and study their properties. In the spirit of Boyarchenko and Drinfeld (2008), we aim to break up the braided monoidal category \(\mathscr {D}_G(G)\) into smaller and more manageable pieces corresponding to these idempotents in \(\mathscr {D}_G(G)\). Drinfeld has conjectured that the idempotent in \(\mathscr {D}_G(G)\) obtained from an Admissible Pair is in fact a minimal idempotent and that any minimal idempotent in \(\mathscr {D}_G(G)\) can be obtained from some Admissible Pair on G. In this paper, we prove this conjecture in the case when the neutral connected component \(G^\circ \subset G\) is a solvable group. For general groups, we prove that this conjecture is in fact equivalent to an a priori weaker conjecture. Using these results, we reduce the problem of defining character sheaves on general algebraic groups to a special case which we call the “Heisenberg case.” Moreover, as we will see in this paper, the study of character sheaves in the Heisenberg case may be considered, in a certain sense, as a twisted version of the theory of character sheaves on reductive groups as developed by Lusztig (Adv Math 56, 57, 59, 61, 1985, 1986).
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minimal idempotents on solvable groups
arXiv: Representation Theory, 2013Co-Authors: Tanmay DeshpandeAbstract:In this paper, we begin to develop a theory of character sheaves on an affine algebraic group $G$ defined over an algebraically closed field $k$ of characteristic $p>0$ using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let $l$ be a prime different from $p$. Following Boyarchenko and Drinfeld, we define the notion of an Admissible Pair on $G$ and the corresponding idempotent in the $\overline{\mathbb{Q}_l}$-linear triangulated braided monoidal category $\mathscr{D}_G(G)$ of conjugation equivariant $\overline{\mathbb{Q}_l}$-complexes (under convolution with compact support) and study their properties. We aim to break up the braided monoidal category $\mathscr{D}_G(G)$ into smaller and more manageable pieces corresponding to these idempotents in $\mathscr{D}_G(G)$. Drinfeld has conjectured that the idempotent in $\mathscr{D}_G(G)$ obtained from an Admissible Pair is in fact a minimal idempotent and that any minimal idempotent in $\mathscr{D}_G(G)$ can be obtained from some Admissible Pair on $G$. We will prove this conjecture in the case when the neutral connected component $G^\circ \subset G$ is a solvable group. For general groups, we prove that this conjecture is in fact equivalent to an a priori weaker conjecture. Using these results, we reduce the problem of defining character sheaves on general algebraic groups to a special case which we call the "Heisenberg case".
Hechmi Ben Messaoud - One of the best experts on this subject based on the ideXlab platform.
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KAC-MOODY LIE ALGEBRAS GRADED BY KAC-MOODY ROOT SYSTEMS
2015Co-Authors: Hechmi Ben Messaoud, Guy Rousseau, G. Benkart, E. ZelmanovAbstract:Abstract. We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C−Admissible Pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If g is a Kac-Moody Lie algebra (with Dynkin diagram indexed by I) and (I, J) is such a C−Admissible Pair, we construct a C−Admissible subalgebra gJ, which is a Kac-Moody Lie algebra of the same type as g, and whose root system Σ grades finitely the Lie algebra g. For an Admissible quotient ρ: I → I we build also a Kac-Moody subalgebra gρ which grades finitely the Lie algebra g. If g is affine or hyperbolic, we prove that the classification of the gradations of g is equivalent to those of the C−Admissible Pairs and of the Admissible quotients. For general Kac-Moody Lie algebras of indefinite type, the situation may be more complicated; it is (less precisely) described by the concept of generalized C−Admissible Pairs. 2000 Mathematics Subject Classification. 17B67. Key words and phrases. Kac-Moody algebra, C−Admissible Pair, gradation. Introduction. The notion of gradation of a Lie algebra g by a finite root system Σ was introduced by S. Berman and R. Moody [7] and further studied b