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Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.


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01 Nov 2005
TL;DR: In this article, the authors defined the notion of equivariant monotone operators as reproducing kernels of H*-ideals of H *-algebras.
Abstract: TOPOLOGICAL LIE ALGEBRAS Fundamentals Universal enveloping algebras The Baker-Campbell-Hausdor series Convergence of the Baker-Campbell-Hausdor series Notes LIE GROUPS AND THEIR LIE ALGEBRAS Definition of Lie groups The Lie algebra of a Lie group Logarithmic derivatives The exponential map Special features of Banach-Lie groups Notes ENLARGIBILITY Integrating Lie algebra homomorphisms Topological properties of certain Lie groups Enlargible Lie algebras Notes Smooth Homogeneous Spaces Basic facts on smooth homogeneous spaces Symplectic homogeneous spaces Some homogeneous spaces related to operator algebras Notes QUASIMULTIPLICATIVE MAPS Supports, convolution, and quasimultiplicativity Separate parts of supports Hermitian maps Notes COMPLEX STRUCTURES ON HOMOGENEOUS SPACES General results Pseudo-Kahler manifolds Flag manifolds in Banach algebras Notes EQUIVARIANT MONOTONE OPERATORS Definition of equivariant monotone operators H*-algebras and L*-algebras Equivariant monotone operators as reproducing kernels H*-ideals of H*-algebras Elementary properties of H*-ideals Notes L*-IDEALS AND EQUIVARIANT MONOTONE OPERATORS From ideals to operators From operators to ideals Parameterizing L*-ideals Representations of automorphism groups Applications to enlargibility Notes HOMOGENEOUS SPACES OF PSEUDO-RESTRICTED GROUPS Pseudo-restricted algebras and groups Complex polarizations Kahler polarizations Admissible pairs of operator ideals Some Kahler homogeneous spaces Notes APPENDICES Differential Calculus and Smooth Manifolds Basic Differential Equations of Lie Theory Topological Groups References Index

52 citations

Posted Content
TL;DR: Ohmoto et al. as discussed by the authors showed that the homogenized, torus equivariant Chern-Schwartz-MacPherson (CSM) class of a constructible function is the restriction of the characteristic cycle of the zero section of the cotangent bundle of a complex projective manifold.
Abstract: Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex projective manifold $X$, the homogenized, torus equivariant CSM class of a constructible function $\varphi$ is the restriction of the characteristic cycle of $\varphi$ via the zero section of the cotangent bundle of $X$. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize $X$ to be a (generalized) flag manifold $G/B$. In this case CSM classes are determined by a Demazure-Lusztig (DL) operator. We prove a `Hecke orthogonality' of CSM classes, determined by the DL operator and its Poincar{\'e} adjoint. We further use the theory of holonomic $\mathcal{D}_X$-modules to show that the characteristic cycle of a Verma module, restricted to the zero section, gives the CSM class of the corresponding Schubert cell. Since the Verma characteristic cycles naturally identify with the Maulik and Okounkov's stable envelopes, we establish an equivalence between CSM classes and stable envelopes; this reproves results of Rim{\'a}nyi and Varchenko. As an application, we obtain a Segre type formula for CSM classes. In the non-equivariant case this formula is manifestly positive, showing that the expansion in the Schubert basis of the CSM class of a Schubert cell is effective. This proves a previous conjecture by Aluffi and Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann manifold case. Finally, we generalize all of this to partial flag manifolds $G/P$.

52 citations

Journal ArticleDOI
TL;DR: In this article, the existence of a non-trivial Kahler-Ricci soliton for Fano manifolds admits an algebraic torus action with general orbit of codimension one.
Abstract: We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahler-Einstein Fano threefolds, and Fano threefolds admitting a non-trivial Kahler-Ricci soliton.

52 citations

Proceedings Article
05 May 2020
TL;DR: It is proved that when the loss is zero, the optimal policy in the abstract MDP can be successfully lifted to the original MDP, and a contrastive loss function is introduced that enforces action equivariance on the learned representations.
Abstract: This work exploits action equivariance for representation learning in reinforcement learning Equivariance under actions states that transitions in the input space are mirrored by equivalent transitions in latent space, while the map and transition functions should also commute We introduce a contrastive loss function that enforces action equivariance on the learned representations We prove that when our loss is zero, we have a homomorphism of a deterministic Markov Decision Process (MDP) Learning equivariant maps leads to structured latent spaces, allowing us to build a model on which we plan through value iteration We show experimentally that for deterministic MDPs, the optimal policy in the abstract MDP can be successfully lifted to the original MDP Moreover, the approach easily adapts to changes in the goal states Empirically, we show that in such MDPs, we obtain better representations in fewer epochs compared to representation learning approaches using reconstructions, while generalizing better to new goals than model-free approaches

52 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid Inc ( N ) of strictly increasing functions.

52 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
2023463
2022888
2021630
2020658
2019526