The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Rudolph Kalveks - One of the best experts on this subject based on the ideXlab platform.
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quiver theories and formulae for Nilpotent orbits of exceptional algebras
Journal of High Energy Physics, 2017Co-Authors: Amihay Hanany, Rudolph KalveksAbstract:We treat the topic of the closures of the Nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal Nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal Nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group Nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different Nilpotent orbits and propose candidates for the constructions of some non-normal Nilpotent orbits of Exceptional algebras.
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quiver theories for moduli spaces of classical group Nilpotent orbits
Journal of High Energy Physics, 2016Co-Authors: Amihay Hanany, Rudolph KalveksAbstract:We approach the topic of Classical group Nilpotent orbits from the perspective of the moduli spaces of quivers, described in terms of Hilbert series and generating functions. We review the established Higgs and Coulomb branch quiver theory constructions for A series Nilpotent orbits. We present systematic constructions for BCD series Nilpotent orbits on the Higgs branches of quiver theories defined by canonical partitions; this paper collects earlier work into a systematic framework, filling in gaps and providing a complete treatment. We find new Coulomb branch constructions for above minimal Nilpotent orbits, including some based upon twisted affine Dynkin diagrams. We also discuss aspects of 3d mirror symmetry between these Higgs and Coulomb branch constructions and explore dualities and other relationships, such as HyperKahler quotients, between quivers. We analyse all Classical group Nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials.
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quiver theories for moduli spaces of classical group Nilpotent orbits
arXiv: High Energy Physics - Theory, 2016Co-Authors: Amihay Hanany, Rudolph KalveksAbstract:We approach the topic of Classical group Nilpotent orbits from the perspective of their moduli spaces, described in terms of Hilbert series and generating functions. We review the established Higgs and Coulomb branch quiver theory constructions for A series Nilpotent orbits. We present systematic constructions for BCD series Nilpotent orbits on the Higgs branches of quiver theories defined by canonical partitions; this paper collects earlier work into a systematic framework, filling in gaps and providing a complete treatment. We find new Coulomb branch constructions for above minimal Nilpotent orbits, including some based upon twisted affine Dynkin diagrams. We also discuss aspects of 3d mirror symmetry between these Higgs and Coulomb branch constructions and explore dualities and other relationships, such as HyperKahler quotients, between quivers. We analyse all Classical group Nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials.
Amihay Hanany - One of the best experts on this subject based on the ideXlab platform.
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quiver theories and formulae for Nilpotent orbits of exceptional algebras
Journal of High Energy Physics, 2017Co-Authors: Amihay Hanany, Rudolph KalveksAbstract:We treat the topic of the closures of the Nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal Nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal Nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group Nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different Nilpotent orbits and propose candidates for the constructions of some non-normal Nilpotent orbits of Exceptional algebras.
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quiver theories for moduli spaces of classical group Nilpotent orbits
Journal of High Energy Physics, 2016Co-Authors: Amihay Hanany, Rudolph KalveksAbstract:We approach the topic of Classical group Nilpotent orbits from the perspective of the moduli spaces of quivers, described in terms of Hilbert series and generating functions. We review the established Higgs and Coulomb branch quiver theory constructions for A series Nilpotent orbits. We present systematic constructions for BCD series Nilpotent orbits on the Higgs branches of quiver theories defined by canonical partitions; this paper collects earlier work into a systematic framework, filling in gaps and providing a complete treatment. We find new Coulomb branch constructions for above minimal Nilpotent orbits, including some based upon twisted affine Dynkin diagrams. We also discuss aspects of 3d mirror symmetry between these Higgs and Coulomb branch constructions and explore dualities and other relationships, such as HyperKahler quotients, between quivers. We analyse all Classical group Nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials.
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quiver theories for moduli spaces of classical group Nilpotent orbits
arXiv: High Energy Physics - Theory, 2016Co-Authors: Amihay Hanany, Rudolph KalveksAbstract:We approach the topic of Classical group Nilpotent orbits from the perspective of their moduli spaces, described in terms of Hilbert series and generating functions. We review the established Higgs and Coulomb branch quiver theory constructions for A series Nilpotent orbits. We present systematic constructions for BCD series Nilpotent orbits on the Higgs branches of quiver theories defined by canonical partitions; this paper collects earlier work into a systematic framework, filling in gaps and providing a complete treatment. We find new Coulomb branch constructions for above minimal Nilpotent orbits, including some based upon twisted affine Dynkin diagrams. We also discuss aspects of 3d mirror symmetry between these Higgs and Coulomb branch constructions and explore dualities and other relationships, such as HyperKahler quotients, between quivers. We analyse all Classical group Nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials.
V. A. Soroka - One of the best experts on this subject based on the ideXlab platform.
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linear odd poisson bracket on grassmann variables
Physics Letters B, 1999Co-Authors: V. A. SorokaAbstract:Abstract A linear odd Poisson bracket (antibracket) realized solely in terms of Grassmann variables is suggested. It is revealed that the bracket, which corresponds to a semi-simple Lie group, has at once three Grassmann-odd Nilpotent Δ -like differential operators of the first, the second and the third orders with respect to Grassmann derivatives, in contrast with the canonical odd Poisson bracket having the only Grassmann-odd Nilpotent differential Δ -operator of the second order. It is shown that these Δ -like operators together with a Grassmann-odd Nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra.
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linear odd poisson bracket on grassmann variables
arXiv: High Energy Physics - Theory, 1998Co-Authors: V. A. SorokaAbstract:A linear odd Poisson bracket (antibracket) realized solely in terms of Grassmann variables is suggested. It is revealed that the bracket, which corresponds to a semi-simple Lie group, has at once three Grassmann-odd Nilpotent $\Delta$-like differential operators of the first, the second and the third orders with respect to Grassmann derivatives, in contrast with the canonical odd Poisson bracket having the only Grassmann-odd Nilpotent differential $\Delta$-operator of the second order. It is shown that these $\Delta$-like operators together with a Grassmann-odd Nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra.
Eric Sommers - One of the best experts on this subject based on the ideXlab platform.
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Generic singularities of Nilpotent orbit closures
Advances in Mathematics, 2017Co-Authors: Daniel Juteau, Paul Levy, Eric SommersAbstract:Abstract According to a theorem of Brieskorn and Slodowy, the intersection of the Nilpotent cone of a simple Lie algebra with a transverse slice to the subregular Nilpotent orbit is a simple surface singularity. At the opposite extremity of the poset of Nilpotent orbits, the closure of the minimal Nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of Nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type . In the present paper, we complete the picture by determining the generic singularities of all Nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an -variety whose normalization is , an -variety whose normalization is , and a two-dimensional variety whose normalization is the simple surface singularity . In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the Nilpotent cone at a point in the subregular orbit.
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generic singularities of Nilpotent orbit closures
arXiv: Representation Theory, 2015Co-Authors: Daniel Juteau, Paul Levy, Eric SommersAbstract:According to a well-known theorem of Brieskorn and Slodowy, the intersection of the Nilpotent cone of a simple Lie algebra with a transverse slice to the subregular Nilpotent orbit is a simple surface singularity. At the opposite extremity of the Nilpotent cone, the closure of the minimal Nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of Nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type $A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all Nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities which do not occur in the classical types. Three of these are unibranch non-normal singularities: an $SL_2(\mathbb C)$-variety whose normalization is ${\mathbb A}^2$, an $Sp_4(\mathbb C)$-variety whose normalization is ${\mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, in analogy with Slodowy's work for the regular Nilpotent orbit.
Martha Precup - One of the best experts on this subject based on the ideXlab platform.
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Affine pavings of Hessenberg varieties for semisimple groups
Selecta Mathematica, 2013Co-Authors: Martha PrecupAbstract:In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to Nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to Nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular Nilpotent elements in the classical cases and arbitrary elements of $$\mathfrak{gl }_n(\mathbb C )$$ are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology.
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affine pavings of hessenberg varieties for semisimple groups
arXiv: Algebraic Geometry, 2012Co-Authors: Martha PrecupAbstract:In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to Nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to Nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular Nilpotent and arbitrary elements of \mathfrak{gl}_n(\C) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases the Hessenberg variety has no odd dimensional cohomology.
