The Experts below are selected from a list of 117 Experts worldwide ranked by ideXlab platform
Karl-hermann Neeb - One of the best experts on this subject based on the ideXlab platform.
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Positive energy representations of double extensions of Hilbert loop algebras
Journal of the Mathematical Society of Japan, 2017Co-Authors: Timothée Marquis, Karl-hermann NeebAbstract:A real Lie algebra with a compatible Hilbert Space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert-Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac-Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a Root Space Decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding Root system yields one of the seven locally affine Root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below.
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Isomorphisms of Twisted Hilbert Loop Algebras
Canadian Journal of Mathematics, 2017Co-Authors: Timothée Marquis, Karl-hermann NeebAbstract:The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert Spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a Root Space Decomposition whose corresponding Root system is a locally affine Root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine Root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the Root Decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.
Tathagata Basak - One of the best experts on this subject based on the ideXlab platform.
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Root Space Decomposition of g2 from octonions
Journal of Pure and Applied Algebra, 2018Co-Authors: Tathagata BasakAbstract:Abstract We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of g 2 . This uses the description of octonions as a twisted group algebra of the finite field F 8 . Generators of Gal ( F 8 / F 2 ) act on the Roots as 120-degree rotations and complex conjugation acts as negation.
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Root Space Decomposition of $\mathfrak{g}_2$ from octonions
arXiv: Representation Theory, 2017Co-Authors: Tathagata BasakAbstract:We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of $\mathfrak{g}_2$. This uses the description of octonions as a twisted group algebra of the finite field $\mathbb{F}_8$. Generators of $\operatorname{Gal}(\mathbb{F}_8/\mathbb{F}_2)$ act on the Roots as $120$-degree rotations and complex conjugation acts as negation.
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Root Space Decomposition of mathfrak g _2 from octonions
arXiv: Representation Theory, 2017Co-Authors: Tathagata BasakAbstract:We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of $\mathfrak{g}_2$. This uses the description of octonions as a twisted group algebra of the finite field $\mathbb{F}_8$. Generators of $\operatorname{Gal}(\mathbb{F}_8/\mathbb{F}_2)$ act on the Roots as $120$-degree rotations and complex conjugation acts as negation.
Timothée Marquis - One of the best experts on this subject based on the ideXlab platform.
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Positive energy representations of double extensions of Hilbert loop algebras
Journal of the Mathematical Society of Japan, 2017Co-Authors: Timothée Marquis, Karl-hermann NeebAbstract:A real Lie algebra with a compatible Hilbert Space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert-Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac-Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a Root Space Decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding Root system yields one of the seven locally affine Root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below.
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Isomorphisms of Twisted Hilbert Loop Algebras
Canadian Journal of Mathematics, 2017Co-Authors: Timothée Marquis, Karl-hermann NeebAbstract:The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert Spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a Root Space Decomposition whose corresponding Root system is a locally affine Root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine Root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the Root Decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.
Giuseppe Marchei - One of the best experts on this subject based on the ideXlab platform.
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A Root Space Decomposition for Finite Vertex Algebras
Documenta Mathematica, 2012Co-Authors: Alessandro D'andrea, Giuseppe MarcheiAbstract:Let L be a Lie pseudoalgebra, a ∈ L. We show that, if a generates a (finite) solvable subalgebra S = hai ⊂L, then one may find a lifting ¯ a ∈ S of (a) ∈ S/Ssuch that hi is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a Decomposition into a semi-direct product V = U⋉N, where U is a subalgebra of V whose underlying Lie conformal algebra U Lie is a nilpotent self-normalizing subalgebra of V Lie , and N = V (∞) is a canonically determined ideal contained in
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A Root Space Decomposition for finite vertex algebras
arXiv: Quantum Algebra, 2011Co-Authors: Alessandro D'andrea, Giuseppe MarcheiAbstract:Let L be a Lie pseudoalgebra, a in L. We show that, if a generates a (finite) solvable subalgebra S= , then one may find a lifting a' in S of [a] in S/S' such that is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a Decomposition into a semi-direct product V = U + N, where U is a subalgebra of V whose underlying Lie conformal algebra U^lie is a nilpotent self-normalizing subalgebra of V^lie, and N is a canonically determined ideal contained in the nilradical Nil V.
Dietrich Burde - One of the best experts on this subject based on the ideXlab platform.
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simple left symmetric algebras with solvable lie algebra
Manuscripta Mathematica, 1998Co-Authors: Dietrich BurdeAbstract:Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group {G} correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector Space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical Root Space Decomposition. A special class of complete LSAs is studied.
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Simple left-symmetric algebras with¶solvable Lie algebra
manuscripta mathematica, 1998Co-Authors: Dietrich BurdeAbstract:Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group { G } correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector Space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical Root Space Decomposition. A special class of complete LSAs is studied.
