Covariant Homogeneous Nets of Standard Subspaces

Morinelli, Vincenzo; Neeb, Karl-Hermann

doi:10.1007/s00220-021-04046-6

Cited by 20 publications

(25 citation statements)

References 56 publications

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“…Example 4.13 Let (V , ω) be a finite dimensional symplectic vector space. We consider g = g(sp(V , ω), V , R, ω) Then the Levi complement is sp(V , ω), and all Euler elements h in sp(V , ω) are conjugate to 1 2 τ , where τ is an antisymplectic involution on V [9,Prop. 3.11(b)].…”

Section: Remark 410mentioning

confidence: 99%

“…Since h is an Euler element, the subalgebra s h ⊆ l generated by l ±1 (h) is a semisimple ideal of l which is a direct sum of hermitian simple ideals of tube type [9,Prop. 3.11(b)].…”

Section: Remark 416 (A)mentioning

confidence: 99%

“…], then it induces on l ∼ = g/u an Euler derivation with l 0 (D) = [l 1 (D), l −1 (D)], and this implies in particular that l is semisimple and a direct sum of simple hermitian ideals of tube type [9,Prop. 3.11(b)].…”

Section: Remark 416 (A)mentioning

confidence: 99%

“…If g is simple hermitian, then every derivation of g is inner and an Euler derivation exists if and only if g is of tube type[9, Prop. 3.11(b)].…”

mentioning

confidence: 99%

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Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

Neeb

Oeh

2021

Bull. Iran. Math. Soc.

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In this note, we study in a finite dimensional Lie algebra $${\mathfrak g}$$ g the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone $$C_x$$ C x . Assuming that $${\mathfrak g}$$ g is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying $$[h,x]=x$$ [ h , x ] = x for which $$C_x$$ C x pointed. Given x, we show that such elements h can be constructed in such a way that $$\mathop {\mathrm{ad}}\nolimits h$$ ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.

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Section: Remark 410mentioning

confidence: 99%

“…Since h is an Euler element, the subalgebra s h ⊆ l generated by l ±1 (h) is a semisimple ideal of l which is a direct sum of hermitian simple ideals of tube type [9,Prop. 3.11(b)].…”

Section: Remark 416 (A)mentioning

confidence: 99%

Section: Remark 416 (A)mentioning

confidence: 99%

“…If g is simple hermitian, then every derivation of g is inner and an Euler derivation exists if and only if g is of tube type[9, Prop. 3.11(b)].…”

mentioning

confidence: 99%

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Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

Neeb

Oeh

2021

Bull. Iran. Math. Soc.

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“…In all these constructions, a good understanding of the class of generalized wedge domains is of crucial importance. This motivated the abstract approach to these domains in [MN20], where the set…”

Section: And the Table Above)mentioning

confidence: 99%

Nets of standard subspaces on Lie groups

Neeb¹,

Ólafsson²

2020

Preprint

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Let G be a Lie group with Lie algebra g, h ∈ g an element for which the derivation ad h defines a 3-grading of g and τG an involutive automorphism of G inducing on g the involution e πi ad h . We consider antiunitary representations (U, H) of the Lie group Gτ = G ⋊ {1, τG} for which the positive cone CU = {x ∈ g : − i∂U (x) ≥ 0} and h span g. To a real subspace E ⊆ H −∞ of distribution vectors invariant under exp(Rh) and an open subset O ⊆ G, we associate the real subspaceFor the real standard subspace V ⊆ H, for which JV = U (τG) is the modular conjugation and ∆ −it/2π V = U (exp th) is the modular group, we obtain sufficient conditions to be of the form H E (S) for an open subsemigroup S ⊆ G. If g is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspaces H E (O), O ⊆ G, satisfying the Bisognano-Wichmann property in a suitable sense. Our construction also yields such nets on simple Jordan space-times and compactly causal symmetric spaces of Cayley type. By second quantization, these nets lead to free quantum fields in the sense of Haag-Kastler on causal homogeneous spaces whose groups are generated by modular groups and conjugations.

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Algebraic Quantum Field Theory and Causal Symmetric Spaces

Neeb

Ólafsson

2023

Trends in Mathematics

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Covariant Homogeneous Nets of Standard Subspaces

Cited by 20 publications

References 56 publications

Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

Nets of standard subspaces on Lie groups

Algebraic Quantum Field Theory and Causal Symmetric Spaces

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