scholarly journals Invariant Cones in Lie Algebras and Positive Energy Representations and Contractions of Conformal Algebras

2013 ◽  
Vol 462 ◽  
pp. 012037 ◽  
Author(s):  
Patrick Moylan
Keyword(s):  
Lie Algebras ◽  
Positive Energy ◽  
Conformal Algebras ◽  
Invariant Cones

scholarly journals Isomorphisms of Twisted Hilbert Loop Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 453-480
Author(s):  
Timothée Marquis ◽  
Karl-Hermann Neeb
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Energy ◽  
Algebra Structure ◽  
Affine Lie Algebras ◽  
Subsequent Work ◽  
Highest Weight ◽  
Loop Algebras ◽  
Highest Weight Representations

Abstract 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 , also called affinisations of . They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families for some infinite set J. To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra , which we call standard. In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra an explicit isomorphism from g to one of the standard affinisations of . The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g. In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of .

scholarly journals Compactly embedded Cartan algebras and invariant cones in Lie algebras

1989 ◽  
Vol 75 (2) ◽  
pp. 168-201 ◽  
Author(s):  
Joachim Hilgert ◽  
Karl Heinrich Hofmann
Keyword(s):  
Lie Algebras ◽  
Invariant Cones

Invariant cones in Lie algebras

1988 ◽  
Vol 37 (1) ◽  
pp. 241-252 ◽  
Author(s):  
Joachim Hilgert ◽  
Karl H. Hofmann
Keyword(s):  
Lie Algebras ◽  
Invariant Cones

scholarly journals Classification of invariant cones in Lie algebras

1988 ◽  
Vol 19 (2) ◽  
pp. 441-447 ◽  
Author(s):  
Joachim Hilgert ◽  
Karl H. Hofmann
Keyword(s):  
Lie Algebras ◽  
Invariant Cones

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

2021 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Daniel Oeh
Keyword(s):  
Convex Hull ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Convex Hull ◽  
Finite Dimensional ◽  
Adjoint Orbits ◽  
Invariant Cones ◽  
Pointed Convex Cone ◽  
Adjoint Orbit

AbstractIn 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.

scholarly journals Gause Symmetry and Howe Duality in 4D Conformal Field Theory Models

10.14311/1193 ◽  
2010 ◽  
Vol 50 (3) ◽  
Author(s):  
I. Todorov
Keyword(s):  
Lie Algebras ◽  
Gauge Group ◽  
Fock Space ◽  
Scalar Fields ◽  
Dual Pair ◽  
Two Dimensions ◽  
Operator Product ◽  
Positive Energy ◽  
Joint Work ◽  
Irreducible Components

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.The paper reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren and the author.

scholarly journals Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Ivan Todorov
Keyword(s):  
Lie Algebras ◽  
Gauge Group ◽  
Fock Space ◽  
Scalar Fields ◽  
Dual Pair ◽  
Two Dimensions ◽  
Operator Product ◽  
Positive Energy ◽  
Dual Pairs ◽  
Irreducible Components

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.

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