Highest weight irreducible unitary representations of Lie algebras of infinite matrices. I. The algebra gl(∞)

1990 ◽  
Vol 31 (3) ◽  
pp. 579-586 ◽  
Author(s):  
Tchavdar D. Palev
Keyword(s):  
Lie Algebras ◽  
Unitary Representations ◽  
Infinite Matrices ◽  
Highest Weight ◽  
Representations Of Lie Algebras ◽  
Irreducible Unitary Representations

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

scholarly journals On representations of lie algebras for quantized hamiltonians

1997 ◽  
Vol 266 ◽  
pp. 69-79 ◽  
Author(s):  
L.A-M. Hanna ◽  
M.E. Khalifa ◽  
S.S. Hassan
Keyword(s):  
Lie Algebras ◽  
Representations Of Lie Algebras

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 .

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