Affine Lie Algebra Modules and Complete Lie Algebras

2006 ◽  
Vol 13 (03) ◽  
pp. 481-486
Author(s):  
Yongcun Gao ◽  
Daoji Meng
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Affine Lie Algebras ◽  
Infinite Dimensional ◽  
Affine Lie Algebra ◽  
Parabolic Subalgebras ◽  
Integrable Modules

In this paper, we first construct some new infinite dimensional Lie algebras by using the integrable modules of affine Lie algebras. Then we prove that these new Lie algebras are complete. We also prove that the generalized Borel subalgebras and the generalized parabolic subalgebras of these Lie algebras are complete.

scholarly journals FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350062 ◽  
Author(s):  
DRAŽEN ADAMOVIĆ ◽  
OZREN PERŠE
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Operator Algebras ◽  
Affine Lie Algebras ◽  
Fusion Rules ◽  
Type Formula ◽  
Complete Reducibility ◽  
Affine Lie Algebra ◽  
Branching Rules ◽  
Level 1

We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.

The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras

2008 ◽  
Vol 51 (2) ◽  
pp. 298-309 ◽  
Author(s):  
Maribel Tocón
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Affine Lie Algebras ◽  
The Core ◽  
Affine Lie Algebra ◽  
Extended Affine Lie Algebra

AbstractIn this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra.

FEIGIN-FUCHS REPRESENTATIONS OF ARBITRARY AFFINE LIE ALEGBRAS

1991 ◽  
Vol 06 (07) ◽  
pp. 581-589 ◽  
Author(s):  
KATSUSHI ITO ◽  
SHIRO KOMATA
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Flag Manifolds ◽  
Affine Lie Algebras ◽  
Canonical Coordinates ◽  
Systematic Method ◽  
Text Type ◽  
Affine Lie Algebra ◽  
Arbitrary Affine

We develop a systematic method to obtain the Feigin-Fuchs representations for arbitrary affine Lie algebras from a geometrical viewpoint. Choosing canonical coordinates for the flag manifolds associated with the Lie algebras, we get the general formulae for the KacMoody currents. In particular, we use this method to construct explicitly the Feigin-Fuchs representations for the affine Lie algebra of [Formula: see text] type.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

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 .

On Weight Systems Derived from Heisenberg Lie Algebras

2003 ◽  
Vol 12 (05) ◽  
pp. 589-604
Author(s):  
Hideaki Nishihara
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Weight System ◽  
Infinite Dimensional ◽  
Conway Polynomial ◽  
Knot Invariant ◽  
Polynomial Representations ◽  
Solvable Lie Algebras

Weight systems are constructed with solvable Lie algebras and their infinite dimensional representations. With a Heisenberg Lie algebra and its polynomial representations, the derived weight system vanishes on Jacobi diagrams with positive loop-degree on a circle, and it is proved that the derived knot invariant is the inverse of the Alexander-Conway polynomial.

Pseudo-Differential Operators $(\psi{\cal D}{\cal O})$ Algebra and Certain Infinite-Dimensional Lie Algebras

1997 ◽  
Vol 12 (22) ◽  
pp. 1589-1595 ◽  
Author(s):  
E. H. El Kinani
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Differential Operators ◽  
Quantum Plane ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators ◽  
Physics Literature

The class of pseudo-differential operators Lie algebra [Formula: see text] on the quantum plane [Formula: see text] is introduced. The embedding of certain infinite-dimensional Lie algebras which occur in the physics literature in [Formula: see text] is discussed as well as the correspondence between [Formula: see text] and [Formula: see text] as k→+∞ is examined.

scholarly journals Integrable modules for twisted toroidal extended affine Lie algebras

2020 ◽  
Vol 556 ◽  
pp. 1057-1072
Author(s):  
S. Eswara Rao ◽  
Sachin S. Sharma ◽  
Punita Batra
Keyword(s):  
Lie Algebras ◽  
Affine Lie Algebras ◽  
Integrable Modules

scholarly journals Quantum black holes, elliptic genera and spectral partition functions

2014 ◽  
Vol 11 (05) ◽  
pp. 1450048 ◽  
Author(s):  
A. A. Bytsenko ◽  
M. Chaichian ◽  
R. J. Szabo ◽  
A. Tureanu
Keyword(s):  
Black Hole ◽  
Lie Algebras ◽  
Modular Forms ◽  
Partition Functions ◽  
Affine Lie Algebras ◽  
Hilbert Schemes ◽  
Spectral Functions ◽  
Infinite Dimensional ◽  
Elliptic Genera ◽  
Bps Invariants

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi–Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.

scholarly journals ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES

2001 ◽  
Vol 03 (04) ◽  
pp. 533-548 ◽  
Author(s):  
NAIHUAN JING ◽  
KAILASH C. MISRA ◽  
CARLA D. SAVAGE
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Mathematical Physics ◽  
Infinite Dimensional ◽  
Algebra Representation ◽  
The Sixties ◽  
Lie Algebra Representation ◽  
New Interpretation

Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

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