The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras

2011 ◽  
Vol 63 (6) ◽  
pp. 1364-1387 ◽  
Author(s):  
Eckhard Meinrenken
Keyword(s):  
Dirac Operator ◽  
Tensor Product ◽  
Lie Algebras ◽  
Clifford Algebras ◽  
Symmetric Bilinear Form ◽  
Affine Lie Algebras ◽  
Weil Algebra ◽  
Casimir Element ◽  
Infinite Dimensional ◽  
Graded Lie Algebra

AbstractLet be an infinite-dimensional graded Lie algebra, with dim , equipped with a non-degenerate symmetric bilinear form B of degree 0. The quantum Weil algebra is a completion of the tensor product of the enveloping and Clifford algebras of g. Provided that the Kac–Peterson class of g vanishes, one can construct a cubic Dirac operator D 2 , whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac– Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl–Kac character formulas for suitable “equal rank” Lie subalgebras of Kac–Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

scholarly journals Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 82 ◽  
Author(s):  
Namhee Kwon
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Affine Lie Algebras ◽  
Series Representations ◽  
Level 2

We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl ^ 2 . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights.

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 THE SPECTRA OF ${{\widehat{sl}\left( N \right)_k } \mathord{\left/ {\vphantom {{\widehat{sl}\left( N \right)_k } {\widehat{sl}\left( N \right)_k }}} \right. \kern-\nulldelimiterspace} {\widehat{sl}\left( N \right)_k }}$ AND WN GRAVITIES

1993 ◽  
Vol 08 (29) ◽  
pp. 5115-5128
Author(s):  
VLADIMIR SADOV
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Hamiltonian Reduction ◽  
Affine Lie Algebras ◽  
Brst Cohomology ◽  
Boundary States ◽  
One To One ◽  
Coset Models

We study the spectra of G/G coset models by computing BRST cohomology of affine Lie algebras with coefficients in tensor product of two modules. One-to-one correspondence between the spectra of [Formula: see text] and that of the minimal matter coupled to gravity (including boundary states of the Kac table) is observed. This phenomenon is discussed from the point of Hamiltonian reduction of BRST complexes of [Formula: see text] Lie algebras.

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

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 Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

1999 ◽  
Vol 67 (2) ◽  
pp. 157-184 ◽  
Author(s):  
A. Caranti ◽  
S. Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Loop Algebra ◽  
Maximal Class ◽  
Graded Lie Algebras ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Wide Range ◽  
Graded Lie Algebra

AbstractWe investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours.

scholarly journals Diamond distances in Nottingham algebras

2021 ◽  
Author(s):  
M. Avitabile ◽  
S. Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Graded Lie Algebras ◽  
Infinite Dimensional ◽  
Graded Lie Algebra ◽  
The Difference ◽  
Dimension One

Nottingham algebras are a class of just-infinite-dimensional, modular, [Formula: see text]-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree [Formula: see text], and the second occurs in degree [Formula: see text], a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to [Formula: see text]. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra [Formula: see text] can be assigned a type, in such a way that the degrees and types of the diamonds completely describe [Formula: see text]. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals [Formula: see text]. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type [Formula: see text].

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