scholarly journals PARAFERMIONS, PARABOSONS AND REPRESENTATIONS OF 𝔰𝔬(∞) AND 𝔬𝔰𝔭(1|∞)

2009 ◽  
Vol 20 (06) ◽  
pp. 693-715 ◽  
Author(s):  
N. I. STOILOVA ◽  
J. VAN DER JEUGT
Keyword(s):  
Lie Algebra ◽  
Fock Space ◽  
Complete Solution ◽  
Lie Superalgebra ◽  
Explicit Construction ◽  
Fock Spaces ◽  
Infinite Rank ◽  
Infinite Set ◽  
Basis Vectors

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra 𝔰𝔬(∞) and of the Lie superalgebra 𝔬𝔰𝔭(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand–Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations.

Some studies on central derivation of nilpotent Lie superalgebra

2018 ◽  
Vol 13 (04) ◽  
pp. 2050068
Author(s):  
Rudra Narayan Padhan ◽  
K. C. Pati
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Nilpotent Lie Algebra ◽  
New Concepts

Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case as the later type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, Saeedi and Sheikh-Mohseni [A characterization of stem algebras in terms of central derivations, Algebr. Represent. Theory 20 (2017) 1143–1150; On [Formula: see text]-derivations of Filippov algebra, to appear in Asian-Eur. J. Math.; S. Sheikh-Mohseni, F. Saeedi and M. Badrkhani Asl, On special subalgebras of derivations of Lie algebras, Asian-Eur. J. Math. 8(2) (2015) 1550032] recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclinism in Lie superalgebra is being recently introduced.

scholarly journals Factorization of the canonical bases for higher-level Fock spaces

2011 ◽  
Vol 55 (1) ◽  
pp. 23-51 ◽  
Author(s):  
Susumu Ariki ◽  
Nicolas Jacon ◽  
Cédric Lecouvey
Keyword(s):  
Fock Space ◽  
Fock Spaces ◽  
Transition Matrices ◽  
Canonical Bases ◽  
Highest Weight ◽  
Highest Weight Modules ◽  
Graphic Module ◽  
Module Structures ◽  
Weight Modules

AbstractThe level l Fock space admits canonical bases $\mathcal{G}_{e}$ and $\smash{\mathcal{G}_{\infty}}$. They correspond to $\smash{\mathcal{U}_{v}(\widehat{\mathfrak{sl}}_{e})}$ and $\mathcal{U}_{v}({\mathfrak{sl}}_{\infty})$-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in ℕ[v]. Restriction to the highest-weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki–Koike algebras.

scholarly journals Interacting Fock spaces and subproduct systems

2020 ◽  
Vol 23 (03) ◽  
pp. 2050017
Author(s):  
Malte Gerhold ◽  
Michael Skeide
Keyword(s):  
Selfadjoint Operator ◽  
Operator Algebras ◽  
Fock Space ◽  
Fock Spaces ◽  
Full Generality ◽  
Necessary And Sufficient Criteria ◽  
Definition Of ◽  
Interacting Fock Space ◽  
Necessary And Sufficient

We present a new more flexible definition of interacting Fock space that allows to resolve in full generality the problem of embeddability. We show that the same is not possible for regularity. We apply embeddability to classify interacting Fock spaces by squeezings. We give necessary and sufficient criteria for when an interacting Fock space has only bounded creators, giving thus rise to new classes of non-selfadjoint and selfadjoint operator algebras.

scholarly journals A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Do Ngoc Diep
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Vector Bundles ◽  
Sigma Model ◽  
Fock Space ◽  
Loop Algebra ◽  
Target Space ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
Geometric Langlands

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry groupGL. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry groupGL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry groupGL. Use the electric-magnetic duality to pass to the Langlands dual Lie groupG. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra=Lie(G). Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groupsG.

scholarly journals Complexity for modules over the classical Lie superalgebra

2012 ◽  
Vol 148 (5) ◽  
pp. 1561-1592 ◽  
Author(s):  
Brian D. Boe ◽  
Jonathan R. Kujawa ◽  
Daniel K. Nakano
Keyword(s):  
Lie Algebra ◽  
Lie Superalgebra ◽  
Projective Resolution ◽  
Lie Superalgebras ◽  
Simple Modules ◽  
Minimal Projective Resolution ◽  
Finite Dimensional ◽  
Reductive Lie Algebra ◽  
Completely Reducible ◽  
Kac Modules

AbstractLet ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

Restricted Envelopes of Lie Superalgebras

2015 ◽  
Vol 22 (02) ◽  
pp. 309-320
Author(s):  
Liping Sun ◽  
Wende Liu ◽  
Xiaocheng Gao ◽  
Boying Wu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Cartan Type ◽  
Modular Lie Algebras ◽  
Modular Lie Superalgebras

Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.

scholarly journals Deformed Fock spaces, Hecke operators and monotone Fock space of Muraki

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Marek Bożejko
Keyword(s):  
Fock Space ◽  
Hecke Operators ◽  
Fock Spaces

AbstractThe main purpose of this paper is to extend our previous construction of

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