scholarly journals The ℤ2 × ℤ2-graded Lie superalgebras pso(2n + 1|2n) and pso(∞|∞), and parastatistics Fock spaces

2021 ◽  
Author(s):  
Neli Ilieva Stoilova ◽  
Joris Van der Jeugt
Keyword(s):  
Infinite Number ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Fock Spaces

Abstract The parastatistics Fock spaces of order p corresponding to an infinite number of parafermions and parabosons with relative paraboson relations are constructed. The Fock spaces are lowest weight representations of the ℤ2 × ℤ2-graded Lie superalgebra pso(∞|∞), with a basis consisting of row-stable Gelfand-Zetlin patterns.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

QUANTUM HAMILTONIAN REDUCTION AND WB ALGEBRA

1992 ◽  
Vol 07 (20) ◽  
pp. 4885-4898 ◽  
Author(s):  
KATSUSHI ITO
Keyword(s):  
Lie Algebras ◽  
Free Field ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Hamiltonian Reduction ◽  
Affine Lie Algebras ◽  
Quantum Hamiltonian Reduction ◽  
Free Field Realization ◽  
Coset Construction ◽  
Quantum Hamiltonian

We study the quantum Hamiltonian reduction of affine Lie algebras and the free field realization of the associated W algebra. For the nonsimply laced case this reduction does not agree with the usual coset construction of the W minimal model. In particular, we find that the coset model [Formula: see text] can be obtained through the quantum Hamiltonian reduction of the affine Lie superalgebra B(0, n)(1). To show this we also construct the Feigin-Fuchs representation of affine Lie superalgebras.

The Finite-dimensional Modular Lie Superalgebra Γ

2010 ◽  
Vol 17 (03) ◽  
pp. 525-540 ◽  
Author(s):  
Xiaoning Xu ◽  
Yongzheng Zhang ◽  
Liangyun Chen
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Explicit Description ◽  
Cartan Type ◽  
Modular Lie Superalgebra ◽  
New Family ◽  
Finite Dimensional ◽  
Derivation Superalgebra ◽  
Modular Lie Superalgebras

A new family of finite-dimensional modular Lie superalgebras Γ is defined. The simplicity and generators of Γ are studied and an explicit description of the derivation superalgebra of Γ is given. Moreover, it is proved that Γ is not isomorphic to any known Lie superalgebra of Cartan type.

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.

Double Derivations of n-Lie Superalgebras

2018 ◽  
Vol 25 (01) ◽  
pp. 161-180
Author(s):  
Bing Sun ◽  
Liangyun Chen ◽  
Xin Zhou
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
General Linear ◽  
Base Field ◽  
Inner Derivation ◽  
Derivation Algebra ◽  
General Linear Lie Superalgebra

Let 𝔤 be an n-Lie superalgebra. We study the double derivation algebra [Formula: see text] and describe the relation between [Formula: see text] and the usual derivation Lie superalgebra Der(𝔤). We show that the set [Formula: see text] of all double derivations is a subalgebra of the general linear Lie superalgebra gl(𝔤) and the inner derivation algebra ad(𝔤) is an ideal of [Formula: see text]. We also show that if 𝔤 is a perfect n-Lie superalgebra with certain constraints on the base field, then the centralizer of ad(𝔤) in [Formula: see text] is trivial. Finally, we give that for every perfect n-Lie superalgebra 𝔤, the triple derivations of the derivation algebra Der(𝔤) are exactly the derivations of Der(𝔤).

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 Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

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.

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