scholarly journals Lie Supergroups Obtained from 3-Dimensional Lie Superalgebras Associated to the Adjoint Representation and Having a 2-Dimensional Derived Ideal

2008 ◽  
Vol 2008 ◽  
pp. 1-9
Author(s):  
I. Hernández ◽  
R. Peniche
Keyword(s):  
Lie Group ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Manifold ◽  
3 Dimensional ◽  
Lie Supergroups

We give the explicit multiplication law of the Lie supergroups for which the base manifold is a 3-dimensional Lie group and whose underlying Lie superalgebrag=g0⊕g1which satisfiesg1=g0,g0acts ong1via the adjoint representation andg0has a 2-dimensional derived ideal.

scholarly journals Explicit Bracket in an Exceptional Simple Lie Superalgebra

1998 ◽  
Vol 08 (04) ◽  
pp. 479-495 ◽  
Author(s):  
Irina Shchepochkina ◽  
Gerhard Post
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Exact Form ◽  
3 Dimensional

This note is devoted to a more detailed description of one of the five simple exceptional Lie superalgebras of vector fields, [Formula: see text] a subalgebra of [Formula: see text]. We derive differential equations for its elements, and solve these equations. Hence we get an exact form for the elements of [Formula: see text]. Moreover we realize [Formula: see text] by "glued" pairs of generating functions on a (3∣3)-dimensional periplectic (odd symplectic) supermanifold and describe the bracket explicitly.

Supersymmetric analogues of the classical theorem on harmonic polynomials

2014 ◽  
Vol 13 (06) ◽  
pp. 1450011 ◽  
Author(s):  
Cuiling Luo ◽  
Xiaoping Xu
Keyword(s):  
Lie Group ◽  
Polynomial Algebra ◽  
Lie Superalgebra ◽  
Free Module ◽  
Lie Superalgebras ◽  
Classical Theorem ◽  
Harmonic Polynomials ◽  
Invariant Polynomials ◽  
Irreducible Modules ◽  
Two Parameter

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of the Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. In this paper, we first establish two-parameter ℤ2-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra gl(n | m). Then we extend the result to two-parameter ℤ-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras osp(2n | 2m) and osp(2n + 1 | 2m).

scholarly journals On Low-Dimensional Complex ω-Lie Superalgebras

2021 ◽  
Author(s):  
Jia Zhou ◽  
Liangyun Chen
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
3 Dimensional ◽  
Finite Dimensional ◽  
Low Dimensional ◽  
Derivation Superalgebra

Abstract Let (g, [−, −], ω) be a finite-dimensional complex ω-Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra Der(g) and the automorphism group Aut(g) of (g, [−, −], ω). We study Derω (g) and Autω (g), which are superalgebra of Der(g) and subgroup of Aut(g), respectively. For any 3-dimensional or 4-dimensional complex ω-Lie superalgebra g, we explicitly calculate Der(g) and Aut(g), and obtain Jordan standard forms of elements in the two sets. We also study representation theory of ω-Lie superalgebras and give a conclusion that all nontrivial non-ω-Lie 3-dimensional and 4-dimensional ω-Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional ω-Lie superalgebra P2,k(k 6= 0, −1) is 1-dimensional.

DERIVATIONS OF THE EVEN PARTS FOR MODULAR LIE SUPERALGEBRAS OF CARTAN TYPE W AND S

2007 ◽  
Vol 17 (04) ◽  
pp. 661-714 ◽  
Author(s):  
WENDE LIU ◽  
YONGZHENG ZHANG
Keyword(s):  
Lie Algebras ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Field ◽  
Characteristic P ◽  
Derivation Algebra ◽  
Finite Dimensional ◽  
Outer Derivation ◽  
Generator Sets

Let 𝔽 be the underlying base field of characteristic p < 3 and denote by [Formula: see text] and [Formula: see text] the even parts of the finite-dimensional generalized Witt Lie superalgebra W and the special Lie superalgebra S, respectively. We first give the generator sets of the Lie algebras [Formula: see text] and [Formula: see text]. Using certain properties of the canonical tori of [Formula: see text] and [Formula: see text], we then determine the derivation algebra of [Formula: see text] and the derivation space of [Formula: see text] to [Formula: see text], where [Formula: see text] is viewed as a [Formula: see text]-module by means of the adjoint representation. As a result, we describe explicitly the derivation algebra of [Formula: see text]. Furthermore, we prove that the outer derivation algebras of [Formula: see text] and [Formula: see text] are abelian Lie algebras or metabelian Lie algebras with explicit structure. In particular, we give the dimension formulas of the derivation algebras and outer derivation algebras of [Formula: see text] and [Formula: see text]. Thus, we may make a comparison between the even parts of the (outer) superderivation algebras of W and S and the (outer) derivation algebras of the even parts of W and S, respectively.

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.

scholarly journals Correction: Çakmak, A. New Type Direction Curves in 3-Dimensional Compact Lie Group Symmetry 2019, 11, 387

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 284
Author(s):  
Ali Çakmak
Keyword(s):  
Lie Group ◽  
Group Symmetry ◽  
Compact Lie Group ◽  
3 Dimensional ◽  
New Type ◽  
Lie Group Symmetry

The authors wish to make the following corrections to their paper [...]

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.

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