Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Naveed Hussain ◽  
Stephen S.-T. Yau ◽  
Huaiqing Zuo
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Complete Intersection ◽  
Direct Product ◽  
Characterization Theorem ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Solvable Lie Algebra ◽  
Simple Singularities

Abstract The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals k-step nilpotent Lie algebras

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michel Goze ◽  
Elisabeth Remm
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Deformation Process ◽  
Nilpotent Lie Algebras ◽  
Early Stages ◽  
Finite Dimensional ◽  
Contraction Process ◽  
Finite Dimensional Lie Algebras

AbstractThe classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other

scholarly journals Hom-structures on semi-simple Lie algebras

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Wenjuan Xie ◽  
Quanqin Jin ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Simple Lie Algebras ◽  
3 Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Reduction Methods

AbstractA Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

scholarly journals Graded modules over simple Lie algebras

2019 ◽  
Vol 53 (supl) ◽  
pp. 45-86
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Necessary And Sufficient

The paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras. In general, a connection between these two objects is given by the so-called loop construction. We review the main features of this construction as well as necessary and sufficient conditions under which finite-dimensional simple modules can be graded. Over the Lie algebra sl2(C), we consider specific gradings on simple modules of arbitrary dimension.

Euclidean Lie Algebras

1969 ◽  
Vol 21 ◽  
pp. 1432-1454 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Cartan Matrix ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Generalized Cartan Matrix

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.

ON THETA PAIRS FOR A MAXIMAL SUBALGEBRA

2012 ◽  
Vol 11 (01) ◽  
pp. 1250001 ◽  
Author(s):  
ALI REZA SALEMKAR ◽  
SARA CHEHRAZI ◽  
SOMAIEH ALIZADEH NIRI
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Nonzero Ideal ◽  
Maximal Subgroups ◽  
Maximal Subalgebra ◽  
Nilpotent Lie Algebras ◽  
Finite Dimensional ◽  
A Finite Group ◽  
Number Of Authors

Given a maximal subalgebra M of a finite-dimensional Lie algebra L, a θ-pair for M is a pair (A, B) of subalgebras such that A ≰ M, B is an ideal of L contained in A ∩ M, and A/B includes properly no nonzero ideal of L/B. This is analogous to the concept of θ-pairs associated to maximal subgroups of a finite group, which has been studied by a number of authors. A θ-pair (A, B) for M is said to be maximal if M has no θ-pair (C, D) such that A < C. In this paper, we obtain some properties of maximal θ-pairs and use them to give some characterizations of solvable, supersolvable and nilpotent Lie algebras.

On Derivations of Lie Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 174-180 ◽  
Author(s):  
Stephen Berman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Cartan Matrices

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

INFINITE-DIMENSIONAL GENERALIZATIONS OF SIMPLE LIE ALGEBRAS

1990 ◽  
Vol 05 (24) ◽  
pp. 1967-1977 ◽  
Author(s):  
E. S. FRADKIN ◽  
V. YA. LINETSKY
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Higher Spin ◽  
Higher Spin Theory

Infinite-dimensional algebras associated with simple finite-dimensional Lie algebra g are considered. Higher-spin generalizations of sl(2) are studied in detail. Those of the Virasoro algebra are viewed as their "analytic continuations". Applications in higher-spin theory and in conformal QFT are discussed.

scholarly journals Abelian Ideals with Given Dimension in Borel Subalgebras

2012 ◽  
Vol 19 (04) ◽  
pp. 755-770
Author(s):  
Li Luo
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subalgebra ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Dimensional Distribution

A well-known Peterson's theorem says that the number of abelian ideals in a Borel subalgebra of a rank-r finite-dimensional simple Lie algebra is exactly 2r. In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite-dimensional simple Lie algebras, which is a refinement of Peterson's theorem capturing more Lie algebra invariants.

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