scholarly journals On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1650
Author(s):  
Naveed Hussain ◽  
Stephen S.-T. Yau ◽  
Huaiqing Zuo
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hypersurface Singularity ◽  
New Class ◽  
Low Dimensional ◽  
Upper Estimate ◽  
Derivatives Of ◽  
Analytic Invariant ◽  
The Ideal

Let (V,0)={(z1,…,zn)∈Cn:f(z1,…,zn)=0} be an isolated hypersurface singularity with mult(f)=m. Let Jk(f) be the ideal generated by all k-th order partial derivatives of f. For 1≤k≤m−1, the new object Lk(V) is defined to be the Lie algebra of derivations of the new k-th local algebra Mk(V), where Mk(V):=On/((f)+J1(f)+…+Jk(f)). Its dimension is denoted as δk(V). This number δk(V) is a new numerical analytic invariant. In this article we compute L4(V) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ4(V). We also verify a sharp upper estimate conjecture for the δ4(V) for large class of singularities. Furthermore, we verify another inequality conjecture: δ(k+1)(V)<δk(V),k=3 for low-dimensional fewnomial singularities.

On Automorphisms and Derivations of a Lie Algebra

2006 ◽  
Vol 13 (01) ◽  
pp. 119-132 ◽  
Author(s):  
V. R. Varea ◽  
J. J. Varea
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Nilpotent Lie Algebra ◽  
Identity Component ◽  
Trivial Center

We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals Homotopical aspects of Lie algebras

1993 ◽  
Vol 54 (3) ◽  
pp. 393-419 ◽  
Author(s):  
Graham J. Ellis
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Crossed Module ◽  
Low Dimensional

AbstractThe Hurewicz theorem, Mayer-Vietoris sequence, and Whitehead's certain exact sequence are proved for simplicial Lie algebras. These results are applied, using crossed module techniques, to obtain information on the low dimensional homology of a Lie algebra, and information on aspherical presentations of Lie algebras.

DERIVATION ALGEBRAS OF RINGS RELATED TO HEISENBERG ALGEBRAS

2004 ◽  
Vol 03 (02) ◽  
pp. 181-191 ◽  
Author(s):  
JEFFREY BERGEN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Quotient Ring ◽  
Finite Codimension ◽  
Enveloping Algebras ◽  
Cartan Type ◽  
Direct Sum ◽  
Heisenberg Algebras ◽  
Martindale Quotient Ring ◽  
The Ideal

In this paper, we will determine the Lie algebra of derivations of rings which are generalizations of the enveloping algebras of Heisenberg Lie algebras. First, we will determine which derivations are X-inner and also determine which elements in the Martindale quotient ring induce X-inner derivations. Then, we will show that the Lie algebra of derivations is the direct sum of the ideal of X-inner derivations and a subalgebra which is isomorphic to a subalgebra of finite codimension in a Cartan type Lie algebra.

The classification of three-dimensional Lie algeb­ras on complex field

2020 ◽  
pp. 143-155
Author(s):  
E. R. Shamardina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Structural Equations ◽  
Complex Numbers ◽  
Complex Field ◽  
Low Dimensional

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation. Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

The Construction of Representations of Lie Algebras of Characteristic Zero

1962 ◽  
Vol 14 ◽  
pp. 304-312
Author(s):  
B. Noonan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Basis Element ◽  
A Value ◽  
Representations Of Lie Algebras ◽  
Explicit Representations ◽  
The Ideal

In this paper a procedure is given whereby, from a representation of an ideal contained in the radical, explicit representations of a Lie algebra by matrices can be constructed in an algebraically closed field of characteristic zero. The construction is sufficiently general to permit one arbitrary eigenvalue to be assigned to the representation of each basis element of the radical not in the ideal. The theorem of Ado is proved as an application of the construction. While Ado's theorem has several proofs (1; 3; 5; 6), the present one has a value in its explicitness and in the fact that the degree of the representation can be given.

scholarly journals Lie Algebras, Structure of Nonlinear Systems and Chaotic Motion

1998 ◽  
Vol 08 (07) ◽  
pp. 1437-1462 ◽  
Author(s):  
S. P. Banks ◽  
D. McCaffrey
Keyword(s):  
Nonlinear Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Approximation Theory ◽  
Chaotic Motion ◽  
Chaotic Systems ◽  
Structure Theory ◽  
New Class ◽  
Levi Decomposition

The structure theory of Lie algebras is used to classify nonlinear systems according to a Levi decomposition and the solvable and semisimple parts of a certain Lie algebra associated with the system. An approximation theory is developed and a new class of chaotic systems is introduced, based on the structure theory of Lie algebras.

scholarly journals Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras

2020 ◽  
Author(s):  
Andrey Lazarev ◽  
Yunhe Sheng ◽  
Rong Tang
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Homotopy Theory ◽  
Lie Bialgebras ◽  
Infinitesimal Deformations ◽  
Baxter Operator

Abstract We determine the $$L_\infty $$ L ∞ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$ Lie Rep  pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie$$_\infty $$ ∞ -algebras.

ON THE DERIVATION LIE ALGEBRAS OF FEWNOMIAL SINGULARITIES

2018 ◽  
Vol 98 (1) ◽  
pp. 77-88 ◽  
Author(s):  
NAVEED HUSSAIN ◽  
STEPHEN S.-T. YAU ◽  
HUAIQING ZUO
Keyword(s):  
Lie Algebras ◽  
Hypersurface Singularity ◽  
Isolated Singularity ◽  
Solvable Algebra ◽  
Finite Dimensional ◽  
Hypersurface Singularities ◽  
Surface Singularities ◽  
Moduli Algebra ◽  
Upper Estimate ◽  
Appl Math

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra, $L(V)$, is the Lie algebra of derivations of the moduli algebra of $V$. It is a finite-dimensional solvable algebra and its dimension $\unicode[STIX]{x1D706}(V)$ is the Yau number. Fewnomial singularities are those which can be defined by an $n$-nomial in $n$ indeterminates. Yau and Zuo [‘A sharp upper estimate conjecture for the Yau number of weighted homogeneous isolated hypersurface singularity’, Pure Appl. Math. Q.12(1) (2016), 165–181] conjectured a bound for the Yau number and proved that this conjecture holds for binomial isolated hypersurface singularities. In this paper, we verify this conjecture for weighted homogeneous fewnomial surface singularities.

scholarly journals The nilpotent variety of W(1;n)p is irreducible

2019 ◽  
Vol 18 (03) ◽  
pp. 1950056
Author(s):  
Cong Chen
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Groups ◽  
Closed Field ◽  
Nilpotent Variety ◽  
Restricted Lie Algebra ◽  
Finite Dimensional ◽  
Restricted Lie Algebras ◽  
Minimal Formula

In the late 1980s, Premet conjectured that the nilpotent variety of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic [Formula: see text] is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series [Formula: see text] and [Formula: see text]. In this paper, with the assumption that [Formula: see text], we confirm this conjecture for the minimal [Formula: see text]-envelope [Formula: see text] of the Zassenhaus algebra [Formula: see text] for all [Formula: see text].

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