scholarly journals Complex structures on the complexification of a real Lie algebra

2018 ◽  
Vol 5 (1) ◽  
pp. 150-157
Author(s):  
Takumi Yamada
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Complex Structures ◽  
Direct Sum ◽  
Direct Sum Decomposition

AbstractLet g = a+b be a Lie algebra with a direct sum decomposition such that a and b are Lie subalgebras. Then, we can construct an integrable complex structure J̃ on h = ℝ(gℂ) from the decomposition, where ℝ(gℂ) is a real Lie algebra obtained from gℂby the scalar restriction. Conversely, let J̃ be an integrable complex structure on h = ℝ(gℂ). Then, we have a direct sum decomposition g = a + b such that a and b are Lie subalgebras. We also investigate relations between the decomposition g = a + b and dim Hs.t∂̄J̃ (hℂ).

scholarly journals VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

2018 ◽  
Vol 61 (3) ◽  
pp. 588-607 ◽  
Author(s):  
Honglei Lang ◽  
Yunhe Sheng ◽  
Aïssa Wade
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Contact Structures ◽  
Complex Structures ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Generalized Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Geometry ◽  
Generalized Complex

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

scholarly journals Extending invariant complex structures

2015 ◽  
Vol 26 (11) ◽  
pp. 1550096 ◽  
Author(s):  
Rutwig Campoamor Stursberg ◽  
Isolda E. Cardoso ◽  
Gabriela P. Ovando
Keyword(s):  
Lie Algebra ◽  
Algebraic Structure ◽  
Symplectic Structure ◽  
Complex Structure ◽  
Complex Structures ◽  
Additional Structure ◽  
Totally Real

We study the problem of extending a complex structure to a given Lie algebra 𝔤, which is firstly defined on an ideal 𝔥 ⊂ 𝔤. We consider the next situations: 𝔥 is either complex or it is totally real. The next question is to equip 𝔤 with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either 𝔥 is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of 𝔤. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.

scholarly journals A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

2021 ◽  
Vol 32 (1) ◽  
pp. 9-32
Author(s):  
C. Choi ◽  
◽  
S. Kim ◽  
H. Seo ◽  
◽  
...  
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Laurent Polynomials ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Weight Multiplicity ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Multiplicity Free

We first present a filtration on the ring Ln of Laurent polynomials such that the direct sum decomposition of its associated graded ring grLn agrees with the direct sum decomposition of grLn, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in the associated graded ring grLn, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).

scholarly journals Some relations between Hodge numbers and invariant complex structures on compact nilmanifolds

2017 ◽  
Vol 4 (1) ◽  
pp. 73-83
Author(s):  
Takumi Yamada
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Complex Structure ◽  
Complex Structures ◽  
Nilpotent Lie Group ◽  
Hodge Numbers ◽  
Compact Nilmanifolds ◽  
Simply Connected ◽  
Invariant Complex Structure

AbstractLet N be a simply connected real nilpotent Lie group, n its Lie algebra, and € a lattice in N. If a left-invariant complex structure on N is Γ-rational, then HƏ̄s,t(Γ/N) ≃ HƏ̄s,t(nC) for each s; t. We can construct different left-invariant complex structures on one nilpotent Lie group by using the complexification and the scalar restriction. We investigate relationships to Hodge numbers of associated compact complex nilmanifolds.

Nijenhuis operators, product structures and complex structures on Lie–Yamaguti algebras

2020 ◽  
pp. 2150146
Author(s):  
Yunhe Sheng ◽  
Jia Zhao ◽  
Yanqiu Zhou
Keyword(s):  
Compatibility Condition ◽  
Complex Structure ◽  
Product Structure ◽  
Vector Spaces ◽  
Complex Structures ◽  
Complex Product ◽  
Direct Sum ◽  
Product Structures

In this paper, first, we study linear deformations of a Lie–Yamaguti algebra and introduce the notion of a Nijenhuis operator. Then we introduce the notion of a product structure on a Lie–Yamaguti algebra, which is a Nijenhuis operator [Formula: see text] satisfying [Formula: see text]. There is a product structure on a Lie–Yamaguti algebra if and only if the Lie–Yamaguti algebra is the direct sum of two subalgebras (as vector spaces). There are some special product structures, each of which corresponds to a special decomposition of the original Lie–Yamaguti algebra. In the same way, we introduce the notion of a complex structure on a Lie–Yamaguti algebra. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Lie–Yamaguti algebra.

scholarly journals Dual-depth Adapted Irreducible Formal Multizeta Values

2013 ◽  
Vol 113 (1) ◽  
pp. 53
Author(s):  
Leila Schneps
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Vector Subspace ◽  
Canonical Isomorphism ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Depth Filtration ◽  
Vector Space Isomorphism ◽  
Canonical Vector ◽  
Standard Weight

Let $\mathfrak{ds}$ denote the double shuffle Lie algebra, equipped with the standard weight grading and depth filtration; we write $\mathfrak{ds}=\oplus_{n\ge 3} \mathfrak{ds}_n$ and denote the filtration by $\mathfrak{ds}^1\supset \mathfrak{ds}^2\supset \cdots$. The double shuffle Lie algebra is dual to the new formal multizeta space $\mathfrak{nfz}=\oplus_{n\ge 3} \mathfrak{nfz}_n$, which is equipped with the dual depth filtration $\mathfrak{nfz}^1\subset \mathfrak{nfz}^2\subset\cdots$ Via an explicit canonical isomorphism $\mathfrak{ds}\buildrel \sim\over\rightarrow\mathfrak{nfz}$, we define the "dual" in $\mathfrak{nfz}$ of an element in $\mathfrak{ds}$. For each weight $n\ge 3$ and depth $d\ge 1$, we then define the vector subspace $\mathfrak{ds}_{n,d}$ of $\mathfrak{ds}$ as the space of elements in $\mathfrak{ds}_n^d-\mathfrak{ds}_n^{d+1}$ whose duals lie in $\mathfrak{nfz}_n^d$. We prove the direct sum decomposition \[ \mathfrak{ds}=\bigoplus_{n\ge 3}\bigoplus_{d\ge 1} \mathfrak{ds}_{n,d}, \] \noindent which yields a canonical vector space isomorphism between $\mathfrak{ds}$ and its associated graded for the depth filtration, $\mathfrak{ds}_{n,d}\simeq \mathfrak{ds}_n^d/ \mathfrak{ds}_n^{d+1}$. A basis of $\mathfrak{ds}$ respecting this decomposition is dual-depth adapted, which means that it is adapted to the depth filtration on $\mathfrak{ds}$, and the basis of dual elements is adapted to the dual depth filtration on $\mathfrak{nfz}$. We use this notion to give a canonical depth 1 generator $f_n$ for $\mathfrak{ds}$ in each odd weight $n\ge 3$, namely the dual of the new formal single zeta value $\zeta(n)\in\mathfrak{nfz}_n$. At the end, we also apply the result to give canonical irreducibles for the formal multizeta algebra in weights up to 12.

STRUKTUR KOMPLEKS DALAM BAHASA INDONESIA ILMIAH DAN MASALAH PENERJEMAHANNYA KE DALAM BAHASA INGGRIS

2018 ◽  
Vol 17 (1) ◽  
Author(s):  
Ni Ketut Mirahayuni ◽  
Susie Chrismalia Garnida ◽  
Mateus Rudi Supsiadji
Keyword(s):  
Complex Structure ◽  
General Concept ◽  
Target Language ◽  
Complex Structures ◽  
Scientific Language ◽  
Logical Relation ◽  
Information Packaging ◽  
Weight Structure ◽  
The Difference ◽  
Target Languages

Abstract. Translating complex structures have always been a challenge for a translator since the structures can be densed with ideas and particular logical relations. The purpose of translation is reproducing texts into another language to make them available to wider readerships. Since language is not merely classification of a set of universal and general concept, that each language articulates or organizes the world differently, the concepts in one language can be radically different from another. One issue in translation is the difference among languages, that the wider gaps between the source and target languages may bring greater problems of transfer of message from the source into the target languages (Culler, 1976). Problematic factors involved in translation include meaning, style, proverbs, idioms and others. A number of translation procedures and strategies have been discussed to solve translation problems. This article presents analysis of complex structures in scientific Indonesian, the problems and effects on translation into English. The study involves data taken from two research article papers in Indonesian to be translated into English. The results of the analysis show seven (7) problems of Indonesian complex structures, whose effect on translation process can be grouped into two: complex structures related to grammar (including: complex structure with incomplete information, run-on sentences, redundancy , sentence elements with inequal semantic relation, and logical relation and choice of conjunctor) and complex structures related to information processing in discourse (including: front-weight- structure and thematic structure with changes of Theme element). Problems related to grammar may be solved with language economy and accuracy while those related to discourse may be solved with understanding information packaging patterns in the target language discourse. Keywords: scientific language, complex structures, translation

scholarly journals A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures

2021 ◽  
Vol 11 (12) ◽  
pp. 5570
Author(s):  
Binbin Wang ◽  
Jingze Liu ◽  
Zhifu Cao ◽  
Dahai Zhang ◽  
Dong Jiang
Keyword(s):  
Structural Characteristics ◽  
Complex Structure ◽  
Complex Structures ◽  
System Matrix ◽  
Substructure Method ◽  
Residual Structure ◽  
The Matrix ◽  
Multi Level ◽  
Fixed Interface ◽  
Selection Of

Based on the fixed interface component mode synthesis, a multiple and multi-level substructure method for the modeling of complex structures is proposed in this paper. Firstly, the residual structure is selected according to the structural characteristics of the assembled complex structure. Secondly, according to the assembly relationship, the parts assembled with the residual structure are divided into a group of substructures, which are named the first-level substructure, the parts assembled with the first-level substructure are divided into a second-level substructure, and consequently the multi-level substructure model is established. Next, the substructures are dynamically condensed and assembled on the boundary of the residual structure. Finally, the substructure system matrix, which is replicated from the matrix of repeated physical geometry, is obtained by preserving the main modes and the constrained modes and the system matrix of the last level of the substructure is assembled to the upper level of the substructure, one level up, until it is assembled in the residual structure. In this paper, an assembly structure with three panels and a gear box is adopted to verify the method by simulation and a rotor is used to experimentally verify the method. The results show that the proposed multiple and multi-level substructure modeling method is not unique to the selection of residual structures, and different classification methods do not affect the calculation accuracy. The selection of 50% external nodes can further improve the analysis efficiency while ensuring the calculation accuracy.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

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