scholarly journals Integrable triples in semisimple Lie algebras

2021 ◽  
Vol 111 (5) ◽  
Author(s):  
Alberto De Sole ◽  
Mamuka Jibladze ◽  
Victor G. Kac ◽  
Daniele Valeri
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Equivalence Class ◽  
Simple Root ◽  
Root Vector ◽  
Integrable Hierarchy ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Kdv Hierarchy ◽  
Hamiltonian Pde

AbstractWe classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in $${\mathfrak {sl}}_2$$ sl 2 corresponds to the KdV hierarchy, and the triple $$(f,0,e_\theta )$$ ( f , 0 , e θ ) , where f is the sum of negative simple root vectors and $$e_\theta $$ e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy.

scholarly journals CLASSIFICATION OF 3-GRADED CAUSAL SUBALGEBRAS OF REAL SIMPLE LIE ALGEBRAS

2021 ◽  
Author(s):  
D. OEH
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Equivalence Class ◽  
Convex Cone ◽  
Closed Convex Cone ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Direct Sum ◽  
Tube Type

AbstractLet ($$ \mathfrak{g} $$ g , τ) be a real simple symmetric Lie algebra and let W ⊂ $$ \mathfrak{g} $$ g be an invariant closed convex cone which is pointed and generating with τ(W) = −W. For elements h ∈ $$ \mathfrak{g} $$ g with τ(h) = h, we classify the Lie algebras $$ \mathfrak{g} $$ g (W, τ, h) which are generated by the closed convex cones $$ {C}_{\pm}\left(W,\tau, h\right):= \left(\pm W\right)\cap {\mathfrak{g}}_{\pm 1}^{-\tau }(h) $$ C ± W τ h ≔ ± W ∩ g ± 1 − τ h , where $$ {\mathfrak{g}}_{\pm 1}^{-\tau }(h):= \left\{x\in \mathfrak{g}:\tau (x)=-x\left[h,x\right]=\pm x\right\} $$ g ± 1 − τ h ≔ x ∈ g : τ x = − x h x = ± x . These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if $$ \mathfrak{g} $$ g (W, τ, h) is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms τ of $$ \mathfrak{g} $$ g with τ(W) = −W a list of possible subalgebras $$ \mathfrak{g} $$ g (W, τ, h) up to isomorphy.

scholarly journals ELEMENTARY PARABOLIC TWIST

2002 ◽  
Vol 01 (04) ◽  
pp. 413-424 ◽  
Author(s):  
V. D. LYAKHOVSKY ◽  
M. E. SAMSONOV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subalgebra ◽  
Main Element ◽  
Two Dimensional ◽  
Parabolic Subalgebra ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra

The twist deformations for simple Lie algebras [Formula: see text] whose twisting elements ℱ are known explicitly are usually defined on the carrier subspace injected in the Borel subalgebra [Formula: see text]. We consider the case where the carrier of the twist intersects nontrivially with both [Formula: see text] and [Formula: see text]. The main element of the new deformation is the parabolic twist ℱ℘ whose carrier is the minimal parabolic subalgebra of simple Lie algebra [Formula: see text]. It has the structure of the algebra of two-dimensional motions, contains [Formula: see text] and intersects nontrivially with [Formula: see text]. The twist ℱ℘ is constructed as a composition of the extended jordanian twist [Formula: see text] and the factor [Formula: see text]. The latter can be considered as a special deformed version of the jordanian twist. The twisted costructure is found for [Formula: see text] and the corresponding universal ℛ-matrix is presented. The parabolic twist can be composed with certain types of chains of extended jordanian twists for algebras A2(n-1). The chains enlarged by the parabolic factor ℱ℘ perform the explicit quantization of the new set of classical r-matrices.

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 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.

Generalized Heisenberg Algebras and Toroidal Lie Algebras

2010 ◽  
Vol 17 (03) ◽  
pp. 375-388 ◽  
Author(s):  
Yingjue Fang ◽  
Liangang Peng
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Heisenberg Algebra ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Heisenberg Algebras ◽  
Toroidal Lie Algebra

In this article we provide two kinds of infinite presentations of toroidal Lie algebras. At first we define generalized Heisenberg algebras and prove that each toroidal Lie algebra is an amalgamation of a simple Lie algebra and a generalized Heisenberg algebra in the sense of Saito and Yoshii. This is one kind of presentations of toroidal Lie algebras given by the generators of generalized Heisenberg algebras and the Chevalley generators of simple Lie algebras with certain amalgamation relations. Secondly by using the generalized Chevalley generators, we give another kind of presentations. These two kinds of presentations are different from those given by Moody, Eswara Rao and Yokonuma.

SERRE RELATIONS AND GRÖBNER-SHIRSHOV BASES FOR SIMPLE LIE ALGEBRAS II

1996 ◽  
Vol 06 (04) ◽  
pp. 401-412 ◽  
Author(s):  
L.A. BOKUT ◽  
A.A. KLEIN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Characteristic 2 ◽  
Cartan Matrices

Gröbner-Shirshov bases for the Lie algebras Bn, Cn, Dn, abstractly defined by generators and the Serre relations for the corresponding Cartan matrices over a field of characteristic ≠2, 3, are constructed. As an application, we obtain that each of the previous algebras is isomorphic to a classical simple Lie algebra so2n+1(k), sp2n(k), so2nk respectively.

scholarly journals A Note on Exponents vs Root Heights for Complex Simple Lie Algebras

10.37236/1160 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Sankaran Viswanath
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Fourier Coefficients ◽  
Combinatorial Proof ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Positive Roots ◽  
Special Case

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra $\mathfrak{ g}$, the partition formed by the exponents of $\mathfrak{ g}$ is dual to that formed by the numbers of positive roots at each height.

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 Decompositions of algebras and post-associative algebra structures

2019 ◽  
Vol 30 (03) ◽  
pp. 451-466
Author(s):  
Dietrich Burde ◽  
Vsevolod Gubarev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Algebra Structure ◽  
Associative Algebras ◽  
Simple Lie Algebra ◽  
Reductive Lie Algebra ◽  
Baxter Operator

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

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.

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