THE CRITICAL REPRESENTATIONS OF AFFINE LIE ALGEBRAS

1986 ◽  
Vol 01 (10) ◽  
pp. 557-564 ◽  
Author(s):  
D. ALTSCHÜLER
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Charge ◽  
Affine Lie Algebras ◽  
Affine Algebra ◽  
Open Problems ◽  
Simple Lie Algebra ◽  
Coxeter Number ◽  
Dual Coxeter Number ◽  
Current Algebras

A critical representation of an affine algebra Ĝ is a representation with central charge k=−g, g being the dual Coxeter number of the underlying simple Lie algebra G. These representations arise naturally in the study of conformal current algebras and BRS cohomology. The author shows how to construct them explicitly in a number of cases, and some intriguing open problems are mentioned.

scholarly journals Coloured Generalised Young Diagrams for Affine Weyl-Coxeter Groups

10.37236/931 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
R. C. King ◽  
T. A. Welsh
Keyword(s):  
Lie Algebras ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Slight Variation ◽  
Affine Lie Algebras ◽  
Young Diagrams ◽  
Bijective Correspondence ◽  
Coxeter Number ◽  
Finite Dimensional ◽  
Dual Coxeter Number

Coloured generalised Young diagrams $T(w)$ are introduced that are in bijective correspondence with the elements $w$ of the Weyl-Coxeter group $W$ of $\mathfrak{g}$, where $\mathfrak{g}$ is any one of the classical affine Lie algebras $\mathfrak{g}=A^{(1)}_\ell$, $B^{(1)}_\ell$, $C^{(1)}_\ell$, $D^{(1)}_\ell$, $A^{(2)}_{2\ell}$, $A^{(2)}_{2\ell-1}$ or $D^{(2)}_{\ell+1}$. These diagrams are coloured by means of periodic coloured grids, one for each $\mathfrak{g}$, which enable $T(w)$ to be constructed from any expression $w=s_{i_1}s_{i_2}\cdots s_{i_t}$ in terms of generators $s_k$ of $W$, and any (reduced) expression for $w$ to be obtained from $T(w)$. The diagram $T(w)$ is especially useful because $w(\Lambda)-\Lambda$ may be readily obtained from $T(w)$ for all $\Lambda$ in the weight space of $\mathfrak{g}$. With $\overline{\mathfrak{g}}$ a certain maximal finite dimensional simple Lie subalgebra of $\mathfrak{g}$, we examine the set $W_s$ of minimal right coset representatives of $\overline{W}$ in $W$, where $\overline{W}$ is the Weyl-Coxeter group of $\overline{\mathfrak{g}}$. For $w\in W_s$, we show that $T(w)$ has the shape of a partition (or a slight variation thereof) whose $r$-core takes a particularly simple form, where $r$ or $r/2$ is the dual Coxeter number of $\mathfrak{g}$. Indeed, it is shown that $W_s$ is in bijection with such partitions.

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 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 Isomorphisms of Twisted Hilbert Loop Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 453-480
Author(s):  
Timothée Marquis ◽  
Karl-Hermann Neeb
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Energy ◽  
Algebra Structure ◽  
Affine Lie Algebras ◽  
Subsequent Work ◽  
Highest Weight ◽  
Loop Algebras ◽  
Highest Weight Representations

Abstract The closest infinite-dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras , also called affinisations of . They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families for some infinite set J. To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra , which we call standard. In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra an explicit isomorphism from g to one of the standard affinisations of . The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g. In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of .

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 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 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 FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350062 ◽  
Author(s):  
DRAŽEN ADAMOVIĆ ◽  
OZREN PERŠE
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Operator Algebras ◽  
Affine Lie Algebras ◽  
Fusion Rules ◽  
Type Formula ◽  
Complete Reducibility ◽  
Affine Lie Algebra ◽  
Branching Rules ◽  
Level 1

We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.

The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras

2008 ◽  
Vol 51 (2) ◽  
pp. 298-309 ◽  
Author(s):  
Maribel Tocón
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Affine Lie Algebras ◽  
The Core ◽  
Affine Lie Algebra ◽  
Extended Affine Lie Algebra

AbstractIn this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra.

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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