Simple Lie Algebra Of Rank Research Articles

Fluxbrane Polynomials and Melvin-like Solutions for Simple Lie Algebras

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions Hs(z) comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters ps>0, s=1,…,n. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks n=1,2,3,4,5 and exceptional algebra E6. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored.

Graded decompositions of fusion products in rank 2

We determine the graded decompositions of fusion products of finite-dimensional irreducible representations for simple Lie algebras of rank 2. Moreover, we give generators and relations for these representations and obtain as a consequence that the Schur positivity conjecture holds in this case. The graded Littlewood–Richardson coefficients in the decomposition are parameterized by lattice points in convex polytopes, and an explicit hyperplane description is given in the various types.

Real forms of embeddings of maximal reductive subalgebras of the complex simple Lie algebras of rank up to 8

We consider the problem of determining the noncompact real forms of maximal reductive subalgebras of complex simple Lie algebras. We briefly describe two algorithms for this purpose that are taken from the literature. We discuss applications in theoretical physics of these embeddings. The supplementary material to this paper contains the tables of embeddings that we have obtained for all real forms of the semisimple Lie algebras of rank up to 8.

The Symmetric Invariants of Centralizers and Slodowy Grading II

Let 𝔤 be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field 𝕜 of characteristic zero. We identify 𝔤 with 𝔤∗ through the Killing form of 𝔤. Let (e, h, f) be an 𝔰𝔩2-triple of 𝔤. Denote by 𝔤 e the centralizer of e in 𝔤 and by $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ the algebra of symmetric invariants of 𝔤 e . We say that e is good if the nullvariety of some l homogeneous elements of $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ in (𝔤 e )∗ has codimension l. In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of Charbonnel and Moreau (Math. Zeitsch. 282, n° 1-2, 273–339 2016) is true. Namely, we prove that e is good if and only if for some homogeneous generating sequence q 1, … , q l of S(𝔤) 𝔤 , the initial homogeneous components of their restrictions to e + 𝔤 f are algebraically independent over 𝕜.

The symmetric invariants of centralizers and Slodowy grading

Let $$\mathfrak {g}$$ be a finite-dimensional simple Lie algebra of rank $$\ell $$ over an algebraically closed field $$\Bbbk $$ of characteristic zero, and let e be a nilpotent element of $$\mathfrak {g}$$ . Denote by $$\mathfrak {g}^{e}$$ the centralizer of e in $$\mathfrak {g}$$ and by $$ \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} $$ the algebra of symmetric invariants of $$\mathfrak {g}^{e}$$ . We say that e is good if the nullvariety of some $$\ell $$ homogenous elements of $$ \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} $$ in $$({\mathfrak g}^{e})^{*}$$ has codimension $$\ell $$ . If e is good then $$ \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} $$ is a polynomial algebra. The main result of this paper stipulates that if for some homogenous generators of $$ \mathrm{S}({\mathfrak g}) ^{{\mathfrak g}} $$ , the initial homogenous components of their restrictions to $$e+\mathfrak {g}^{f}$$ are algebraically independent, with (e, h, f) an $$\mathfrak {sl}_2$$ -triple of $$\mathfrak {g}$$ , then e is good. As applications, we pursue the investigations of Panyushev et al. (J. Algebra 313:343–391, 2007) and we produce (new) examples of nilpotent elements that satisfy the above polynomiality condition, in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type $$\mathbf{D}_{7}$$ .

Invertible linear maps on Borel subalgebras preserving zero Lie products

Let g be a simple Lie algebra of rank l over an algebraically closed field of characteristic zero, b a Borel subalgebra of g. An invertible linear map φ on b is said preserving zero Lie products in both directions if for x,y∈b, [x,y]=0 if and only if [φ(x),φ(y)]=0. In this paper, it is shown that an invertible linear map φ on b preserving zero Lie products in both directions if and only if it is a composition of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism, which extends the main result in [8] from a linear solvable Lie algebra to far more general cases.

Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ϕ on 𝔤 is said to preserve solvability in both directions if ϕ, as well as ϕ−1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ϕ on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.

Triple automorphisms of simple Lie algebras

An invertible linear map φ on a Lie algebra L is called a triple automorphism of it if φ([x, [y, z]]) = [φ(x), [φ(y), φ(z)]] for ∀ x, y, z ∈ L. Let g be a finite-dimensional simple Lie algebra of rank l defined over an algebraically closed field F of characteristic zero, p an arbitrary parabolic subalgebra of g. It is shown in this paper that an invertible linear map φ on p is a triple automorphism if and only if either φ itself is an automorphism of p or it is the composition of an automorphism of p and an extremal map of order 2.

Biderivations of the Parabolic Subalgebras of Simple Lie Algebras

Let 𝔤 be a simple Lie algebra of rank l over an algebraically closed field of characteristic zero, 𝔭 an arbitrary parabolic subalgebra of 𝔤. A bilinear map ϕ: 𝔭 × 𝔭 → 𝔭 is called a biderivation of 𝔭 if it is a derivation with respect to both components, meaning that for all x, y, z ∈ 𝔭. It is shown in this article that a bilinear map ϕ: 𝔭 × 𝔭 → 𝔭 is a biderivation if and only if it is a sum of an inner and an extremal biderivation.

Lie triple derivations on the parabolic subalgebras of simple Lie algebras

A linear map ϕ on a Lie algebra L is called a Lie triple derivation if ϕ([x, [y, z]]) = [ϕ(x), [y, z]] + [x, [ϕ(y), z]] + [x, [ y, ϕ(z)]] for all x, y, z ∈ L. Let be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field F of characteristic zero, and an arbitrary parabolic subalgebra of . It is shown in this note that a linear map ϕ on is a Lie triple derivation if and only if it is an inner derivation, which generalizes the main result of Tolpygo [A.K. Tolpygo, On chomology of parabolic Lie algebras, Math. Notes 12 (1973), pp. 585–587] saying that every derivation of is inner derivation.

Quasi-Automorphisms of Lie Algebras

An invertible linear transformation ϕ on a Lie algebra L is called a quasi-automorphism of it if there exists an invertible linear transformation on L such that for ∀ x, y ∈ L. The group of all quasi-automorphisms of L is denoted by Q Aut(L). Let 𝔤 be a finite-dimensional simple Lie algebra of rank l defined over an algebraically closed field F of characteristic zero, 𝔭 an arbitrary parabolic subalgebra of 𝔤. It is shown in this article that, if l = 1, then Q Aut(𝔭) = GL(𝔭); otherwise, Q Aut(𝔭) = Aut(𝔭) × F*I 𝔭, where F*I 𝔭 denotes the group of all non-zero scalar multiplication maps on 𝔭.

Le Groupe des Traces de Poisson de Certaines Algèbres D'invariants

Let 𝔤 be a simple Lie algebra of rank l, 𝔥 a Cartan subalgebra and W the corresponding Weyl group. The space 𝔥⊕𝔥* is naturally equipped with a symplectic structure and the invariant algebra S(𝔥⊕𝔥*) W inherits a Poisson structure which can be deformed by the invariant algebra where A l denotes the Weyl algebra of rank l. The purpose of this article is to compute the Poisson homology group of degree zero of S(𝔥⊕𝔥*) W and to compare it to the Hochschild homology group of degree zero of . These two groups have the same dimensions in rank 2 which are equal respectively to 1.2 and 3 in the types A 2, B 2 and G 2.

Confluence theory for graphs

We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie algebra of rank at most 2, gives rise to a confluent system of reduction rules of graphs (via Kuperberg's spiders) in an arbitrary surface. As a further consequence of this result, we find canonical bases of SU_3-skein modules of cylinders over orientable surfaces.

On symmetric invariants of centralisers in reductive Lie algebras

Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g e be the centraliser of e in g . In this paper we study the algebra S ( g e ) g e of symmetric invariants of g e . We prove that if g is of type A or C, then S ( g e ) g e is always a graded polynomial algebra in l variables, and we show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type A we prove that the invariant algebra S ( g e ) g e is freely generated by a regular sequence in S ( g e ) and describe the tangent cone at e to the nilpotent variety of g .

Generalized Derivations of Lie Algebras

Suppose L is a finite-dimensional Lie algebra with multiplication μ: L∧L→L. Let Δ(L) denote the set of triples (f,f′,f″), with f, f′, f″∈Hom(L,L), such that μ∘(f∧IL+IL∧f′)=f″∘μ. We consider the Lie algebra GenDer(L)={f∈Hom(L,L)|∃f′, f″: (f,f′, f″)∈Δ(L)}. Well-researched subalgebras of GenDer(L) include the derivation algebra, Der(L)={f∈Hom(L, L)|(f, f, f)∈Δ(L)}, and the centroid, C(L)={f∈Hom(L,L)|(f,0,f)∈δ(L)}. We now study the subalgebra QDer(L)={f∈Hom(L,L)|∃f′: (f,f,f′)∈Δ(L)}, and the subspace QC(L)={f∈Hom(L,L)|(f,−f,0)∈Δ(L)}. In characteristic ≠2, GenDer(L)=QDer(L)+QC(L) and we are concerned with the inclusions Der(L)⊆QDer(L) and C(L)⊆QC(L)∩QDer(L). If Z(L)=0 then C(L)=QC(L)∩QDer(L) and, under reasonable conditions on Lie algebras with toral Cartan subalgebras, we show QDer(L)=Der(L)+C(L); if L is a parabolic subalgebra of a simple Lie algebra of rank >1 in characteristic 0, then we even have GenDer(L)=ad(L)+(IL). In general QC(L) is not closed under composition or Lie bracket; however, if Z(L)=0 then QC(L) is a commutative, associative algebra, and we describe conditions that force QC(L)=C(L) or, equivalently, GenDer(L)=QDer(L). We show that, in characteristic 0, GenDer(L) preserves the radical of L, thus generalizing the classical result for Der(L). We also discuss some applications of the main results to the study of functions f∈Hom(L,L) such that f∘μ or μ∘(f∧IL) defines a Lie multiplication.

Roots of unity: Representations of quantum groups

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases.

On basic Cycles of An, Bn, Cn and Dn

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

Some noteworthy spin plethysms

The spin plethysms lambda G(X) Delta that arise in the reduction of Delta under SO(N) to G when (1) to lambda G are considered. It is shown that, for the simple Lie algebras of rank k, if lambda G= phi G, the adjoint representation of G, then phi G(X) Delta =2(k2)/ delta G where delta G is the representation of G whose highest weight is half the sum of the positive roots. Certain results for other representations are described. A remarkable series of S-functions is introduced leading to a new dimensional equality between certain representations of O(2k) and Sp(2k).

Representation of zero curvature for the system of nonlinear partial differential equations $$x_{\alpha ,z\bar z} = \exp (kx)_\alpha $$ and its integrability

Representation of zero curvature for the system of essentially nonlinear partial differential equations \(x_{\alpha ,z\bar z} = exp\left( {\sum\nolimits_{\beta = 1}^r {k_{\alpha \beta } x_\beta } } \right),1 \leqslant \alpha \leqslant r\), with an arbitrary numeral matrix k is constructed in an explicit form. On the basis of this representation we give an invariant integration method for the system when k coincides with the Cartan matrix of simple Lie algebra of rank r. The final solutions depend on 2r arbitrary functions.

Simple Lie algebras of toral rank one

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 7 p > 7 . Let L have Cartan decomposition L = H + ∑ γ ∈ Γ L γ L = H + {\sum _{\gamma \in \Gamma }}{L_\gamma } . If Γ \Gamma generates a cyclic group then L is isomorphic to sl ( 2 , F ) {\text {sl}}(2,F) or to one of the simple Lie algebras of generalized Cartan type W ( 1 : n ) W(1:{\mathbf {n}}) or H ( 2 : n : Φ ) ( 2 ) H{(2:{\mathbf {n}}:\Phi )^{(2)}} .