Finite-dimensional Semisimple Lie Algebra Research Articles

3-Dimensional mixed BF theory and Hitchin’s integrable system

The affine Gaudin model, associated with an untwisted affine Kac–Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological–holomorphic Chern–Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

Root components for tensor product of affine Kac-Moody Lie algebra modules

Let g \mathfrak {g} be an affine Kac-Moody Lie algebra and let λ , μ \lambda , \mu be two dominant integral weights for g \mathfrak {g} . We prove that under some mild restriction, for any positive root β \beta , V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) contains V ( λ + μ − β ) V(\lambda +\mu -\beta ) as a component, where V ( λ ) V(\lambda ) denotes the integrable highest weight (irreducible) g \mathfrak {g} -module with highest weight λ \lambda . This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) . Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G \mathcal {G} -Schubert varieties in the product partial flag variety G / P × G / P \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of G \mathcal {G} -sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.

Partial actions on reductive Lie algebras

In this paper, we study partial group actions on Lie algebras. We describe the structure of the inverse semigroup of all partial automorphisms (isomorphisms between ideals) of a finite-dimensional reductive Lie algebra. Also, we show that every partial group action on a finite-dimensional semisimple Lie algebra admits a globalization, unique up to isomorphism. As a consequence, we obtain that the globalization problem for partial group actions on reductive Lie algebra is equivalent to the globalization problem on its center.

On the tensor semigroup of affine Kac-Moody lie algebras

The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra g \mathfrak {g} defines a semigroup of triples of weights. Namely, given λ \lambda in the set P + P_+ of dominant integral weights, V ( λ ) V(\lambda ) denotes the irreducible representation of g \mathfrak {g} with highest weight λ \lambda . We are interested in the tensor semigroup Γ N ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + 3 | V ( μ ) ⊂ V ( λ 1 ) ⊗ V ( λ 2 ) } , \begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+}^3\,|\, V(\mu )\subset V(\lambda _1)\otimes V(\lambda _2)\}, \end{equation*} and in the tensor cone Γ ( g ) \Gamma (\mathfrak {g}) it generates: Γ ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + , Q 3 | ∃ N ≥ 1 V ( N μ ) ⊂ V ( N λ 1 ) ⊗ V ( N λ 2 ) } . \begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu )\subset V(N\lambda _1)\otimes V(N\lambda _2)\}. \end{equation*} Here, P + , Q P_{+,{\mathbb {Q}}} denotes the rational convex cone generated by P + P_+ . In the special case when g \mathfrak {g} is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone Γ ( g ) \Gamma (\mathfrak {g}) is described in Belkale and Kumar [Invent. Math. 166 (2006), pp. 185–228] by an explicit finite list of inequalities. In general, Γ ( g ) \Gamma (\mathfrak {g}) is neither polyhedral, nor closed. In this article we describe the closure of Γ ( g ) \Gamma (\mathfrak {g}) by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar’s conjecture in this case (see Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]). The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer d d is called a saturation factor, if V ( N λ 1 ) ⊗ V ( N λ 2 ) V(N\lambda _1)\otimes V(N\lambda _2) contains V ( N μ ) V(N\mu ) for some positive integer N N then V ( d λ 1 ) ⊗ V ( d λ 2 ) V(d\lambda _1)\otimes V(d\lambda _2) contains V ( d μ ) V(d\mu ) , assuming that μ − λ 1 − λ 2 \mu -\lambda _1-\lambda _2 belongs to the root lattice. For g = s l n \mathfrak {g}={\mathfrak {sl}}_n , the famous Knutson-Tao theorem asserts that d = 1 d=1 is a saturation factor (see Knutson and Tao [J. Amer. Math. Soc. 12 (1999), pp. 1055–1090]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case, Γ N ( g ) \Gamma _{\mathbb {N}}(\mathfrak {g}) is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type A ~ n \tilde A_n , we prove that any integer d ≥ 2 d\geq 2 is a saturation factor, generalizing the case A ~ 1 \tilde A_1 shown in Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936].

Borel–de Siebenthal Theory for Affine Reflection Systems

We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.

Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagovi\'c the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data first considered by A. Heck. A generalized Satake diagram naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.

Logarithmic conformal field theories of type B n, ℓ = 4 and symplectic fermions

There are important conjectures about logarithmic conformal field theories (LCFT’s), which are constructed as a kernel of screening operators acting on the vertex algebra of the rescaled root lattice of a finite-dimensional semisimple complex Lie algebra. In particular, their representation theory should be equivalent to the representation theory of an associated small quantum group. This article solves the case of the rescaled root lattice Bn/2 as a first working example beyond A1/p. We discuss the kernel of short screening operators, its representations, and graded characters. Our main result is that this vertex algebra is isomorphic to a well-known example: The even part of n pairs of symplectic fermions. In the screening operator approach, this vertex algebra appears as an extension of the vertex algebra associated with A1n/2, which are n copies of the even part of one pair of symplectic fermions. The new long screenings give the new global Cn-symmetry. The extension is due to a degeneracy in this particular case: Rescaled long roots still have an even integer norm. For the associated quantum group of divided powers, the first author has previously encountered matching degeneracies: It contains the small quantum group of type A1n and the Lie algebra Cn. Recent results by Farsad, Gainutdinov, and Runkel on symplectic fermions suggest finally the conjectured category equivalence to this quantum group. We also study the other degenerate cases of a quantum group, giving extensions of LCFT’s of type Dn, D4, A2 with larger global symmetry Bn, F4, G2.

ON CATEGORIES OF ADMISSIBLE ( g $$ \mathfrak{g} $$ , sl(2))-MODULES

Let \( \mathfrak{g} \) be a complex finite-dimensional semisimple Lie algebra and \( \mathfrak{k} \) be any sl(2)-subalgebra of \( \mathfrak{g} \). In this paper we prove an earlier conjecture by Penkov and Zuckerman claiming that the first derived Zuckerman functor provides an equivalence between a truncation of a thick parabolic category \( \mathcal{O} \) for \( \mathfrak{g} \) and a truncation of the category of admissible (\( \mathfrak{g} \) , \( \mathfrak{k} \))-modules. This latter truncated category consists of admissible (\( \mathfrak{g} \) , \( \mathfrak{k} \))-modules with sufficiently large minimal \( \mathfrak{k} \)-type. We construct an explicit functor inverse to the Zuckerman functor in this setting. As a corollary we obtain an estimate for the global injective dimension of the inductive completion of the truncated category of admissible (\( \mathfrak{g} \) , \( \mathfrak{k} \))-modules.

On flux integrals for generalized Melvin solution related to simple finite-dimensional Lie algebra

A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra mathcal G is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of mathcal G. It is governed by a set of n moduli functions H_s(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials—the so-called fluxbrane polynomials. These polynomials depend upon integration constants q_s, s = 1,dots ,n. In the case when the conjecture on the polynomial structure for the Lie algebra mathcal G is satisfied, it is proved that 2-form flux integrals Phi ^s over a proper 2d submanifold are finite and obey the relations q_s Phi ^s = 4 pi n_s h_s, where the h_s > 0 are certain constants (related to dilatonic coupling vectors) and the n_s are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, s = 1,dots ,n. The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra mathcal G. Examples of polynomials and fluxes for the Lie algebras A_1, A_2, A_3, C_2, G_2 and A_1 + A_1 are presented.

On Brane Solutions with Intersection Rules Related to Lie Algebras

The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form M = M0 x M1 x . . . x Mn , where all Mi with i >= 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, M0 is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M0 . The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold M0 of dimension d0 ≠ 2 . By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For d0 = 1 , the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank—non-singular Kac–Moody (KM) algebras—the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: H2(q, q) , AE3, HA(1)2, E10 and Lorentzian KM algebra P10 , are presented.

An algorithm for computing weight multiplicities in irreducible modules for complex semisimple Lie algebras

Let g be a finite-dimensional semisimple Lie algebra over C having rank l and let V be an irreducible finite-dimensional g-module having highest weight λ. Computations of weight multiplicities in V, usually based on Freudenthal's formula, are in general difficult to carry out in large ranks or for λ with large coefficients (in terms of the fundamental weights). In this paper, we first show that in some situations, these coefficients can be “lowered” in order to simplify the calculations. We then investigate how this can be used to improve the aforementioned formula of Freudenthal, leading to a more efficient version of the latter in terms of complexity as well as to a way of dealing with certain computations in unbounded ranks. We conclude by illustrating the last assertion with a concrete example.

2-Local automorphisms on finite-dimensional Lie algebras

We prove that every 2-local automorphism on a finite-dimensional semi-simple Lie algebra L over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie algebra L with dim⁡L≥2 admits a 2-local automorphism which is not an automorphism.

A Frobenius homomorphism for Lusztig’s quantum groups for arbitrary roots of unity

For a finite-dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite-dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite-dimensional Hopf kernel and with the image of the universal enveloping algebra. In this article, we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases, the Frobenius–Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.

Faces and maximizer subsets of highest weight modules

In this paper we study general highest weight modules Vλ over a complex finite-dimensional semisimple Lie algebra g. We present three formulas for the set of weights of a large family of modules Vλ, which include but are not restricted to all simple modules and all parabolic Verma modules. These formulas are direct and do not involve cancellations, and were not previously known in the literature. Our results extend the notion of the Weyl polytope to general highest weight g-modules Vλ.We also show that for all simple modules, the convex hull of the weights is a WJ-invariant polyhedron for some parabolic subgroup WJ. We compute its vertices, faces, and symmetries – more generally, we also do this for all parabolic Verma modules, and for all modules Vλ with highest weight λ not on a simple root hyperplane. To show our results, we extend the notion of convexity to arbitrary additive subgroups A⊂(R,+) of coefficients. Our techniques enable us to completely classify “weak A-faces” of the support sets wt(Vλ), in the process extending classical results of Satake, Borel–Tits, Vinberg, and Casselman, as well as modern variants by Chari–Dolbin–Ridenour and Cellini–Marietti, to general highest weight modules.

On Ellipsoids Attached to Root Systems

For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.

Local derivations on finite-dimensional Lie algebras

We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras L with dim⁡L≥3 which admit local derivations which are not derivations.

Floer cohomology of -equivariant Lagrangian branes

Building on Seidel and Solomon’s fundamental work [Symplectic cohomology and$q$-intersection numbers, Geom. Funct. Anal.22(2012), 443–477], we define the notion of a$\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold$M$, where$\mathfrak{g}\subset SH^{1}(M)$is a sub-Lie algebra of the symplectic cohomology of$M$. When$M$is a (symplectic) mirror to an (algebraic) homogeneous space$G/P$, homological mirror symmetry predicts that there is an embedding of$\mathfrak{g}$in$SH^{1}(M)$. This allows us to study a mirror theory to classical constructions of Borel, Weil and Bott. We give explicit computations recovering all finite-dimensional irreducible representations of$\mathfrak{sl}_{2}$as representations on the Floer cohomology of an$\mathfrak{sl}_{2}$-equivariant Lagrangian brane and discuss generalizations to arbitrary finite-dimensional semisimple Lie algebras.

2-Local derivations on finite-dimensional Lie algebras

We prove that every 2-local derivation on a finite-dimensional semi-simple Lie algebra L over an algebraically closed field of characteristic zero is a derivation. We also show that a finite-dimensional nilpotent Lie algebra L with dim⁡L≥2 admits a 2-local derivation which is not a derivation.

Hom-structures on semi-simple Lie algebras

Abstract A 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.

Restrictions of Symmetric Invariants of a Complex Semisimple Lie Algebra and Some Applications

Let 𝔤 be a finite-dimensional complex semisimple Lie algebra and σ an arbitrary semisimple automorphism of 𝔤. Let 𝔱 be a Cartan subalgebra of 𝔨 = 𝔤σ and 𝔥 =Z 𝔤(𝔱) be the centralizer of 𝔱 in 𝔤. Then 𝔥 is a σ-invariant Cartan subalgebra of 𝔤 and 𝔱 = 𝔥σ. Let W(𝔤, 𝔥) be the Weyl group. One knows that Δ(𝔤, 𝔱), the set of roots of 𝔤 in 𝔱, is also a root system. It is proved that the corresponding Weyl group W(𝔤, 𝔱) is isomorphic to W(𝔤, 𝔥)σ, which is the subgroup of W(𝔤, 𝔥) consisting of those elements commuting with σ. It is also shown that the image of the restriction map S(𝔥*) W(𝔤, 𝔥) → S(𝔱*) W(𝔨, 𝔱), where S(𝔥*) and S(𝔱*) are the polynomial algebras on 𝔥 and 𝔱, respectively, is exactly S(𝔱*) W(𝔤, 𝔱). Based on the above result, we also get a complete classification of the pairs (𝔤, σ) such that 𝔤σ is noncohomologous to zero in 𝔤.