Abelian Ideals Research Articles

Abelian ideals and the variety of Lagrangian subalgebras

For a semisimple algebraic group G of adjoint type with Lie algebra g over the complex numbers, we establish a bijection between the set of closed orbits of the group G⋉g⁎ acting on the variety of Lagrangian subalgebras of g⋉g⁎ and the set of abelian ideals of a fixed Borel subalgebra of g. In particular, the number of such orbits equals 2rkg by Peterson's theorem on abelian ideals.

Abelian subalgebras and ideals of maximal dimension in Poisson algebras

This paper studies the abelian subalgebras and ideals of maximal dimension of Poisson algebras P of dimension n. We introduce the invariants α and β for Poisson algebras, which correspond to the dimension of an abelian subalgebra and ideal of maximal dimension, respectively. We prove that these invariants coincide if α(P)=n−1. We characterize the Poisson algebras with α(P)=n−2 over arbitrary fields. In particular, we characterize Lie algebras L with α(L)=n−2. We also show that α(P)=n−2 for nilpotent Poisson algebras implies β(P)=n−2. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.

Fino–Vezzoni conjecture on Lie algebras with abelian ideals of codimension two

Fino–Vezzoni conjecture on Lie algebras with abelian ideals of codimension two

Abelian Subalgebras and Ideals of Maximal Dimension in Solvable Leibniz Superalgebras

Abelian Subalgebras and Ideals of Maximal Dimension in Solvable Leibniz Superalgebras

Complex symplectic Lie algebras with large Abelian subalgebras

We present two constructions of complex symplectic structures on Lie algebras with large Abelian ideals. In particular, we completely classify complex symplectic structures on almost Abelian Lie algebras. By considering compact quotients of their corresponding connected, simply connected Lie groups we obtain many examples of complex symplectic manifolds which do not carry (hyper)kähler metrics. We also produce examples of compact complex symplectic manifolds endowed with a fibration whose fibers are Lagrangian tori.

Hermitian geometry of Lie algebras with abelian ideals of codimension 2

Hermitian geometry of Lie algebras with abelian ideals of codimension 2

Abelian Subalgebras and Ideals of Maximal Dimension in Solvable Leibniz Algebras

Abelian Subalgebras and Ideals of Maximal Dimension in Solvable Leibniz Algebras

Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension 1 and supersolvable Zinbiel algebras in which such subalgebras have codimension 2, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for α and β for the low dimensional Zinbiel algebras over the complex field that have been classified. Communicated by Alberto Elduque

Abelian Non Jordan-Lie Inner Ideals of the Orthogonal Finite Dimensional Lie Algebras

Let A associative algebra (finite dimensional over ) of any characteristic with involution * and let K = skew(A) = {a ∈ A|a* = -a} be its corresponding sub-algebra under the Lie product [a, b] = ab - ba for all a, b ∈ A. If A = End V for some finite dimensional vector space over and * is an adjoint involution with a symmetric non-alternating bilinear form on V, then * is said to be orthogonal. In this paper, abelian inner ideals which are non Jordan-Lie of such Lie algebras were defined, considered, studied, and classified. Some examples and results were provided. It is proved that every abelian inner ideal which is non Jordan-Lie B of K can be expressed as B = {v, H ⊥}, where v is an isotropic vector of a hyperbolic plane H ⊆ V and H ⊥ is the orthogonal subspace of H.

Commutative Polarisations and the Kostant Cascade

Let \(\mathfrak {g}\) be a complex simple Lie algebra. We classify the parabolic subalgebras \(\mathfrak {p}\) of \(\mathfrak {g}\) such that the nilradical of \(\mathfrak {p}\) has a commutative polarisation. The answer is given in terms of the Kostant cascade. It requires also the notion of an optimal nilradical and some properties of abelian ideals in a Borel subalgebra of \(\mathfrak {g}\).

Lie algebra fermions

We define a supersymmetric quantum mechanics of fermions that take values in a simple Lie algebra . We summarize what is known about the spectrum and eigenspaces of the Laplacian which corresponds to the Koszul differential d. Firstly, we concentrate on the zero eigenvalue eigenspace which coincides with the Lie algebra cohomology. We provide physical insight into useful tools to compute the cohomology, namely Morse theory and the Hochschild–Serre spectral sequence. We list explicit generators for the Lie algebra cohomology ring. Secondly, we concentrate on the eigenspaces of the supersymmetric quantum mechanics with maximal eigenvalue at given fermion number. These eigenspaces have an explicit description in terms of abelian ideals of a Borel subalgebra of the simple Lie algebra. We also introduce a model of Lie algebra valued fermions in two dimensions, where the spaces of maximal eigenvalue acquire a cohomological interpretation. Our work provides physical interpretations of results by mathematicians, and simplifies the proof of a few theorems. Moreover, we recall that these mathematical results play a role in pure supersymmetric gauge theory in four dimensions, and observe that they give rise to a canonical representation of the four-dimensional chiral ring.

Abelian subalgebras and ideals of maximal dimension in supersolvable and nilpotent lie algebras

In this paper, we continue the study of abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable and nilpotent Lie algebras. We show that supersolvable Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two, and that the same is true for nilpotent Lie algebras with an abelian subalgebra of codimension 4, provided that the characteristic of the field is greater than five.

On the speciality of Jordan algebras and subquotients of Lie algebras

In [7] Jordan algebras were attached to ad-nilpotent elements of index less than or equal to three of Lie algebras. In this paper we study conditions on the own structure of the Lie algebras that imply the speciality of these Jordan algebras. Similar results are obtained when dealing with subquotients associated to abelian inner ideals.

Glorious pairs of roots and Abelian ideals of a Borel subalgebra

Let $${{\mathfrak {g}}}$$ be a simple Lie algebra with a Borel subalgebra $${{\mathfrak {b}}}$$ . Let $$\Delta ^+$$ be the corresponding (po)set of positive roots and $$\theta $$ the highest root. A pair $$\{\eta ,\eta '\}\subset \Delta ^+$$ is said to be glorious, if $$\eta ,\eta '$$ are incomparable and $$\eta +\eta '=\theta $$ . Using the theory of abelian ideals of $${{\mathfrak {b}}}$$ , we (1) establish a relationship of $$\eta ,\eta '$$ to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin diagram. In types $${{\mathbf {\mathsf{{{DE}}}}}}_{}$$ , we prove that if $$\{\eta ,\eta '\}$$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $$\eta \wedge \eta '$$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type $${{\mathbf {\mathsf{{{A}}}}}}_{}$$ . As an application, we describe the minimal non-abelian ideals of $${{\mathfrak {b}}}$$ .

Jordan supersystems related to Lie superalgebras

Given a Lie superalgebra and an even ad-nilpotent element of index one can obtain a Jordan superalgebra attached to that element; inspired by that construction we build a Jordan superpair attached to an odd ad-nilpotent element of index We introduce inner ideals for Lie superalgebras, and we prove that the associated subquotients are Jordan superpairs. All three constructions agree when considering abelian inner ideals generated by one element.

Recollements of abelian categories and ideals in heredity chains—a recursive approach to quasi-hereditary algebras

Recollements of abelian categories are used as a basis of a homological and recursive approach to quasi-hereditary algebras. This yields a homological proof of Dlab and Ringel’s characterisation of idempotent ideals occurring in heredity chains, which in turn characterises quasi-hereditary algebras recursively. Further applications are given to hereditary algebras and to Morita context rings.

Abelian Ideals of a Borel Subalgebra and Root Systems, II

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$ and $\mathfrak{Ab}$ the set of abelian ideals of $\mathfrak b$. Let $\Delta^+$ be the corresponding set of positive roots. We continue our study of combinatorial properties of the partition of $\mathfrak{Ab}$ parameterised by the long positive roots. In particular, the union of an arbitrary set of maximal abelian ideals is described, if $\mathfrak g\ne\mathfrak{sl}_n$. We also characterise the greatest lower bound of two positive roots, when it exists, and point out interesting subposets of $\Delta^+$ that are modular lattices.

The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture

We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the $S_n$-representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case, and generalizes results obtained by Shareshian-Wachs and Teff. Our proof uses previous work of Stanley, Gasharov, Shareshian-Wachs, and Brosnan-Chow, as well as results of the second author on the geometry and combinatorics of Hessenberg varieties. As part of our arguments, we obtain inductive formulas for the Poincare polynomials of regular abelian Hessenberg varieties.

Abelian ideals and amazing roots

Let [Formula: see text] be a simple Lie algebra with a Borel subalgebra [Formula: see text]. To any long positive root [Formula: see text], one associates two ideals of [Formula: see text]: the abelian ideal [Formula: see text] and not necessarily abelian ideal [Formula: see text]. It is known that [Formula: see text], and [Formula: see text] is said to be amazing if the equality holds. The set of amazing roots, [Formula: see text], is closed under the operation “∨” in [Formula: see text], and [Formula: see text] is said to be primitive, if it cannot be written as [Formula: see text] with incomparable amazing roots [Formula: see text]. We classify the amazing roots and notice that the number of primitive roots equals [Formula: see text]. Moreover, if [Formula: see text] (respectively, [Formula: see text]) is the set of simple (respectively, primitive) roots, then there is a natural bijection [Formula: see text]. We also study the subset [Formula: see text] of [Formula: see text].

Strongly prime algebraic Lie PI-algebras

ABSTRACTIn a recent paper by the author and Golubkov, it was proved that a strongly prime Lie PI-algebra with an algebraic adjoint representation over an algebraically closed field of characteristic 0 is simple and finite dimensional. In this note, we derive this result from a more general one on strongly prime Lie PI-algebras with abelian minimal inner ideals, which is closely related to the intrinsic characterization of simple finitary Lie algebras with abelian minimal inner ideals.