Satake Diagrams Research Articles

Double Satake Diagrams and Canonical Forms in Compact Symmetric Triads

In this paper, we first introduce the notion of double Satake diagrams for compact symmetric triads. In terms of this notion, we give an alternative proof for the classification theorem for compact symmetric triads, which was originally given by Toshihiko Matsuki. Secondly, we introduce the notion of canonical forms for compact symmetric triads, and prove the existence of canonical forms for compact simple symmetric triads. We also give some properties for canonical forms.

Universal K-matrices for quantum Kac-Moody algebras

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H H endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U q L s l 2 U_qL\mathfrak {sl}_{2} .

Nilpotent orbits and mixed gradings of semisimple Lie algebras

Let σ be an involution of a complex semisimple Lie algebra g and g=g0⊕g1 the related Z2-grading. We study relations between nilpotent G0-orbits in g0 and the respective G-orbits in g. If e∈g0 is nilpotent and {e,h,f}⊂g0 is an sl2-triple, then the semisimple element h yields a Z-grading of g. Our main tool is the combined Z×Z2-grading of g, which is called a mixed grading. We prove, in particular, that if eσ is a regular nilpotent element of g0, then the weighted Dynkin diagram of eσ, D(eσ), has only isolated zeros. It is also shown that if G⋅eσ∩g1≠∅, then the Satake diagram of σ has only isolated black nodes and these black nodes occur among the zeros of D(eσ). Using mixed gradings related to eσ, we define an inner involution σˇ such that σ and σˇ commute. Here we prove that the Satake diagrams for both σˇ and σσˇ have isolated black nodes.

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.

Factorisation of quasi K-matrices for quantum symmetric pairs

The theory of quantum symmetric pairs provides a universal K-matrix which is an analog of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this paper we restrict to the cases where the underlying Lie algebra is mathfrak {sl}_n or the Satake diagram has no black dots. In these cases we give an explicit formula for the quasi K-matrix as a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the underlying symmetric Lie algebra in the same way as the factorisation of the quasi R-matrix depends on the Weyl group of the Lie algebra. We conjecture that our formula holds in general.

Classification of simple linearly compact Kantor triple systems over the complex numbers

Simple finite-dimensional Kantor triple systems over the complex numbers are classified in terms of Satake diagrams. We prove that every simple and linearly compact Kantor triple system has finite dimension and give an explicit presentation of all the classical and exceptional systems.

A simplicial complex of Nichols algebras

We translate the concept of restriction of an arrangement in terms of Hopf algebras. In consequence, every Nichols algebra gives rise to a simplicial complex decorated by Nichols algebras with restricted root systems. As applications, some of these Nichols algebras provide Weyl groupoids which do not arise for diagonal Nichols algebras and in fact we realize all root systems of finite Weyl groupoids of rank greater than three. Further, our result explains the root systems of the folded Nichols algebras over nonabelian groups and of generalized Satake diagrams.

Satake diagrams and real structures on spherical varieties

With each antiholomorphic involution [Formula: see text] of a connected complex semisimple Lie group [Formula: see text] we associate an automorphism [Formula: see text] of its Dynkin diagram. The definition of [Formula: see text] is given in terms of the Satake diagram of [Formula: see text]. Let [Formula: see text] be a self-normalizing spherical subgroup. If [Formula: see text] then we prove the uniqueness and existence of a [Formula: see text]-equivariant real structure on [Formula: see text] and on the wonderful completion of [Formula: see text].

Local orbit types of the isotropy representations for semisimple pseudo-Riemannian symmetric spaces

We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. Our method to determine the local orbit types is based on the theory of restricted root systems and Satake diagrams associated with the spaces.

A note on Clifford-Klein forms

We consider the problem of finding Clifford-Klein forms in a class of homogeneous spaces determined by inclusions of real Lie algebras of a special type which we call strongly regular. This class of inclusions is described in terms of their Satake diagrams. For example, the complexifications of such inclusions contain the class of subalgebras generated by automorphisms of finite order. We show that the condition of strong regularity implies the restriction on the real rank of subalgebras. This in part explains why the known examples of Clifford-Klein forms are rare. We make detailed calculations of some known examples from the point of view of the Satake diagrams.

Clifford–Klein Forms and a-Hyperbolic Rank

The porpose of this article is to introduce and investigate properties of a tool (the a-hyperbolic rank) which enables us to obtain new examples of homogeneous spaces G/H which admit and do not admit almost compact Clifford-Klein forms. We achieve this goal by exploring in greater detail the technique of adjoint orbits developed by Okuda combined with the well-known conditions of Benoist. We find easy-to-check conditions on G and H expressed directly in terms of the Satake diagrams of the corresponding Lie algebras, in cases when G is a real form of a complex Lie group of types A, D or E6. One of the advantages of this approach is the fact that we don't need to know the embedding of H into G. Using the a-hyperbolic rank we also show that the homogeneous space G/H of reductive type of the real form of E6 (of the IV type) admits compact Clifford-Klein forms if and only if H is compact. Inspired by the work of Okuda on symmetric spaces G/H we classify all 3-symmetric spaces admitting almost compact Clifford-Klein forms.

Satake Diagrams and Restricted Root Systems of Semisimple Pseudo-Riemannian Symmetric Spaces

In this paper, we complete the lists of Satake diagrams and restricted root systems (including signatures of roots) for all classical semisimple pseudo-Riemannian symmetric spaces, which were classified by M. Berger. We also complete the list of the cohomogeneities of the linear isotropy representations of the spaces.

Bi-isotropic decompositions of semisimple Lie algebras and homogeneous bi-Lagrangian manifolds

Let g be a real semisimple Lie algebra with Killing form B and k a B-nondegenerate subalgebra of g of maximal rank. We give a description of all ad k -invariant decompositions g = k + m + + m − such that B | m ± = 0 , B ( k , m + + m − ) = 0 and k + m ± are subalgebras. It is reduced to a description of parabolic subalgebras of g with given reductive part k . This is obtained in terms of crossed Satake diagrams. As an application, we get a classification of invariant bi-Lagrangian (or equivalently para-Kähler) structures on homogeneous manifolds G / K of a semisimple group G.

Satake diagrams of affine Kac–Moody algebras

Satake diagrams of affine Kac–Moody algebras (untwisted and twisted) are obtained from their Dynkin diagrams. These diagrams give a classification of restricted root systems associated with these algebras. In the case of simple Lie algebras, these root systems and Satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems, quantum transport problems, random matrix theories etc. We hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine Kac–Moody algebras if they exist.

The group theory of oxidation II: cosets of non-split groups

The oxidation program given in the first article of this series (see preceding article in this issue) is extended to cover oxidation of 3d sigma model theories on a coset G/ H, with G non-compact (but not necessarily split), and H the maximal compact subgroup. We recover the matter content, the equations of motion and Bianchi identities from group lattice and Cartan involution. Satake diagrams provide an elegant tool for the computations, the maximal oxidation dimension, and group disintegration chains can be directly read off. We give a complete list of theories that can be recovered from oxidation of a 3-dimensional coset sigma model on G/ H, where G is a simple non-compact group.

Satake diagrams and Iwasawa decomposition of twisted Kac–Moody algebras

The involutive automorphism of twisted Kac–Moody algebras of A2(2) and D4(3) are computed from the Satake diagrams of these algebras, which are then used to furnish a general procedure of their Iwasawa decomposition.

Involutive automorphisms and Iwasawa decomposition of some hyperbolic Kac–Moody algebras

The involutive automorphisms of hyperbolic Kac–Moody algebras are computed from Satake diagrams of these algebras which are then used to furnish a general procedure of their Iwasawa decomposition. In particular Â1(1) and Â2(1) are considered as representative examples to illustrate the efficacy of the underlying treatment.

Iwasawa decomposition of affine Kac–Moody algebras using Satake diagrams

We carry out the Iwasawa decomposition of the affine Kac–Moody algebras within the general framework of involutive automorphisms which are determined with the help of the Satake diagrams corresponding to these algebras. The parabolic subalgebras essential for the construction of induced representations are computed from the Langlands decomposition. In particular, we consider the untwisted versions, viz., A1(1) and A2(1), as representative examples for the purpose of illustration.

A family of simple groups associated with the Satake diagrams

AbstractUsing the theory of the Satake diagrams associated with the non-compact simple Lie algebras over the real number field R, we shall construct a family of simple groups over a field K which are called the simple groups associated with the Satake diagrams. The list of these simple groups includes all Chevalley groups and twisted groups, and all simple algebraic groups of adjoint type defined over R if K is the complex number field C (except two types given by Table II′). Furthermore, the simple groups associated with the Satake diagrams of type AIII, BI, DI are identified with the simple groups obtained from the unitary or orthogonal groups of non-zero indices. The quasi-Bruhat decomposition of the “non-split” simple groups associated with the Satake diagrams which are not Chevalley groups or twisted groups will be given in this paper.

Satake diagrams, Iwasawa decompositions, and representations of the exceptional Lie group F4(−20)

The parabolic subgroups of the noncompact, exceptional Lie group F4(−20) are computed from a systematic analysis of Iwasawa and Langlands decompositions. Satake diagrams have been used to determine the involutive automorphisms of F4(−20) which facilitate the Iwasawa decompositions. The polarizations associated with noncompact orbits with parabolic subgroups are computed. The representations so obtained for these polarizations using Kostant’s induction scheme yield the principal series representations of F4(−20).