Root Subgroups Research Articles

On pairs of root groups and reflections associated to the Chevalley group E 6(K), in characteristic two

This paper is concerned with computing the number of orbits under the action of the algebraic group E = E 6 ( K ) on the set of all pairs of root subgroups of itself. Furthermore, the number of orbits of the action of the group generated by a pair of certain reflections, on the 27 dimensional module A, has been computed. For the computations done in this paper, an explicit representation of E inside GL ( A ) has been fixed right at the onset.

Decompositions of Hyperbolic Kac–Moody Algebras with Respect to Imaginary Root Groups

We propose a novel way to define imaginary root subgroups associated with (timelike) imaginary roots of hyperbolic Kac–Moody algebras. Using in an essential way the theory of unitary irreducible representation of covers of the group SO(2, 1), these imaginary root subgroups act on the complex Kac–Moody algebra viewed as a Hilbert space. We illustrate our new view on Kac–Moody groups by considering the example of a rank-two hyperbolic algebra that is related to the Fibonacci numbers. We also point out some open issues and new avenues for further research, and briefly discuss the potential relevance of the present results for physics and current attempts at unification.

Groups with 𝖠_{ℓ}-commutator relations

If A A is a unital associative ring and ℓ ≥ 2 \ell \geq 2 , then the general linear group GL ⁡ ( ℓ , A ) \operatorname {GL}(\ell , A) has root subgroups U α U_\alpha and Weyl elements n α n_\alpha for α \alpha from the root system of type A ℓ − 1 \mathsf A_{\ell - 1} . Conversely, if an arbitrary group has such root subgroups and Weyl elements for ℓ ≥ 4 \ell \geq 4 satisfying natural conditions, then there is a way to recover the ring A A . A generalization of this result not involving the Weyl elements is proved, so instead of the matrix ring M ⁡ ( ℓ , A ) , \operatorname {M}(\ell , A), a nonunital associative ring with a well-behaved Peirce decomposition is provided.

Sacrifice of Involved Nerve Root during Surgical Resection of Foraminal and/or Dumbbell Spinal Neurinomas.

Even if usually needed to achieve the gross total resection (GTR) of spinal benign nerve sheath tumors (NSTs), nerve root sacrifice remains controversial regarding the risk of neurological deficit. For foraminal NSTs, we hypothesize that the involved root is poorly functional and thus can be safely sacrificed. All spinal benign NSTs with foraminal extension that underwent surgery from 2013 to 2021 were reviewed. The impacts of preoperative clinical status and patient and tumor characteristics on long-term outcomes were analyzed. Twenty-six patients were included, with a mean follow-up (FU) of 22.4 months. Functional motor roots (C5-T1, L3-S1) were involved in 14 cases. The involved nerve root was routinely sacrificed during surgery and GTR was obtained in 84.6% of cases. In the functional root subgroup, for patients with a pre-existing deficit (n = 5/14), neurological aggravation persisted in one case at last FU (n = 1/5), whereas for those with no preop deficit (n = 9/14), a postoperative deficit persisted in one patient only (n = 1/9). Preoperative radicular pain was the only characteristic significantly associated with an immediate postoperative motor deficit (p = 0.03). The sacrifice of an involved nerve root in foraminal NSTs seems to represent a reasonable and relevant option to resect these tumors, permitting one to achieve tumor resection in an oncologic fashion with a high rate of GTR.

Defining $R$ and $G(R)$

We show that for Chevalley groups G(R) of rank at least 2 over an integral domain R each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that R and G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is (finitely) axiomatizable in the appropriate class of groups provided R is (finitely) axiomatizable in the corresponding class of rings.

Root subgroups on affine spherical varieties

Given a connected reductive algebraic group $G$ and a Borel subgroup $B \subseteq G$, we study $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical $G$-variety $X$ we obtain a complete description of all $G$-normalized one-parameter additive group actions on $X$ and show that the open $G$-orbit in $X$ can be connected with every $G$-stable prime divisor via a suitable choice of a $B$-normalized one-parameter additive group action. Finally, when $G$ is of semisimple rank $1$, we obtain a complete description of all $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties having an open orbit of a maximal torus $T \subseteq B$.

Homeomorphisms of S1 and Factorization

Abstract For each $n>0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations “conjugated by $z \to z^n$”. We show that these families are free of relations, which determines the structure of “the group of homeomorphisms of finite type”. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup factorization is related to triangular factorization (i.e., conformal welding) and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients.

Subsystem subgroups of the group of type $\mathrm {F}_4$ generated by short root subgroups

Subsystem subgroups of the group of type $\mathrm {F}_4$ generated by short root subgroups

The Existence of Root Subgroup Translated by a Given Element into its Opposite

Let Φ be a simply laced root system, K an algebraically closed field, and G = Gad(Φ,K) the adjoint group of type Φ over K. Then for every nontrivial element g ∈ G there exists a root element x of the Lie algebra of G such that x and gx are opposite.

Loops in SL(2, ℂ) and root subgroup factorization

In previous work, we proved that for a [Formula: see text]-valued loop having the critical degree of smoothness (one half of a derivative in the [Formula: see text] Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. For a loop [Formula: see text] satisfying these conditions, the Toeplitz determinant [Formula: see text] and shifted Toeplitz determinant [Formula: see text] factor as products in root subgroup coordinates. In this paper, we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in [Formula: see text]. The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set and associated uniqueness issues, and (2) the noncompactness of [Formula: see text] entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.

On the Normalizer of a Unipotent Root Subgroup in a Chevalley Group

In the present paper, the normalizer of a unipotent short and long root subgroup in a Chevalley group over an arbitrary field is calculated in detail. Surely this result is known to specialists. However the author could not find a reference to it.

Gale duality and homogeneous toric varieties

ABSTRACTA non-degenerate toric variety X is called S-homogeneous if the subgroup of the automorphism group Aut(X) generated by root subgroups acts on X transitively. We prove that maximal S-homogeneous toric varieties are in bijection with pairs (P,𝒜), where P is an abelian group and 𝒜 is a finite collection of elements in P such that 𝒜 generates the group P and for every a∈𝒜 the element a is contained in the semigroup generated by 𝒜∖{a}. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal S-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is S-homogeneous.

Reduction Theorems for Triples of Short Root Subgroups in Chevalley Groups

Reduction theorems for triples of short root unipotent subgroups in Chevalley groups of type B l and C l are proved. The main result asserts that except for one case, any subgroup generated by such a triple is conjugate to a subgroup of G(B4, K)U(B5, K) or G(C4, K)U(C5, K), respectively.

NONCOMPACT GROUPS OF HERMITIAN SYMMETRIC TYPE AND FACTORIZATION

We investigate Birkhoff (or triangular) factorization and (what we propose to call) root subgroup factorization for elements of a noncompact simple Lie group G0 of Hermitian symmetric type. For compact groups root subgroup factorization is related to Bott–Samelson desingularization, and many striking applications have been discovered by Lu ([5]). In this paper, in the noncompact Hermitian symmetric case, we obtain parallel characterizations of the Birkhoff components of G0 and an analogous construction of root subgroup coordinates for the Birkhoff components. As in the compact case, we show that the restriction of Haar measure to the top Birkhoff component is a product measure in root subgroup coordinates.

交换环上线性群中含根子群的极大子群

本文验证了交换环上线性群中三种含根子群的子群的极大性，并讨论了交换环上线性群中含根子群的极大子群的可能类型。 The maximality of three types of subgroups containing root subgroups in linear groups is verified in this paper. And the possible types of maximal subgroups containing root subgroups in linear groups over commutative ring are discussed.

Overgroups of root groups in classical groups

We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this form. We also determine the maximal overgroups of short root subgroups in finite classical groups, and the maximal overgroups in finite orthogonal groups of c-root subgroups.

Loops in SU(2), Riemann Surfaces, and Factorization, I

In previous work we showed that a loop $g\colon S^1 \to {\rm SU}(2)$ has a triangular factorization if and only if the loop $g$ has a root subgroup factorization. In this paper we present generalizations in which the unit disk and its double, the sphere, are replaced by a based compact Riemann surface with boundary, and its double. One ingredient is the theory of generalized Fourier-Laurent expansions developed by Krichever and Novikov. We show that a ${\rm SU}(2)$ valued multiloop having an analogue of a root subgroup factorization satisfies the condition that the multiloop, viewed as a transition function, defines a semistable holomorphic ${\rm SL}(2,\mathbb C)$ bundle. Additionally, for such a multiloop, there is a corresponding factorization for determinants associated to the spin Toeplitz operators defined by the multiloop.

Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization

In previous work with Pittmann-Polletta, we showed that a loop in a simply connected compact Lie group has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group of Hermitian symmetric type. In previous work we showed that for a constant loop there is a unique Birkhoff factorization if and only if there is a root subgroup factorization. However for loops, while a root subgroup factorization implies a unique Birkhoff factorization, the converse is false. As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.

On the Invariants of Root Subgroups of Finite Classical Groups

We show that the invariant fields F q (X 1 , . . . ,X n ) G are purely transcendental over F q if G are root subgroups of finite classical groups. The key step is to find good similar groups of our groups. Moreover, the invariant rings of the root subgroups of special linear groups are shown to be polynomial rings and their corresponding Poincaré series are presented.

Abstract simplicity of locally compact Kac–Moody groups

AbstractIn this paper, we establish that complete Kac–Moody groups over finite fields are abstractly simple. The proof makes essential use of Mathieu and Rousseau’s construction of complete Kac–Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.