Automorphisms Of Lie Algebras Research Articles

A classification of automorphic Lie algebras on complex tori

Abstract We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$ . For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2({\mathbb{Z}})$ , apart from four cases, which are all isomorphic to Onsager’s algebra.

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.

2-term averaging L∞-algebras and non-abelian extensions of averaging Lie algebras

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L∞-algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context.

Автомофизмы нильтреугольных подколец алгебр Шевалле типа $G_2$ над областями целостности. I

Let $N\Phi(K)$ be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring $K$ with the identity associated with a root system $\Phi$. This paper studies the well-known problem of describing automorphisms of Lie algebras and rings $N\Phi(K)$. Automorphisms of the Lie algebra $N\Phi(K)$ under restrictions $K=2K=3K$ on ring $K$ are described by Y. Cao, D. Jiang, J. Wang (2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type $A_{n}$, a description of automorphisms of Lie rings $N\Phi(K)$ over $K$ was obtained by V.M. Levchuk (1983). Automorphisms of the Lie ring $N\Phi(K)$ are described by V.M. Levchuk (1990) for type $D_4$ over $K$, and for other types by A.V. Litavrin (2017), excluding types $G_2$ and $F_4$. In this paper we describe automorphisms of a nil-triangular Lie ring of type $G_2$ over an integrity domain $K$ under the following restrictions on $K$: either $K=2K=3K$, or $3K = 0$. To study automorphisms, the upper and lower central series described in this paper, are essentially used.

Автоморфизмы нильтреугольных подколец алгебр Шевалле типа G2 над полем характеристики 2

Let NΦ(K) be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring K with the identity associated with a root system Φ (The basis of NΦ(K) consists of all elements er ∈ Φ+ of the Chevalley basis). This paper studies the well-known problem of describing automorphisms of Lie algebras and rings NΦ(K). Automorphisms of the Lie algebra NΦ(K) under restrictions K = 2K = 3K on ring K are described by Y. Cao, D. Jiang, J. Wang (Intern. J. Algebra and Computation, 2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type An, a description of automorphisms of Lie rings NΦ(K) over K was obtained by V.M. Levchuk (Siberian Mathematical Journal, 1983). Automorphisms of the Lie ring NΦ(K) are described by V.M. Levchuk (Algebra and Logic, 1990) for type D4 over K, and for other types by A.V. Litavrin (Thesis for: Cand. Sc. (Physics and Mathematics) – 01.01.06. Siberian Federal University, 2017), excluding types G2 and F4. The author (2022) obtained a description of automorphisms of Lie rings NΦ(K) of type G2 when K is an integrity domain and K = 2K = 3K or 3K = 0. In this paper we describe automorphisms of a niltriangular Lie ring of type G2 over a field K under restriction 2K = 0. To study automorphisms, the upper and lower central series described in this work are essentially used. A new non-standard automorphism was found, called an S-automorphism.

Local automorphisms of complex solvable Lie algebras of maximal rank

This paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism.

Wild Local Structures of Automorphic Lie Algebras

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

Weight-one elements of vertex operator algebras and automorphisms of categories of generalized twisted modules

Given a weight-one element u of a vertex operator algebra V, we construct an automorphism of the category of generalized g-twisted modules for automorphisms g of V fixing u. We apply this construction to the case that V is an affine vertex operator algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules.

ON AUTOMORPHISMS OF LIE ALGEBRA OF SYMMETRIC POLYNOMIALS

Let $L_{n}$ be the free Lie algebra of rank $n$ over a field $K$ of characteristic zero, $L_{n,c}=L_{n}/(L_{n}''+\gamma_{c+1}(L_{n}))$ be the free metabelian nilpotent of class $c$ Lie algebra, and $F_{n}=L_{n}/L_{n}''$ be the free metabelian Lie algebra generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ in these Lie algebras {\it symmetric} if $p(x_1,\ldots,x_n)=p(x_{\pi(1)},\ldots,x_{\pi(n)})$ for each element of the symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, and $L_{n,c}^{S_n}$ of symmetric polynomials coincides with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determine the groups $\text{\rm Inn}(L_{n,c}^{S_n})\cap \text{\rm Inn}(L_{n,c})$ and $\text{\rm Inn}(F_{n}^{S_n})\cap \text{\rm Inn}(F_{n})$ of inner automorphisms of the algebras $L_{n,c}^{S_n}$ and $F_{n}^{S_n}$ in the groups $\text{\rm Inn}(L_{n,c})$ and $\text{\rm Inn}(F_{n})$, respectively. In particular, we obtain the descriptions of the groups $\text{\rm Aut}(L_{2}^{S_2})\cap \text{\rm Aut}(L_{2})$ and $\text{\rm Aut}(F_{2}^{S_2})\cap \text{\rm Aut}(F_{2})$ of automorphisms of the algebras $L_{2}^{S_2}$ and $F_{2}^{S_2}$ in the groups $\text{\rm Aut}(L_{2})$ and $\text{\rm Aut}(F_{2})$, respectively.

Nonlinearizable CR Automorphisms for Polynomial Models in $$\mathbb C^N$$

The Lie algebra of infinitesimal CR automorphisms is a fundamental local invariant of a CR manifold. Motivated by the Poincaré local equivalence problem, we analyze its positively graded components, containing nonlinearizable holomorphic vector fields. The results provide a complete description of invariant weighted homogeneous polynomial models in $$\mathbb C^N$$ , which admit symmetries of degree higher than two. For homogeneous polynomial models, symmetries with quadratic coefficients are also classified completely. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove that such automorphisms arise from one common source, by pulling back via a holomorphic mapping a suitable symmetry of a hyperquadric in some (typically high dimensional) complex space.

Левоинвариантная парасасакиева структура на групповой модели вещественного расширения плоскости де-Ситтера

In this paper, a group model for a real extension of the de Sitter plane is pro-posed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.

Duality defects in E8

We classify all non-invertible Kramers-Wannier duality defects in the E8 lattice Vertex Operator Algebra (i.e. the chiral (E8)1 WZW model) coming from ℤm symmetries. We illustrate how these defects are systematically obtainable as ℤ2 twists of invariant sub-VOAs, compute defect partition functions for small m, and verify our results against other techniques. Throughout, we focus on taking a physical perspective and highlight the important moving pieces involved in the calculations. Kac’s theorem for finite automorphisms of Lie algebras and contemporary results on holomorphic VOAs play a role. We also provide a perspective from the point of view of (2+1)d Topological Field Theory and provide a rigorous proof that all corresponding Tambara-Yamagami actions on holomorphic VOAs can be obtained in this manner. We include a list of directions for future studies.

POINTWISE INNER AUTOMORPHISMS OF RELATIVELY FREE LIE ALGEBRAS

Let $K$ be a field of characteristic zero, and $L_{m,c}$ be the free metabelian nilpotent Lie algebra of class $c$ of rank $m$ over $K$. We call an automorphism $\phi$ pointwise inner, if there exists an inner automorphism $\xi_i$ for each generator $x_i$, $i=1,\ldots,m$, such that $\phi(x_i)=\xi_i(x_i)$. In this study, we exemine the group $PI(L_{m,c})$ of pintwise inner automorphisms of the Lie algebra $L_{m,c}$, and we provide a set of generators for this group.

CR-manifolds of infinite type in the sense of Bloom and Graham

An analogue of the Bloom–Graham theorem for germs of real analytic CR-manifolds of infinite type is devised, and a certain standard form to which they can be transformed (a reduced form) is described. The concept of Bloom–Graham type is refined (as a stratified type). The refined type is also holomorphically invariant. The concept of a quasimodel surface is introduced and it is shown that for biholomorphically equivalent manifolds such surfaces are quasilinearly equivalent. A criterion for the Lie algebra of infinitesimal holomorphic automorphisms to be finite-dimensional is obtained in the case when the type is uniformly infinite (that is, infinite at all points). In combination with the criterion of a finite-dimensional automorphism algebra for manifolds of finite type almost everywhere, this yields a complete criterion for this algebra to be finite-dimensional. The sets of fixed Blooom–Graham type are shown to be semi-analytic and the type of a generic point (lying outside a proper analytic subset) is minimal in a certain sense.

3-Derivations and 3-Automorphisms on Lie Algebras

In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms. Then we study the derivations and automorphisms of the standard embedding Lie algebra of a Lie triple system. We prove that derivations and automorphisms of a Lie triple system give rise to derivations and automorphisms of the corresponding standard embedding Lie algebra. Finally we compute the 3-derivations and 3-automorphisms of 3-dimensional real Lie algebras.

Automorphic Lie Algebras and Modular Forms

AbstractWe introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma $ be a finite index subgroup of $\textrm {SL}(2,\mathbb Z)$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\textrm {SL}(2,{\mathbb {C}})$. We show that the Lie algebra of the corresponding $\mathfrak {g}$-valued modular forms is isomorphic to the extension of $\mathfrak {g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $\Gamma (N), \, N\leq 6$, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras.

Matsushima–Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type

In K\"ahler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized K\"ahler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized K\"ahler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that $H^{1}(M)=0$. Then we show that the Lie algebra of the automorphisms is a reductive Lie algebra if a generalized complex manifold admits a generalized K\"ahler structure of symplectic type with constant scalar curvature. This is a generalization of Matsushima and Lichnerowicz theorem in K\"ahler geometry. We explicitly calculate the Lie algebra of the automorphisms of a generalized complex structure given by a cubic curve on $\Bbb C P^2$. Cubic curves are classified into nine cases (see Figure.$1 -- 9$). In the three cases as in Figures. 7, 8 and 9, the Lie algebra of the automorphisms is not reductive and there is an obstruction to the existence of generalized K\"ahler structures of symplectic type with constant scalar curvature in the three cases. We also discuss deformations starting from an ordinary K\"ahler manifold $(X,\omega)$ with constant scalar curvature and show that nontrivial generalized K\"ahler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms of $X$ is trivial.

On the infinitesimal automorphisms of principal bundles

We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra of infinitesimal automorphisms of $L$. If the representation of $\mathfrak{g}_L$ on the space of holomorphic sections of $L$ is irreducible then $Z$ is rational; (2) Let $P$ be a holomorphic principal bundle over the Riemann sphere, with structural group $G$ whose Lie algebra is not equal to its nilpotent radical. Then there exists a Lie subgroup $H$ of $G$ which is a quotient of a Borel subgroup of ${\rm SL}(2)$ and such that $P$ admits a reduction to $H$.

Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

Symmetries of quaternionic Kähler manifolds with S1‐symmetry

We study symmetry properties of quaternionic K\"ahler manifolds obtained by the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal automorphisms of the initial hyper-K\"ahler data we associate a central extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the resulting quaternionic K\"ahler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the $c$-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special K\"ahler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed $c$-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic K\"ahler metric on $SU(n,2)/S(U(n)\times U(2))$, which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet ($n=1$), for which we determine the full isometry group.