Solvable Lie algebras, products by generators and some of its applications

简单有限维向量李超代数的极大分次可解子代数

Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras

本文在复数域上，对标准分次下所有简单有限维向量李超代数的极大分次可解子代数进行了完整分类。

Representations of solvable Lie algebras with filtrations

We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.

On a Subclass of 5-Dimensional Solvable Lie Algebras Which Have 3-Dimensional Commutative Derived Ideal

The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e., five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbit) are orbit of zero or maximal dimension. The main results of the paper is the classification up to an isomorphism of all MD5-algebras $\mathcal{G}$ with the derived ideal ${\mathcal{G}}^{1} := [\mathcal{G}, \mathcal{G}]$ is a 3-dimensional commutative Lie algebra.

具有开余伴随拟轨道的可解李群的正则表示研究

On the regular representation of solvable Lie groups with open coadjoint quasi-orbits

本研究对正则表示为因子表示的连通单连通可解李群给出李理论内蕴刻画，得到超有限II$_\text{∞}$因子标准表示的几何模型，并证明具有开余伴随轨道的这类群其正则表示为I型。

Indecomposable modules of 2-step solvable Lie algebras in arbitrary characteristic

Let $F$ be an algebraically closed field and consider the Lie algebra ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, where $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Refer to a ${\mathfrak g}$-module as admissible if $[{\mathfrak g},{\mathfrak g}]$ acts via nilpotent operators on it, which is automatic if $\mathrm{char}(F)=0$. In this paper we classify all indecomposable ${\mathfrak g}$-modules $U$ which are admissible as well as uniserial, in the sense that $U$ has a unique composition series.

Superrigid subgroups of solvable Lie groups

Let $Γ$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_GΓ$ has the same Zariski closure as $\Ad G$. If $α\colon Γ\to \GL_n(\real)$ is any finite-dimensional representation of~$Γ$,we show that $α$ virtually extends to a continuous representation~$σ$ of~$G$. Furthermore, the image of~$σ$ is contained in the Zariski closure of the image of~$α$. When $Γ$ is not discrete, the same conclusions are true if we make the additional assumption that the closure of $[Γ, Γ]$ is a finite-index subgroup of $[G,G] \cap Γ$ (and $Γ$ is closed and $α$ is continuous).

Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

Let {\Lnk} be the class of all $n$-dimensional real solvable Lie algebras having $k$-dimensional derived ideals. In 2020 the authors et al. gave a classification of all non 2-step nilpotent Lie algebras of {\Li}. We propose in this paper to study representations of these Lie algebras as well as their corresponding connected and simply connected Lie groups. That is, for each algebra, we give an upper bound of the minimal degree of a faithful representation. Then, we give a geometrical description of coadjoint orbits of corresponding groups. Moreover, we show that the characteristic property of …

$C^*$-dynamical systems of solvable Lie groups

In this paper we develop a groupoid approach to some basic topological properties of dual spaces of solvable Lie groups using suitable dynamical systems related to the coadjoint action. One of our main results is that the coadjoint dynamical system of any exponential solvable Lie group is a piecewise pullback of group bundles. Our dynamical system approach to solvable Lie groups also allows us to construct some new examples of connected solvable Lie groups whose $C^*$-algebras admit faithful irreducible representations.

Quasidiagonality of $C^*$-algebras of solvable Lie groups

We characterize the solvable Lie groups of the form ${\mathbb R}^m\rtimes {\mathbb R}$, whose $C^*$-algebras are quasidiagonal. Using this result, we determine the connected simply connected solvable Lie groups of type~I whose $C^*$-algebras are strongly quasidiagonal. As a by-product, we give also examples of amenable Lie groups with non-quasidiagonal $C^*$-algebras.

Cohomologically rigid solvable Lie superalgebras with model filiform and model nilpotent nilradical

In this paper, we find a family $SL^{n,m}$, in any arbitrary dimensions, of cohomologically rigid solvable Lie superalgebras with nilradical the model filiform Lie superalgebra $L^{n,m}$. Moreover, we exhibit a family of cohomologically rigid solvable Lie superalgebras with nilradical the model nilpotent Lie superalgebra of generic characteristic sequence. Both cases correspond to solvable Lie superalgebras of maximal dimension for a given nilradical. Contrariwise, we will show that the family of Lie superalgebras $SL^{n,m}$ can be deformed if defined over a field of odd characteristic.

On stably finiteness for $C^*$-algebras of exponential solvable Lie groups

We study the link between stably finiteness and stably projectionless-ness for $C^*$-algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by $4$; otherwise, they are not necessarily equivalent. To provide examples proving the last assertion, we study exponential solvable Lie groups that have nonempty finite open sets in their unitary dual.

光滑向量的指数可解李群表示的强因子分解定理

Strong factorization theorem for smooth vectors of exponential solvable Lie group representations

本文改进了Dixmier-Malliavin定理，为指数可解李群在弗雷歇空间上的光滑向量建立了新的强因子分解性质。

On $C^*$-algebras of exponential solvable Lie groups and their real ranks

For any solvable Lie group whose exponential map $\exp_G\colon{\mathfrak g}\to G$ is bijective, we prove that the real rank of $C^*(G)$ is equal to $\dim({\mathfrak g}/[{\mathfrak g},{\mathfrak g}])$. We also indicate a proof of a similar formula for the stable rank of $C^*(G)$, as well as some estimates on the ideal generated by the projections in $C^*(G)$.

Solvable Complemented Lie Algebras

In this paper a characterisation is given of solvable complemented Lie algebras. They decompose as a direct sum of abelian subalgebras and their ideals relate nicely to this decomposition. The class of such algebras is shown to be a formation whose residual is the ideal closure of the prefrattini subalgebras.

On certain decompositions of solvable Lie algebras

The author has previously shown that solvable Lie A-algebras and complemented solvable Lie algebras decompose as a vector space direct sum of abelian subalgebras, and their ideals relate nicely to this decomposition. However, neither of these classes is contained in the other. The object of this paper is to find a larger class of algebras, containing each of these classes, in which these same results hold.

Nilpotency degree of the nilradical of a solvable Lie algebra on two generators

Given a sequence $\vec d=(d_1,\dots,d_k)$ of natural numbers, we consider the Lie subalgebra $\mathfrak{h}$ of $\mathfrak{gl}(d,\mathbb{F})$, where $d=d_1+\cdots +d_k$ and $\mathbb{F}$ is a field of characteristic 0, generated by two block upper triangular matrices $D$ and $E$ partitioned according to $\vec d$, and study the problem of computing the nilpotency degree $m$ of the nilradical $\mathfrak{n}$ of $\mathfrak{h}$. We obtain a complete answer when $D$ and $E$ belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain so…

Noncommutative rational functions invariant under the action of a finite solvable group

This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups $G$ with a well-behaved representation theory it is shown that the invariant skew fields are free on $|G|(d-1)+1$ generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.