A construction of finite-dimensional faithful representation of Lie algebra

Faithful representations of infinite-dimensional nilpotent Lie algebras

For locally convex, nilpotent Lie algebras we construct faithful representations by nilpotent operators on a suitable locally convex space. In the special case of nilpotent Banach-Lie algebras we get norm continuous representations by bounded operators on Banach spaces.

Computing faithful representations for nilpotent Lie algebras

We describe three methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods in finding bounds for the smallest dimension $μ(\Lg)$ of a faithful $\Lg$-module for some nilpotent Lie algebras $\Lg$. In particular, we describe an infinite family of filiform nilpotent Lie algebras $\Lf_n$ of dimension $n$ over $\Q$ and conjecture that $μ(\Lf_n) > n+1$. Experiments with our algorithms suggest that $μ(\Lf_n)$ is polynomial in $n$.

Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor

For a finite dimensional Lie algebra $\g$ over a field $\k$ of characteristic zero, the $μ$-function (respectively $μ_{nil}$-function) is defined to be the minimal dimension of $V$ such that $\g$ admits a faithful representation (respectively a faithful nilrepresentation) on $V$. Let $\h_m$ be the Heisenberg Lie algebra of dimension $2m + 1$ and let $\mathfrak{a}_n$ be the abelian Lie algebra of dimension $n$. The aim of this paper is to compute $μ(\h_m \oplus \mathfrak{a}_n)$ and $μ_{nil}(\h_m \oplus \mathfrak{a}_n)$ for all $m,n \in \mathbb{N}$.

Faithful Representations of Minimal Dimension of 6-dimensional nilpotent Lie algebras

The main goal of this paper is to compute $μ(\g)$ and $μ_{nil}(\g)$ for each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic zero $\k$. Here $μ(\g)$ and $μ_{nil}(\g)$ is the minimal dimension of a faithful representation of $\g$ and the minimal dimension of a faithful nilrepresentation of $\g$, respectively. We also give a minimal faithful representation and nilrepresentation for each nilpotent Lie algebra of dimension $\leq 6$.

Faithful irreducible representations of modular Lie algebras

Let L be a finite-dimensional Lie algebra over a field of non-zero characteristic. By a theorem of Jacobson, L has a finite-dimensional faithful module which is completely reducible. We show that if the field is not algebraically closed, then L has an irreducible such module. We also give a necessary and sufficient condition for a finite-dimensional Lie algebra over an algebraically closed field of non-zero characteristic to have a faithful irreducible module.

Faithful representations of Lie algebras and Homogeneous Spaces

The faithful representations of real Lie algebras g with dimensions $n\leq 4$ acting on the dual space of the universal covering algebra of an ideal s of g, are calculated. With the help of left and right translations of the Lie group, we define the homogeneous manifolds of dimension n and we calculate the Killing fields (generators of left translations), the invariant fields (generators of right translations) and the left invariant 1-forms.

Faithful representations of minimal dimension of current Heisenberg Lie algebras

Given a Lie algebra $\mathfrak{g}$ over a field of characteristic zero $k$, let $μ(\mathfrak{g})=\min\{\dim π: π\text{is a faithful representation of}\mathfrak{g}\}$. Let $\mathfrak{h}_{m}$ be the Heisenberg Lie algebra of dimension $2m+1$ over $k$ and let $k[t]$ be the polynomial algebra in one variable. Given $m\in\mathbb{N}$ and $p\in k[t]$, let $\mathfrak{h}_{m,p}=\mathfrak{h}_m\otimes k[t]/(p)$ be the current Lie algebra associated to $\mathfrak{h}_m$ and $k[t]/(p)$, where $(p)$ is the principal ideal in $k[t]$ generated by $p$. In this paper we prove that $ mu(\mathfrak{h}_{m,p}) = m °p …

D型情形中Iwasawa代数的最高权模的忠实性

Faithfulness of highest-weight modules for Iwasawa algebras in type D

本文研究D型单李代数对应Iwasawa代数上无限维最高权模的忠实性，并据此证明其所有非零双边理想均具有有限余维数。

A Generalization of Lie's Theorem

We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra (Borel subalgebra) is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L.

Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank $r$

Given a finite dimensional Lie algebra $\mathfrak{g}$, let $\mathfrak{z}(\mathfrak{g})$ denote the center of $\mathfrak{g}$ and let $μ(\mathfrak{g})$ be the minimal possible dimension for a faithful representation of $\mathfrak{g}$. In this paper we obtain $μ(\mathcal{L}_{r,2})$, where $\mathcal{L}_{r,k}$ is the free $k$-step nilpotent Lie algebra of rank $r$. In particular we prove that $μ(\mathcal{L}_{r,2})= \left\lceil \sqrt{2r(r-1)} \right\rceil + 2$ for $r \geq 4$. It turns out that $μ(\mathcal{L}_{r,2}) \simμ\big(\mathfrak{z}(\mathcal{L}_{r,2})\big) \sim 2\sqrt{\dim\mathcal{L}_{r,2}} $ (…

Faithful Cyclic Modules for Enveloping Algebras and Sklyanin Algebras

Let $U$ be the enveloping algebra of a finite dimensional nonabelian Lie algebra $\mathfrak{g}$ over a field of characteristic zero. We show that there is an open nonempty open subset $X$ of $U_1 = \mathfrak{g}\oplus K$ such that $U/Ux$ is faithful for all $x \in X$. We prove similar results for homogenized enveloping algebras and for the three dimensional Sklyanin algebras at points of infinite order. It would be interesting to know if there is a common generalization of these results.

Khovanov弧代数的忠实覆盖

Faithful covers of Khovanov arc algebras

本文证明了扩展Khovanov代数$K^m_n$构成Khovanov弧代数$H^m_n$的$(|n-m|-1)$-忠实覆盖，揭示了二者之间的覆盖关系。

Verma modules for Iwasawa algebras are faithful

We establish the faithfulness of Verma modules for rational Iwasawa algebras of split semisimple compact $L$-analytic groups. We also prove the algebraic independence of Arens-Michael envelopes over Iwasawa algebras and compute the centre of affinoid enveloping algebras of semisimple $p$-adic Lie algebras.

Faithfulness of generalised Verma modules for Iwasawa algebras

We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type $A_{2}$. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.

Hopf actions on vertex operator algebras

The Hopf actions on vertex operator algebras are investigated. If the action is semisimple, a Schur-Weyl type decomposition is obtained. When the Hopf algebra is finite dimensional and the action is faithful, the action is a group action. Moreover if the Hopf algebra is finite dimensional and the action is semisimple and inner faithful, the action is also a group action. In this case, inner faithfulness is equivalent to faithfulness.

通过B型之字形代数证明2-辫子群的忠实性

Faithfulness of the 2-Braid Group via Zigzag Algebra in Type B

本文通过证明B型之字形代数上的双模范畴是B型Soergel双模范畴的商范畴，为Rouquier关于B型2-辫子群忠实性的猜想提供了另一种证明。

On faithfully balanced modules, F-cotilting and F-Auslander algebras

We revisit faithfully balanced modules. These are faithful modules having the double centralizer property. For finite-dimensional algebras our main tool is the category ${\rm cogen}^1(M)$ of modules with a copresentation by summands of finite sums of $M$ on which ${\rm Hom}(-,M)$ is exact. For a faithfully balanced module $M$ the functor ${\rm Hom}(-,M)$ is a duality on these categories - for cotilting modules this is the Brenner-Butler theorem. We also study new classes of faithfully balanced modules combining cogenerators and cotilting modules. Then we turn to relative homological algebra in…

Extensions of almost faithful prime ideals in virtually nilpotent mod-$p$ Iwasawa algebras

Let $G$ be a nilpotent-by-finite compact $p$-adic analytic group for some $p>2$, and $H = \mathbf{FN}_p(G)$ its finite-by-(nilpotent $p$-valuable) radical. Fix a finite field $k$ of characteristic $p$, and write $kG$ for the completed group ring of $G$ over $k$. We show that almost faithful $G$-stable prime ideals $P$ of $kH$ extend to prime ideals $PkG$ of $kG$.

$K_0$-invariance of the completely faithful property of Iwasawa modules

Let $H$ be a compact $p$-adic analytic group without torsion element, whose Lie algebra is split semisimple and $\mathfrak{N}_H(G)$ be the full subcategory of the category of finitely generated modules over the Iwasawa algebra $Λ_G$ that are also finitely generated as $Λ_H$-modules, where $G = \mathbb{Z}_{p} \times H$. We show that if the class of a module $N$ in the Grothendieck group of $\mathfrak{N}_H(G)$ equals to the class of a completely faithful module, then $q(N)$ is also completely faithful, where $q(N)$ denotes the image of $N$ via the quotient functor modulo the full subcategory of …

温和代数上倾斜模的自同态代数的几何模型

Geometric models for endomorphism algebras of tilting modules over gentle algebras

研究温和代数上倾斜模及其自同态代数，利用标记曲面与平铺框架，证明平铺忠实剖分对应倾斜模，定义辅助代数并证其与自同态代数同构，构造新平铺，引入保持翻转的倾斜模。

Invariant bilinear forms of algebras given by faithfully flat descent

The existence of nondegenerate invariant bilinear forms is one of the most important tools in the study of Kac-Moody Lie algebras and extended affine Lie algebras. In practice, these forms are created, or shown to exist, either by assumption or in an ad hoc basis. The purpose of this work is to describe the nature of the space of invariant bilinear forms of certain algebras given by faithfully flat descent (which includes the affine Kac-Moody Lie algebras, as well as Azumaya algebras and multiloop algebras) within a functorial framework. This will allow us to conclude the existence, uniqueness…

微分分次代数上的忠实作用决定一大类群的同构类型

Faithful actions on Differential Graded Algebras determine the isomorphism type of a large class of groups

本研究证明了包含有限群、可数阿廷群、特定曲面映射类群在内的一大类群的同构类型，可由其忠实作用的微分分次$\boldsymbol{\rm Q}$-代数集合确定，为群同构判定提供了新的代数视角。

Modular representations of Heisenberg algebras

Let $F$ be be an arbitrary field and let $h(n)$ be the Heisenberg algebra of dimension $2n+1$ over $F$. It was shown by Burde that if $F$ has characteristic 0 then the minimum dimension of a faithful $h(n)$-module is $n+2$. We show here that his result remains valid in prime characteristic $p$, as long as $(p,n)\neq (2,1)$. We construct, as well, various families of faithful irreducible $h(n)$-modules if $F$ has prime characteristic, and classify these when $F$ is algebraically closed. Applications to matrix theory are given.

Finitistic Dimension of Faithfully Flat Weak Hopf-Galois Extension

Let $H$ be a finite-dimensional weak Hopf algebra over a field $k$ and $A/B$ be a right faithfully flat weak $H$-Galois extension. We prove that if the finitistic dimension of $B$ is finite, then it is less than or equal to that of $A$. Moreover, suppose that $H$ is semisimple. If the finitistic dimension conjecture holds, then the finitistic dimension of $B$ is equal to that of $A$.

Representing Lie algebras using approximations with nilpotent ideals

We prove a refinement of Ado's theorem for Lie algebras over an algebraically-closed field of characteristic zero. We first define what it means for a Lie algebra $L$ to be approximated with a nilpotent ideal, and we then use such an approximation to construct a faithful representation of $L$. The better the approximation, the smaller the degree of the representation will be. We obtain, in particular, explicit and combinatorial upper bounds for the minimal degree of a faithful $L$-representation. The proofs use the universal enveloping algebra of Poincaré-Birkhoff-Witt and the almost-algebraic…

Degrees of Faithful Irreducible Representations of Certain Metabelian Groups and a Question of Sim

In this paper, we answer affirmatively a question of H S Sim on representations in characteristic $0$, for a class of metabelian groups. Moreover, we provide examples to point out that the analogous answer is no longer valid if the solvable group has derived length larger than 2. Let $F$ be a field of characteristic $0$ and $\overline{F}$ be its algebraic closure. We prove that if $G$ is a finite metabelian group containing a maximal abelian normal subgroup which is a p-group with abelian quotient, all possible faithful irreducible representations over $F$ have the same degree and that the Sch…

Twisted symplectic reflection algebras

In this paper we introduce the notion of twisted symplectic reflection algebras and describe the category of representations of such an algebra associated to a non-faithful G-action in terms of those for faithful actions of G.

New applications of graded Lie algebras to Lie algebras, generalized Lie algebras and cohomology

We give new applications of graded Lie algebras to: identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, $2k$-Lie algebras, generalized Poisson brackets and so on.

Classical derived functors as fully faithful embeddings

Given associative unital algebras $A$ and $B$ and a complex $T^\bullet$ of $B-A-$bi\-modules, we give necessary and sufficient conditions for the total derived functors, $\Rh_A(T^\bullet,?):\D(A)\longrightarrow\D(B)$ and $?\Lt_BT^\bullet:\D(B)\longrightarrow\D(A)$, to be fully faithful. We also give criteria for these functors to be one of the fully faithful functors appearing in a recollement of derived categories. In the case when $T^\bullet$ is just a $B-A-$bimodule, we connect the results with (infinite dimensional) tilting theory and show that some open question on the fully faithfulness …

Reduced contragredient Lie algebras and PC Lie algebras

The first aim of this paper is to show that any finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation can be embedded into some PC Lie algebra. The second aim is to find the structure of a PC Lie algebra.

Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra

It is shown that any generalized Kac-Moody Lie algebra g that has no mutually orthogonal imaginary simple roots can be written as the vector space direct sum of a Kac-Moody subalgebra and subalgebras isomorphic to free Lie algebras over certain modules for the Kac-Moody subalgebra. Also included is a detailed discussion of Borcherds' construction of the Monster Lie algebra from a vertex algebra and an elementary proof of Borcherds' theorem relating Lie algebras with `an almost positive definite bilinear form' to generalized Kac-Moody algebras. (Preprint version 1996)

On the maximal abelian subalgebras of the general linear Lie color algebras

Let $Γ$ be a finite group and $V$ a finite-dimensional $Γ$-graded space over an algebraically closed field of characteristic not equal to 2. In the sense of conjugation, we classify all the so-called pre-nil or nil maximal abelian subalgebras for the general linear Lie color algebra $\frak{gl}(V,Γ)$. In the situation of $Γ$ being a cyclic group, we determine the minimal dimensions of pre-nil or nil faithful representations for any finite-dimensional abelian Lie color algebra.

Contragredient Lie algebras and Lie algebras associated with a standard pentad

We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.

Faithful representations of SL(2) over truncated Witt vectors

Let G be the six dimensional linear algebraic k-group SL_2(W_2), where W_2 is the ring of Witt vectors of length two over the algebraically closed field k of characteristic p>2. Then the minimal dimension of a faithful rational k-representation of G is p+3.

A characterisation of Morita algebras in terms of covers

A pair $(A, P)$ is called a cover of $\operatorname{End}_A(P)^{op}$ if the Schur functor $\operatorname{Hom}_A(P, -)$ is fully faithful on the full subcategory of projective $A$-modules, for a given projective $A$-module $P$. By definition, Morita algebras are the covers of self-injective algebras and then $P$ is a faithful projective-injective module. Conversely, we show that $A$ is a Morita algebra and $\operatorname{End}_A(P)^{op}$ is self-injective whenever $(A, P)$ is a cover of $\operatorname{End}_A(P)^{op}$ for a faithful projective-injective module $P$.

Ado theorem for nilpotent Hom-Lie algebras

We prove an analog of the Ado theorem - the existence of a finite-dimensional faithful representation - for a certain kind of finite-dimensional nilpotent Hom-Lie algebras.

亚辛表示是忠实的

The Metaplectic Representation is Faithful

本文提出证明一致 pro-p 群的 Iwasawa 代数上无限维模忠实性的方法，并据此证明辛群的亚辛表示是忠实的。

Non Complete Affine Connections on Filiform Lie Groups

We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular, on the 3 dimensional Heisenberg algebra.

李型秩1群及相关群的具有少量若尔当块的模

Modules with few Jordan blocks for rank $1$ groups of Lie type and related groups

本文针对具有强p嵌入子群的有限群X，研究其k-活跃忠实FF_pX模的非平凡合成因子，其中k不超过X的p-秩m（m≥2）。

Finite-dimensional Lie subalgebras of algebras with continuous inversion

We investigate the finite-dimensional Lie groups whose points are separated by the continuous homomorphisms into groups of invertible elements of locally convex algebras with continuous inversion that satisfy an appropriate completeness condition. We find that these are precisely the linear Lie groups, that is, the Lie groups which can be faithfully represented as matrix groups. Our method relies on proving that certain finite-dimensional Lie subalgebras of algebras with continuous inversion commute modulo the Jacobson radical.

Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras

The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.

Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras

The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced $3$-Lie algebras.

The center of the affine nilTemperley-Lieb algebra

We give a description of the center of the affine nilTemperley-Lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations. We present a normal form for monomials, hence construct a basis of the algebra, and use this basis to show that the affine nilTemperley-Lieb algebra is finitely generated over its center. As an application, we obtain a natural embedding of the affine nilTemperley-Lieb algebra on N generators into the affine nilTemperley-Lieb algebra on N + 1 generators.

有限截断赋值环上幂零群的忠实维数的多项式性

Polynomiality of the faithful dimension of nilpotent groups over finite truncated valuation rings

本文研究有限截断赋值环上幂零群的忠实维数，证明在有限域且素数充分大时其由有限多项式列表给出，并提出一般情形的多项式性猜想并证明特殊情形，最后精确计算自由亚交换幂零Lie代数的忠实维数。

Extended affine Lie algebras and other generalizations of affine Lie algebras - a survey

This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.

$3$-$Lie_\infty$-algebras and $3$-Lie 2-algebras

In this paper, we introduce the notions of a $3$-$Lie_\infty$-algebra and a 3-Lie 2-algebra. The former is a model for a 3-Lie algebra that satisfy the fundamental identity up to all higher homotopies, and the latter is the categorification of a 3-Lie algebra. We prove that the 2-category of 2-term $3$-$Lie_\infty$-algebras is equivalent to the 2-category of 3-Lie 2-algebras. Skeletal and strict 3-Lie 2-algebras are studied in detail. A construction of a 3-Lie 2-algebra from a symplectic 3-Lie algebra is given.

Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators

In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_\infty$-algebra, whose Maurer-Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie $3$-algebra whose Maurer-Cartan elements are O-operators (also called relative Rota-Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of t…