Efim Zelmanov, Zezhou Zhang
In this paper we discuss finite presentability of the universal central extensions of Lie algebras ${\mathfrak{sl}_n(R)}$, where $n\geq 3$ and $R$ is a unital associative $k$-algebra. We show that a universal central extension is finitely presented if and only if the algebra $R$ is finitely presented.
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In this note we connect finite presentability of a Jordan algebra to finite presentability of its Tits-Kantor-Koecher algebra. Through this we complete our discussion of finite presentability of universal central extensions of ${\mathfrak{sl}_n(A)}$, $A$ a $k$-algebra, initiated in our previous paper, and answer a question raised by Shestakov-Zelmanov in the positive.
Subalgebras of the Universal Central Extension of $\mathfrak{sl}(n)[u^{\pm},v]$ from Two Involutions
本文提出了一种关于型$D$的扭 affine Yangian 的定义，并展示了其与相关代数结构的关系及映射构造。
On Graded Deformations of The universal Enveloping Algebra of a Color Lie Algebra
本文研究了色李代数的universal enveloping algebra的graded deformations，提出了显式构造方法并给出了相关结果。
The key ingredient of this paper is the universal central extension of the generalized orthosymplectic Lie superalgebra $\mathfrak{osp}_{m|2n}(R,{}^-)$ coordinatized by a unital associative superalgebra $(R,{}^-)$ with superinvolution. Such a universal central extension will be constructed via a Steinberg orthosymplectic Lie superalgebra coordinated by $(R,{}^-)$. The research on the universal central extension of $\mathfrak{osp}_{m|2n}(R,{}^-)$ will yield an identification between the second homology group of the generalized orthosymplectic Lie superalgebra $\mathfrak{osp}_{m|2n}(R,{}^-)$ and…
Universal Coacting Hopf Algebra of a finite-dimensional Algebra over an Operad
本文将泛共作用 Hopf 代数的构造推广到对称 operad 上的任意有限维代数，证明其范畴富集性并刻画自同构与分次，覆盖现有代数情形且扩展至分次对称 operad 上的代数。
Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\Re$ is the unital associative $\mathbb{F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$. The relations assert that \begin{equation*} [A,B]=[B,C]=[C,A]=2D \end{equation*} and each of the elements \begin{gather*} α=[A,D]+AC-BA, \qquad β=[B,D]+BA-CB, \qquad γ=[C,D]+CB-AC \end{gather*} is central in $\Re$. Additionally the element $δ=A+B+C$ is central in $\Re$. In this paper we explore the relationship between the Racah algebra $\Re$ and the universal …
Around 1992 A. Zhedanov introduced the Askey-Wilson algebra AW(3). Recently we introduced a central extension $Δ$ of AW(3) called the universal Askey-Wilson algebra. In this paper we discuss a connection between $Δ$ and the quantum algebra $U_q(\mathfrak{sl}_2)$. Our main result is an algebra injection from $Δ$ into a relative of $U_q(\mathfrak{sl}_2)$; the relative is obtained from $U_q(\mathfrak{sl}_2)$ by adjoining three mutually commuting indeterminates. We describe the injection using the equitable presentation of $U_q(\mathfrak{sl}_2)$
Ideals in Enveloping Algebras of Loop Algebras
本文研究环代数L𝔤的泛包络代数U(L𝔤)的双边理想结构，证明其不满足升链条件但存在典范嵌套理想族，还探讨对称代数S(L𝔤)的泊松结构及其在表示论中的应用。
Centers of Universal Enveloping Algebras
本文研究特征$p\ge0$的代数闭域上流/圈（超）代数的泛包络代数中心结构，明确零特征与素特征下的生成元，并探讨半单李（超）代数的对应中心结构。
This work is part of the overarching question of whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. The main result of this paper is that the universal enveloping algebra of any Krichever-Novikov algebra is not noetherian, extending a result of Sierra and Walton on the Witt (or classical Krichever-Novikov) algebra. As a subsidiary result, which may be of independent interest, we show that if $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$ of finite codimension, then the noetherianity of $U(\mathfrak{h})$ is equivalent to th…
The goal of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1},u]$ with finite dimensional simple Lie algebra $\mathfrak{g}$ and $u^m=p(t)$.
Functors between representation categories. Universal modules
该研究针对李代数的表示范畴关联问题，通过构造泛代数模与李代数模的张量积的李模结构，导出两类范畴间的函子，在有限维假设下证明函子有左伴随，进而构造出泛模。
On Cohomology group of current Lie algebras
本文通过类似模的泛系数定理的短正合序列，建立了李代数与流李代数的上同调群之间的联系，并确定了半单李代数情形下流李代数的上同调群。
Let $\mathfrak{g}$ be a finite dimensional complex simple classical Lie superalgebra and $A$ be a commutative, associative algebra with unity over $\mathbb{C}$. In this paper we define an integral form for the universal enveloping algebra of the map superalgebra $\mathfrak{g}\otimes A$, and exhibit an explicit integral basis for this integral form.
Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\mathfrak{sl}(2,\mathbb{C})$
本文完整分类了特殊线性李代数sl(2,C)通用包络代数中的所有非零有限维复数李子代数，证明幂零子代数必为阿贝尔、不可解子代数为约化且半单部分同构于sl(2,C)，非幂零可解子代数为阿贝尔中心与可解族r(d,m)的直和，并讨论了实斜埃尔米特情形及其与外尔代数实现的关系。
For an $n$-dimensional Leibniz/Lie algebra $\mathfrak{h}$ over a field $k$ we introduce a new invariant ${\mathcal A}(\mathfrak{h})$, called the \emph{universal algebra} of $\mathfrak{h}$, as a quotient of the polynomial algebra $k[X_{ij} \, | \, i, j = 1, \cdots, n]$ through an ideal generated by $n^3$ polynomials. We prove that ${\mathcal A}(\mathfrak{h})$ admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on $\mathfrak{h}$. The new object ${\mathcal A} (\mathfrak{h})$ is the key tool in answering two open problems in Lie algebra th…
Assume that $\mathbb F$ is an algebraically closed field and $q$ is a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra generated by $A,B, C$ and the relations state that each of $$ A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $\triangle_q$. The universal DAHA $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$ and the relation…
A Lie group corresponding to the free Lie algebra and its universality
本文研究自由李代数完备化对应的李群，证明其稠密子群可唯一积分任意同态，并具有普适性，与Pestov的Banach--Lie群结果相关。
For a commutative algebra $A$ over $\mathbb{C}$,denote $\mathfrak{g}=\text{Der}(A)$. A module over the smash product $A\# U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.In the present paper, we study jet modules in the case of $A=\mathbb{C}[t_1^{\pm 1},t_2]$.We show that $A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the Weyl algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of $A\# U(\mathfrak{g})$ called…
Notes on a homomorphism from the affine Yangian associated with $\hat{\mathfrak{sl}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$
本文利用先前构造的affine Yangian同态，构造了从affine Yangian到A型非矩形W-代数泛包络代数的同态，探讨了affine shifted Yangian的新定义，并证明了其与量子仿射代数同态的关联。
Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (on the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S…
In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, there is a correspondence between Verma modules and certain line modules that associates a pair $(\mathfrak{h},\,ϕ)$, where $\mathfrak{h}$ is a two-dimensional Lie subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ and $ϕ\in \mathfrak{h}^*$ sati…
Universal deformation rings of a special class of modules over generalized Brauer tree algebras
本文计算了广义Brauer树代数上周期弦模的泛变形环，并对一类特定广义Brauer树代数中含单模的Ω-稳定分支中的模完成了泛变形环分类，综合运用了Brauer图代数的表示论工具及导出等价理论。
The Askey--Wilson algebras, the Lie algebra $\mathfrak{so}_{3}$, and their fermionic realizations
本文通过代数同构将Lie代数$\mathfrak{so}_{3}$与Askey-Wilson代数联系起来，并扩展到费米子设定，揭示了量子群与费米子类比之间的统一对应关系。
The purpose of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, $\mathfrak g\otimes \mathbb C[t,t^{-1},u|u^2=(t^2-b^2)(t^2-c^2)]$, appearing in the work of Date, Jimbo, Kashiwara and Miwa in their study of integrable systems arising from Landau-Lifshitz differential equation.
In this paper we study properties of a homomorphism $ρ$ from the universal enveloping algebra $U=U(\mathfrak{gl}(n+1))$ to a tensor product of an algebra $\mathcal D'(n)$ of differential operators and $U(\mathfrak{gl}(n))$. We find a formula for the image of the Capelli determinant of $\mathfrak{gl}(n+1)$ under $ρ$, and, in particular, of the images under $ρ$ of the Gelfand generators of the center $Z(\mathfrak{gl}(n+1))$ of $U$. This formula is proven by relating $ρ$ to the corresponding Harish-Chandra isomorphisms, and, alternatively, by using a purely computational approach. Furthermore, we…
Representation theory of mirabolic quantum $\mathfrak{sl}_n$
本文通过证明mirabolic量子群MU(n)是量子 enveloping代数U_v(sl_n)的comodule代数，完成其有限维表示的分类，并明确描述所有不可约有限维表示的构造，证明该类别是半单的。
Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_ρ$ for each irreducible $\mathfrak{g}$-representation $ρ$, one relation $g_π = g_ρ^{-1}$ whenever $π$ is weakly contained in the dual representation $ρ^*$ (i.e. the kernel of $π$ in the enveloping algebra $U(\mathfrak{g})$ contains that of $ρ^*$), and one relation $g_ρ = g_{ρ'}g_{ρ"}$ whenever $ρ$ is weakly contained in $ρ'\otimesρ"$. The main result is that attaching to an irreducible representation its…
Symplectic Differential Reduction Algebras and Generalized Weyl Algebras
本文研究秩为2的辛李代数在局部化的第二个Weyl代数与其泛包络代数张量中的典范实现所对应的微分约化代数，证明其为Bavula意义下新型skew-affine广义Weyl代数，为利用标准GWA技术研究不可约权模奠定基础。
Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $φ$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $φ^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that …
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