Bogdan Ion
We show that over fields of characteristic zero a Hopf algebra with central Hopf algebra coradical has a PBW basis as a module over the coradical.
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A Hom-type algebra is called involutive if its Hom map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Hom-associative algebra on a Hom-module. We then apply this construction to obtain the universal enveloping algebra of an involutive Hom-Lie algebra. Finally we obtain a Poincaré-Birkhoff-Witt theorem for the enveloping associative algebra of an involutive Hom-Lie algebra.
Rota-Baxter associative algebras and Rota-Baxter Lie algebras are both important in mathematics and mathematical physics, with the former a basic structure in quantum field renormalization and the latter a operator form of the classical Yang-Baxter equation. An outstanding problem posed by Gubarev is to determine whether there is a Poincaré-Birkhoff-Witt theorem for the universal enveloping Rota-Baxter associative algebra of a Rota-Baxter Lie algebra. This paper resolves this problem positively, working with operated algebras and applying the method of Gröbner-Shirshov bases.
In this paper we introduce the notion of weak $Σ$-rigid ring which extends $α$-rigid rings and $Σ$-rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak $Σ$-skew Armendariz ring which extends that of $α$-skew Armendariz ring. In this way we generalize several results in the literature, from Ore extensions of injective type to the more general setting of skew PBW extensions.
Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincare-Birkhoff-Witt (PBW) deformation of the smash product algebra B#H. Many examples of these deformations are given.
PBW bases for a class of braided Hopf algebras
证明了P（R）的辫子子空间V生成的辫子霍普夫代数R的Poincare-Birkhoff-Witt型基的存在性，如果V上的辫子满足三角条件.我们将所得结果应用于具有交换余根的点霍普夫代数和低维单模U_q（sl_2）的Nichols代数。
Poincare-Birkhoff-Witt Deformations of Calabi-Yau Algebras
最近，Bocklandt证明了Van den Bergh在其分次版本中提出的一个猜想，指出一个3维Calabi-Yau的分次代数（有关系）是由齐次势W定义的。本文证明了，如果我们给W加上任意一个较小次的势，我们得到A的Poincare-Birkhoff-Witt形变。这种PBW变形是卡-丘变形，并且在A的所有PBW变形中具有特征。提出了各种例子。
On SAGBI bases theory of skew Poincaré-Birkhoff-Witt extensions
本文首次探讨了斜 Poincaré-Birkhoff-Witt 扩展的 SAGBI 基理论，并研究了这些扩展子代数的 SAGBI 基的多项式复合问题。
In this paper we review some classical results on the algebraic dependence of commuting elements in several noncommutative algebras as differential operator rings and Ore extensions. Then we extend all these results to a more general setting, the family of noncommutative rings known as skew Poincaré-Birkhoff-Witt extensions.
A quantum analog of the Poincare-Birkhoff-Witt theorem
我们减少的基础作图题的霍普夫代数产生的斜本原半不变量的特殊元素的研究，称为"超级字母“，这是定义的Shirshov标准字。通过这种方式，我们证明了上述霍普夫代数总是有PBW生成元集（“硬”超级字母）。它还表明，这些霍普夫代数不超过1000个“硬”超字母共享的泛包络代数的有限维李代数的一些性质。证明的背景是一个过滤的建设，使相关的分次代数是通过迭代的斜多项式建设，可能其次是因式分解。
Differential Calculi of Poincare-Birkhoff-Witt type on Universal Enveloping Algebras
定义了李代数g的泛包络代数上的Poincare-Birkhoff-Witt型微分演算。这个定义与g中所选的基无关。解释了g的自同构的作用。证明了半单李代数上不存在Poincare-Birkhoff-Witt型微分。给出了gl_n、Abel李代数、Heisenberg代数、Witt代数和维拉索罗代数的例子。完全处理的是2维可解李代数，和3维海森堡代数。
The Poincare-Birkhoff-Witt theorem for quadratic algebras of Koszul type
本文将经典的PBW-定理推广到Koszul代数上。我们的结果是A.Polischuk和L.Positselsky所得结果的一个轻微推广，但证明是不同的，并使用变形理论。
A Poincaré-Birkhoff-Witt theorem for generalized color Lie algebras
给出了一类与Gurevich S-李代数密切相关的广义李代数的Poincaré-Birkhoff-Witt定理的证明.作为具体例子，我们构造了A_n型量子泛包络代数的正（负）部分和GL（n）的非标准量子变形M_{p,q,e}（n，K）.特别是，我们得到，这两个代数，一个统一的证明的庞加莱-伯克霍夫-维特定理，我们表明，他们是真正的普遍包络代数的某些广义李代数。
Poincare'-Birkhoff-Witt property for bicovariant differential algebras on simple quantum groups
研究了在量子群SO_q（N）和Sp_q（N）上构造双共变微分演算的可能性，证明了与GL（N）和SL（N）-情形相反，当微分形式为李代数值时，可能的分次SO-和Sp-双共变括号（与拟三角r-矩阵相关）都不服从Jacobi恒等式.经典泊松结构的缺乏给出了一个指示，即描述量子正交群和辛群上的双协变微分演算的微分代数不是Poincar 'e-Birkhoff-Witt型的。
We prove that to every inclusion $A\hookrightarrow L$ of Lie algebroids over the same base manifold $M$ corresponds a Kapranov dg-manifold structure on $A[1]\oplus L/A$, which is canonical up to isomorphism. As a consequence, $Γ(Λ^\bullet A^\vee\otimes L/A)$ carries a canonical $L_\infty[1]$ algebra structure whose unary bracket is the Chevalley--Eilenberg differential corresponding to the Bott representation of $A$ on $L/A$ and whose binary bracket is a cocycle representative of the Atiyah class of the Lie pair $(L,A)$. To this end, we construct explicit isomorphisms of $C^\infty(M)$-coalgebr…
The construction of HNN-extensions of involutive Hom-associative algebras and involutive Hom-Lie algebras is described. Then, as an application of HNN-extension, by using the validity of Poincaré-Birkhoff-Witt theorem for involutive Hom-Lie algebras, we provide an embedding theorem.
A PBW commutator lemma for U_q[gl(m|n)]
给出并详细证明了量子超代数U_q[gl（m）]的一个Poincare-Birkhoff-Witt换位子引理|n）]。
We give a Poincare-Birkhoff-Witt type basis for local Weyl modules of the current algebra of type $C$. As a consequence, we get a fermionic character formula for these modules.
$PBW$-deformations of graded rings
本文在一般框架下证明了经典Poincaré-Birkhoff-Witt定理的多个版本，扩展了已有结果，并讨论了相关应用与例子。
We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.
Structure theorems for braided Hopf algebras
本文在辫子 Hopf 代数背景下发展了 Poincaré-Birkhoff-Witt 和 Cartier-Milnor-Moore 定理的版本，并利用辫子算子在辫子幺半范畴中引入了 Lie 代数的新类似物。
Finite dimensional rank 2 Nichols algebras of diagonal type I: Examples
Nichols代数自然地出现在有限维尖霍普夫代数的分类中。在基域特征为零的条件下，给出了几个新的有限维秩2对角型Nichols代数。每一个都用生成器和关系来描述。给出了所有这些Nichols代数的Poincaré-Birkhoff-Witt基，并计算了它们的维数.连同本文的延续，这给出了一个问题的答案Andruskiewitsch。关键词：霍普夫代数，林登字，满二叉树
A Hopf monoid (in Joyal's category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a self-contained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and …
We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincaré-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal enveloping algebra can be endowed with a structure of Hopf algebra. The study of the dual of the universal enveloping algebra enables to define the parameters of the transformation of a Lie algebra of order three. It turns out that these variables are the variables which generate the three-exterior algebra.
On the structure of cofree Hopf algebras
本文证明了非余交换霍普夫代数的Poincare '-Birkhoff-Witt定理和Cartier-Milnor-Moore定理的一个类似定理。上自由霍普夫代数的本原部分是B-无限代数。我们构造了一个从B-无限代数到2-结合代数（即具有两个结合运算的代数）的泛包络函子U 2。证明了任何余自由霍普夫代数H都是U ~ 2（Prim H）型.我们利用平面树对自由2as-代数的简单描述来揭示运算子B-无穷大的结构。
Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $ U_q(\mathfrak{g}) \, $, generated by $ q $-divided powers of Chevalley generators over a suitable subring $ R $ of $ K(q) \, $. We prove a Poincaré-Birkhoff-Witt like theorem for $ \mathfrak{U}_q(\mathfrak{g}) \, $, yielding a basis over $ R $ made of ordered products of $ q $-divided powers of suitable quantum root vectors.
The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras sp(2n). The representations are parameterized by their highest weights or by their Drinfeld polynomials. In the simplest case of sp(2) we give an explicit description of all the representations as tensor products of evaluation modules. We prove analogues of the Poincare-Birkhoff-Witt theorem for the quantum affine algebra and for the twisted q…
On the R-matrix realization of Yangians and their representations
本文研究了B、C、D型单李代数a的Yangians Y（a）。代数Y（a）可以看作是广义杨数X（a）的商，其定义关系用R-矩阵的形式表示。本文研究了代数X（a）的代数结构和表示。我们证明了X（a）的Poincare-Birkhoff-Witt定理的一个类似定理，并证明了Yangian Y（a）可以实现为X（a）的一个子代数.进一步，我们给出了X（a）的有限维不可约表示的分类定理的独立证明，它蕴涵了Yangians Y（a）的Drinfeld相应定理.我们也给出了明确的建设的所有基本代表性的Yangians。
Infinitesimal or cocommutative dipterous bialgebras and good triples of operads
Poincare，Birkhoff，Witt和Cartier，Milnor，Moore关于连通余交换霍普夫代数的工作翻译成算符的语言意味着赋予霍普夫相容关系的算符三元组（Com，As，Lie）是好的。在本文中，我们专注于左双翅（分别。右双翅）代数，它们是具有额外左（或右双翅）的结合代数.右）模块上，并寻找好的三元组被$As$替换为dipterous operad Dipt。自Loday和Ronco的工作以来，具有半Hopf相容关系的三元组（As，Dipt，B_ \infty）被认为是好的。本文证明了具有所谓（非单位）半无穷小相容关系的三元组（As、Dipt、Grove）是好的。为此，给出了K-向量空间V上自由双翅代数和自由Grove-代数的显式构造。这些结构原来是相关的有根平面树和小的一个大的施罗德数。给出了许多双翅代数的例子，特别是自由L-双翅代数。作为我们结果的推论，我们还得到了同时具有单位半Hopf和单位半无穷小相容关系的运算三元组（2As，Dipt，Vect）是好的，这里2As表示2-结合代数的运算.本文还讨论了与量子场论中的Connes-Kreimer霍普夫代数（Com，Dipt，Prim_{Com} Dipt）有关的具有Hopf相容关系的好三元组，并给出了一个从纠缠双翅类算符中给出好三元组的一般定理，称之为结合分子.
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