Tatsuro Ito, Paul Terwilliger, Chih-wen Weng
We show that the quantum algebra $U_q(sl_2)$ has a presentation with generators $x,x^{-1},y,z$ and relations $x x^{-1}=1$, $x^{-1} x=1$, $\frac{qxy-q^{-1}yx}{q-q^{-1}}=1$, $\frac{qyz-q^{-1}zy}{q-q^{-1}}=1$, $\frac{qzx-q^{-1}xz}{q-q^{-1}}=1$. We call this the equitable presentation. We investigate the action of $x,x^{-1},y,z$ on finite-dimensional $U_q(sl_2)$-modules.
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We consider the quantum group $U_q(g)$ associated with a symmetrizable Kac-Moody algebra $g$. We display a presentation for $U_q(g)$ that we find attractive; we call this the equitable presentation. For $g=sl_2$ the equitable presentation has generators $X,X^{-1},Y,Z$ and relations $X X^{-1} =1$, $X^{-1} X=1$, $\frac{qXY-q^{-1}YX}{q-q^{-1}}=1$, $\frac{qYZ-q^{-1}ZY}{q-q^{-1}}=1$, $\frac{qZX-q^{-1}XZ}{q-q^{-1}}=1$.
The equitable presentation of the quantum superalgebra $\mathfrak{osp}_q(1|2)$, in which all generators appear on an equal footing, is exhibited. It is observed that in their equitable presentations, the quantum algebras $\mathfrak{osp}_q(1|2)$ and $\mathfrak{sl}_q(2)$ are related to one another by the formal transformation $q\rightarrow -q$. A $q$-analog of the Bannai-Ito algebra is shown to arise as the covariance algebra of $\mathfrak{osp}_q(1|2)$.
A unified framework for the Chevalley and equitable presentation of $U_q(sl_2)$ is introduced. It is given in terms of a system of Freidel-Maillet type equations satisfied by a pair of quantum K-operators ${\mathcal K}^\pm$, whose entries are expressed in terms of either Chevalley or equitable generators. The Hopf algebra structure is reconsidered in light of this unified framework, and interwining relations for each pair of ${\mathcal K}^\pm$ are obtained. A K-operator solving a spectral parameter dependent Freidel-Maillet type equation is also considered. Different specializations of this K-…
The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type $U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ are introduced and studied. A presentation for $U_q^{T,+}$ is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-oper…
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 $U_q(\mathfrak{sl}_2)$
We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ in the equitable presentation. From this point of view, we describe the Lusztig automorphism and the corresponding Lusztig operator.
A quantum shuffle approach to quantum affine super algebra of type $C(2)^{(2)}$ and its equitable presentation
本文提出利用量子洗牌方法和卡塔兰词，以闭式表达式呈现类型 $C(2)^{(2)}$ 的量子拟叠代数的两个基，并给出了其玻色化表示。
We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ with $q$ not a root of unity. We describe the finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules from the point of view of the equitable presentation.
Automorphisms of the infinite dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical analog of the Askey-Wilson algebra. In the general case, generalizations of the classical Askey-Wilson algebra are obtained. The corresponding class of solutions of the non-standard classical Yang-Baxter algebra are constructed, from which a generating function of elements in the commutative subalgebra is derived. We provide also another presentation of the Ons…
We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra.
An infinite dimensional algebra denoted $\bar{\cal A}_q$ that is isomorphic to a central extension of $U_q^+$ - the positive part of $U_q(\widehat{sl_2})$ - has been recently proposed by Paul Terwilliger. It provides an `alternating' Poincaré-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of $\bar{\cal A}_q$ in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for $\bar{\cal A}_q$ are constructed; (b) explicit isomorphisms from $\bar…
We construct a wide class of finite W-algebras as truncations of Yangians. These truncations correspond to algebra homomorphisms and allow to construct the W-algebras as exchange algebras, the R-matrix being the Yangian's one. As an application, we classify all irreducible finite dimensionnal representations of these W-algebras and determine their center.
Starting from a finite-dimensional representation of the Yangian $Y(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$ in Drinfeld's original presentation, we construct a Hopf algebra $X_\mathcal{I}(\mathfrak{g})$, called the extended Yangian, whose defining relations are encoded in a ternary matrix relation built from a specific $R$-matrix $R(u)$. We prove that there is a surjective Hopf algebra morphism $X_\mathcal{I}(\mathfrak{g})\twoheadrightarrow Y(\mathfrak{g})$ whose kernel is generated as an ideal by the coefficients of a central matrix $\mathcal{Z}(u)$. When the underlying represe…
A Drinfeld type presentation of twisted Yangians of quasi-split type
本文提出了一类扭曲杨代数的Drinfeld型呈现，并展示了其与反射代数的同构关系。
Affine $\imath$quantum groups and twisted Yangians in Drinfeld presentations
本文通过仿射 ı 量子群 Drinfeld 呈现的退化构造了分裂型扭 Yangian，证明其具有 PBW 型基且为扭流代数的形变，并猜想其与 RTT 呈现中的对应代数同构。
Following the approach of Ding and Frenkel [Comm. Math. Phys. 156 (1993), 277-300] for type $A$, we showed in our previous work [J. Math. Phys. 61 (2020), 031701, 41 pages] that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type $C$ were given therein, while the present paper deals with types $B$ and $D$. The arguments for all classical types are quite similar so we mostly concentrate on necessary additional details specific to the underlying orthogonal Lie…
We give a presentation in terms of generators and relations of Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. As a consequence we get an analog presentation of Nichols algebras of diagonal type.
We give integral presentations of quantum lattice Heisenberg algebras by viewing them as Heisenberg doubles. Our presentations generalize those appearing previously in the literature.
In this paper the surjective homomorphism from the Drinfeld realization to the Drinfeld and Jimbo presentation of affine quantum algebras is proved to be injective. A consequence of the arguments used in the paper is the triangular decomposition of the Drinfeld realization of affine quantum algebras also in the twisted case. A presentation of the affine Kac-Moody algebras in terms of the "Drinfeld generators" is also provided.
Presentations of skein algebras of genus 2
该研究给出亏格2的闭或单洞定向曲面的考夫曼括号纽结代数的表示，为亏格大于1的相关问题提供了首个此类结果。
In this note presentations are given for the nilHecke algebras implicit in the work of Bressler and Evens on Schubert calculus for generalized cohomology theories. Such algebras do not usually satisfy the braid relation. Here the obstruction to the braid relation being satisfied is given explicitly and a connection with the deformation theory of algebras is made.
The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. We investigate the alternating central extension $\mathcal O_q$ of $O_q$. The algebra $\mathcal O_q$ was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal O_q$ that involves a subset of the o…
The equitable presentation of $U_q(\mathfrak{sl}_2)$ was introduced in 2006 by Ito, Terwilliger, and Weng. This presentation involves some generators $x, y, y^{-1}, z$. It is known that $\{x^r y^s z^t : r, t \in \mathbb{N}, s \in \mathbb{Z}\}$ is a basis for the $\mathbb{K}$-vector space $U_q(\mathfrak{sl}_2)$. In 2013, Bockting-Conrad and Terwilliger introduced a subalgebra $\mathcal{A}$ of $U_q(\mathfrak{sl}_2)$ spanned by the elements $\{x^r y^s z^t : r, s, t \in \mathbb{N}, r+s+t \ {\rm even}\}$. We give a presentation of $\mathcal{A}$ by generators and relations. We also classify up to is…
Finite presentations for stated skein algebras and lattice gauge field theory
该研究给出stated skein代数的有限表示，证明其为Koszul代数且与格点规范场论的量子模代数同构，推广了前人相关研究结果。
We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin. 26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive formula and the original definition of Jucys-Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys-Murphy elements given in this paper are used to construct the seminormal representations for the partition alge…
R-Matrix presentation of quantum affine algebra in type $A_{2n-1}^{(2)}$
本文首次为具有非平凡中心元素的扭量子仿射代数建立 RTT 呈现，并利用 L-算子的高斯分解证明其与 Drinfeld 新实现同构。
We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based in Kharchenko's theory of PBW basis of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for such PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.
Motivated by work of R.M. Green, we obtain a presentation of Schur algebras (both the classical and quantized versions) in terms of generators and relations. The presentation is compatible with the usual presentation of the (quantized or classical) enveloping algebra of gl(n). As a result, we obtain a new ``integral'' basis for Schur algebras which is a subset of Kostant's basis of the integral form of the enveloping algebra (or its q-analogue). Projection onto an appropriate component gives a new "integral" basis and a presentation for the Hecke algebra, compatible with the basis and presenta…
Quiver presentations and Schur--Weyl duality for Khovanov arc algebras
本文通过给出Khovanov弧代数的Ext-箭图与关系表示，证明了Kleshchev–Martin猜想在该框架下的精确类比，揭示了其与Schur–Weyl对偶的联系。
Let $\mathfrak{g}$ be the derived subalgebra of a Kac-Moody Lie algebra of finite type or affine type, $μ$ a diagram automorphism of $\mathfrak{g}$ and $L(\mathfrak{g},μ)$ the loop algebra of $\mathfrak{g}$ associated to $μ$. In this paper, by using the vertex algebra technique, we provide a general construction of current type presentations for the universal central extension $\widehat{\mathfrak{g}}[μ]$ of $L(\mathfrak{g},μ)$. The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras ([Dr]) and the Moody-Rao-Yokonuma presen…
We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmuller space and generalizes the construction of the Kauffman bracket skein algebra. In this paper, we also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory.
Presentations for small reflection equation algebras of type A
本研究通过生成元和关系，给出A型GLₙ和SLₙ反射方程小代数的表现形式，将其表为无限维对应代数的商，且该形式对代数的适当整形式有效。
The quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$ was defined as a generalization of the Drinfeld's new realization of the quantum affine algebra to the loop algebra of any Kac-Moody algebra $\mathfrak{g}$. It has been shown by Schiffmann that the Hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$, for some $\mathfrak{g}$ with a star-shaped Dynkin diagram. In this paper we study Drinfeld's presentation of $U_{v}(\mathcal{L}\mathfrak{g})$ in the double Hall algebra setting, based …
Feigin-Stoyanovsky's type subspace $W(Λ)$ of a standard $\tilde{\mathfrak g}$-module $L(Λ)$ is a $\tilde{\mathfrak g}_1$-submodule of $L(Λ)$ generated by the highest-weight vector $v_Λ$, where $\tilde{\mathfrak g}_1$ is a certain commutative subalgebra of $\tilde{\mathfrak g}$. Based on the description of basis of $W(Λ)$ for $\tilde{\mathfrak g}$ of type $C_\ell^{(1)}$, we give a presentation of this subspace in terms of generators and relations $$W(Λ)\simeq U(\tilde{\mathfrak g}_1^-)/J.$$
Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type $X\not=A,C,BC$. We follow a unified approach for the types under consideration.
A quiver presentation of the type D arc algebras and intrinsic formality of $\mathbb{D}_4$
本文构造了 D 型 Khovanov 弧代数的箭图表示，并利用其 Koszul 对偶计算 Hochschild 上同调，证明 $\mathbb{D}_4$ 内在形式，无非平凡 $A_\infty$ 形变。
An explicit isomorphism between the $R$-matrix and Drinfeld presentations of the quantum affine algebra in type $A$ was given by Ding and I. Frenkel (1993). We show that this result can be extended to types $B$, $C$ and $D$ and give a detailed construction for type $C$ in this paper. In all classical types the Gauss decomposition of the generator matrix in the $R$-matrix presentation yields the Drinfeld generators. To prove that the resulting map is an isomorphism we follow the work of E. Frenkel and Mukhin (2002) in type $A$ and employ the universal $R$-matrix to construct the inverse map. A …
A Disproof of Santharoubane's Conjecture on Presentations of Generic Skein Algebras
本文针对亏格至少3且至多1个边界分支的紧连通定向曲面的一般Kauffman括号skein代数，通过证明其不存在单位特征而Santharoubane构造的对应代数存在增广特征，给出了Santharoubane猜想的反例。
We derive some new presentations for the Yangian associated to the Lie algebra gl_n(C) that are adapted to parabolic subalgebras. At one extreme, the presentation is just the usual RTT presentation, whilst at the other extreme it is a variation on Drinfeld's presentation. All these presentations play an important role in our subsequent article "Shifted Yangians and finite W-algebras".
We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gl_n. Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto.
In this paper, we define a number of closely related isomorphisms. On one side of these isomorphisms sit a number of of algebras generalizing the Hecke and affine Hecke algebras, which we call the "Hecke family"; on the other, we find generalizations of KLR algebras in finite and affine type A, the "KLR family." We show that these algebras have compatible isomorphisms generalizing those between Hecke and KLR algebras given by Brundan and Kleshchev. This allows us to organize a long list of algebras and categories into a single system, including (affine/cyclotomic) Hecke algebras, (affine/cyclo…
We provide an alternative approach to the Faddeev-Reshetikhin-Takhtajan presentation of the quantum group U_q(g), with L-operators as generators and relations ruled by an R-matrix. We look at U_q(g) as being generated by the quantum Borel subalgebras U_q(b_+) and U_q(b_-), and we use the standard presentation of the latters as quantum function algebras. When g = gl(n) these Borel quantum function algebras are generated by the entries of a triangular q-matrix, thus eventually U_q(gl(n)) is generated by the entries of an upper triangular and a lower triangular q-matrix, which share the same diag…
Presentations of Kauffman bracket skein algebras of planar surfaces
本文在含特定可逆元的交换环上，通过显式生成元与关系展示了平面曲面的Kauffman括号skein代数，该展示独立于环且可视为幺模矩阵迹代数的量子化。
We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We also locate the ``integral'' form of the quantum Schur algebra within the presented algebra and show it has a basis which is closely related to Lusztig's basis of the integral form of the quantized enveloping algebra.
We prove the equivalence of two presentations of the Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ and we also show the equivalence with a third presentation when $\mathfrak{g}$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.
We develop a method to give presentations of quantized function algebras of complex reductive groups. In particular, we give presentations of quantized function algebras of automorphism groups of finite dimensional simple complex Lie algebras.
We define the Drinfeld generators for $Y_3^+$, the twisted Yangian associated to the Lie algebra $\mathfrak{so}_3(\mathbb{C})$. This allows us to define shifted twisted Yangians, which are certain subalgebras of $Y_3^+$. We show that there are families of homomorphisms from the shifted twisted Yangians in $Y_3^+$ to the universal enveloping algebras of various orthogonal and symplectic Lie algebras, and we conjecture that the images of these homomorphisms are isomorphic to various finite $W$-algebras.
We obtain a presentation of Schur algebras (and q-Schur algebras) by generators and relations which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra gl(n) of n x n matrices. We also find several new bases of Schur algebras and their corresponding integral forms.
We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A case. The presentation is compatible with Lusztig's modified form of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from the modified form of the quantized enveloping algebra, that this gives them a cellular structure, and thus they are quasihereditary over a field.
Skein Theory for Affine A Subfactor Planar Algebras
本研究推进Kuperberg纲领，给出指标为4的仿射A子因子平面代数的生成元与关系表示，以纯图解方式新证这类平面代数的存在数量及对应范畴的幺半等价性，提供全新图解方法。
The Nichols algebras of diagonal type with finite root system are either of standard, super or (yet) unidentified type. A concrete description of the defining relations of all those Nichols algebras was given in \cite{A-exp presentation}. In the present paper we use this result to give an explicit presentation of all Nichols algebras of unidentified type.
We give an algebraic construction of the topological graph-tree configuration pairing of Sinha and Walter beginning with the classical presentation of Lie coalgebras via coefficients of words in the associative Lie polynomial. Our work moves from associative algebras to preLie algebras to graph complexes, justifying the use of graph generators for Lie coalgebras by iteratively expanding the set of generators until the set of relations collapses to two simple local expressions. Our focus is on new computational methods allowed by this framework and the efficiency of the graph presentation in pr…
A compact presentation for the alternating central extension of the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
本文为量子群正部分的交错中心扩张 $\mathcal U^+_q$ 构造了一个仅用少量生成元和简洁关系的紧凑表示，简化了原有表示。
We give a presentation of the Kauffman (BMW) skein algebra of the torus, which is the "type BCD" analogue of the Homflypt skein algebra of torus which was computed by the first and third authors. In the appendix we show this presentation is compatible with the Frohman-Gelca description of the Kauffman bracket (Temperley-Lieb) skein algebra of the torus [FG00].
MSc thesis of the author offering an introduction to the operator algebraic approach to noncommutative geometry, with a treatment of some more advanced elements such as the noncommutative geometry of quantum groups, fuzzy physics, and compact quantum metric spaces.
Shifted twisted Yangians and finite $W$-algebras of classical type
本文通过引入抛物型表示和截断平移扭曲 Yangian，建立其 PBW 基，并证明其同构于经典型有限 $W$-代数，从而为 B、C 型偶幂零元及 D 型双 Jordan 块幂零元提供表示。
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