C. Briot, E. Ragoucy
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.
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We give a unified RTT presentation of (super)-Yangians Y(g) for so(n), sp(2n) and osp(m|2n).
RTT presentation of coideal subalgebra of quantized enveloping algebra of type CI
本文利用R矩阵表示构造CI型量子包络代数的余理想子代数，通过A形式方法得到其PBW基，并构造相关Poisson代数及辫群作用。
We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which we explicitly evaluate for the free field representations of the quantum toroidal algebra. This cocycle is responsible for the braiding, which permutes the external legs in the q-deformed conformal block and its 5d/6d gauge theory counterpart, i.e. the non-perturbative Nekrasov functions. Thus, it defines their modular properties and symmetry. We show how to…
A Drinfeld Presentation of the Queer Super-Yangian
本文通过块高斯分解和 RTT 关系，提出 queer 超 Yangian 的 Drinfeld 展开，并获得其中心系列的新表达式。
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".
R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A.~Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e.\ of refined topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2,Z) group of DIM algebra automorphisms. We also construct the T-oper…
Yangian-like algebras, associated with current R-matrices, different from the Yang ones, are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangians are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, quantum determinants, are introduced. …
The Cayley-Hamilton-Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras M(R,F) defined by a pair of compatible solutions of the Yang-Baxter equation. This class includes the RTT-algebras as well as the Reflection equation algebras.
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 prove how the Yangian of $\mathfrak{gl}_N$ in its RTT presentation and Olshanski's twisted Yangians for the orthogonal and symplectic Lie algebras can be obtained by a degeneration process from the corresponding quantum loop algebra and some of its twisted analogues.
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。
Parabolic presentations of Yangian in types $B$ and $C$
本文通过对称合成参数化和块矩阵高斯分解，建立了B、C型扩展Yangian的抛物型表示，并由此得到其中心的新公式。
In this note we are dealing with a particular class of quadratic algebras -- the so-called quantum matrix algebras. The well-known examples are the algebras of quantized functions on classical Lie groups (the RTT algebras). We consider the problem of constructing some projectors on homogenous components of such algebras, which are analogs of the usual symmetrizers. The main objective of this note is to present a method, which hopefully enables one to construct symmetrizers on all homogenous components of the RTT algebras. We illustrate the construction by two low-dimensional examples. A way of…
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.
On the $h$-adic quantum vertex algebras associated with Hecke symmetries
本研究围绕与赫克对称性相关的$h$进量子顶点代数展开，构建RTT型及辫子杨ian的模族，推导对应量子顶点代数与模结构，利用量子行列式获中心元与不变量，并探究相关表示论代数。
The Cayley-Hamilton-Newton theorem - which underlies the Newton identities and the Cayley-Hamilton identity - is reviewed, first, for the classical matrices with commuting entries, second, for two q-matrix algebras, the RTT-algebra and the RLRL-algebra. The Cayley-Hamilton-Newton identities for these q-algebras are related by the factorization map. A class of algebras M(R,F) is presented. The algebras M(R,F) include the RTT-algebra and the RLRL-algebra as particular cases. The algebra M(R,F) is defined by a pair of compatible matrices R and F. The Cayley-Hamilton-Newton theorem for the algebra…
RE-algebras, quasi-determinants and the full Toda system
本文针对n=2,3的RE-代数构造了一族关于反对角线为主且相互交换的拟行列式，其乘积给出量子行列式，从而将全Toda系统积分的构造推广到量子情形，并揭示了RE-代数作为量子齐性空间的意义。
The q-generalizations of the two fundamental statements of matrix algebra -- the Cayley-Hamilton theorem and the Newton relations -- to the cases of quantum matrix algebras of an "RTT-" and of a "Reflection equation" types have been obtained in [2]-[6]. We construct a family of matrix identities which we call Cayley-Hamilton-Newton identities and which underlie the characteristic identity as well as the Newton relations for the RTT- and Reflection equation algebras, in the sence that both the characteristic identity and the Newton relations are direct consequences of the Cayley-Hamilton-Newton…
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…
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 $\mathfrak{osp}_q(1|2)$.
The Yangian of the Lie algebra sl_n is known to have different presentations, in particular the RTT realisation and the Drinfel'd realisation. Using the isomorphism between them, the explicit expressions of the comultiplication, the antipode and the counit in the Drinfel'd realisation of the Yangian Y(sl_n) are given. As examples, the cases of Y(sl_2) and Y(sl_3) are worked out.
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 $X X^{-1} =1$, $X^{-1} X=1$, $\frac{qXY-q^{-1}YX}{q-q^{-1}}=1$, $\frac{qYZ-q^{-1}ZY}{q-q^{-1}}=1$, $\frac{qZX-q^{-1}XZ}{q-q^{-1}}=1$.
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.
By a generalized Yangian we mean a Yangian-like algebra of one of two classes. One of these classes consists of the so-called braided Yangians, introduced in our previous paper. The braided Yangians are in a sense similar to the reflection equation algebra. The generalized Yangians of second class, called the Yangians of RTT type, are defined by the same formulae as the usual Yangians are but with other quantum $R$-matrices. If such an $R$-matrix is the simplest trigonometrical $R$-matrix, the corresponding Yangian of RTT type is the so-called q-Yangian. We claim that each generalized Yangian …
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…
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。
The centralizer construction and Yangian-type algebras
本文通过推广中心化子构造得到一类新的Yangian型代数族，并证明其保留二次-线性定义关系等Yangian性质，同时为自由结合代数上的特殊双重Poisson括号提供量子化。
We give integral presentations of quantum lattice Heisenberg algebras by viewing them as Heisenberg doubles. Our presentations generalize those appearing previously in the literature.
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.
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…
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…
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…
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。
In the framework of the Drinfeld theory of twists in Hopf algebras we construct quantum matrix algebras which generalize the Reflection Equation and the RTT algebras. Finite-dimensional representations of these algebras related to the theory of nonultralocal spin chains are presented. The Cayley-Hamilton-Newton identities are demonstrated. These identities allow to define the quantum spectrum for the quantum matrices. We mention possible applications of the new quantum matrix algebras to constructions of noncommutative analogs of Minkowski space and quantum Poincaré algebras.
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.
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。
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.
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…
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 $U_q(\mathfrak{sl}_2)$
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 …
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。
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 $$W(Λ)\simeq U(\tilde{\mathfrak g}_1^-)/J.$$
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 …
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 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。
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 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.
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 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…
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.
The notion of compatible braidings was introduced by Isaev, Ogievetsky and Pyatov. On the base of this notion they defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogs of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach we introduce certain new algebras called generalized Yangians and define analogs of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutat…
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.
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