The R-matrix for (twisted) affine quantum algebras

Classification of bases of twisted affine root supersystems

Following the definition of a root basis of an affine root system, we define a base of the root system of an affine Lie superalgebra to be a linearly independent subset $B$ of its root system such that each root can be written as a linear combination of elements of $B$ with integral coefficients such that all coefficients are nonnegative or all coefficients are nonpositive. Characterization and classification of bases of root systems of affine Lie algebras are known in the literature; in fact, up to $\pm 1$-multiple, each base of an affine root system is conjugate with the standard base under …

The twisted quantum affine algebra $U_q(A^{(2)_2)$ and correlation functions of the Izergin-Korepin model

We derive the exchange relations of the vertex operators of $U_q(A_2^{(2)})$ and show that these vertex operators give the bosonization of the Izergin-Korepin model. We give an integral expression of the correlation functions of the Izergin-Korepin model and derive the difference equations which they satisfy.

Two-parameter twisted quantum affine algebras

We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.

Twisted Quantum Affine Algebras

We classify the irreducible finite-dimensional representations of the twisted quantum affine algebras.

Construction of perfect crystals conjecturally corresponding to Kirillov-Reshetikhin modules over twisted quantum affine algebras

Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.

Twisted representations of vertex operator algebras associated to affine Lie algebras

In this paper, we prove the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalization of Zhu's algebra for these vertex operator algebras, constructed in \cite{HY}. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.

扭曲 q-Yangian 与 Sklyanin 行列式

Twisted q-Yangians and Sklyanin determinants

本文通过将量子仿射代数表示为两个 q-Yangian 副本的乘积，研究其及扭曲版本的不变量理论，并引入扭曲 Sklyanin 行列式、建立相关恒等式。

Lower-bounded and grading-restricted twisted modules for affine vertex (operator) algebras

We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules.

$Q\widetilde{Q}$-systems for twisted quantum affine algebras

We establish the $Q \widetilde{Q}$-systems for the twisted quantum affine algebras that were conjectured in arXiv:1606.05301. We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their prefundamental representations. We also propose a general conjecture on the relations between twisted and non-twisted types. We prove this conjecture for some particular classes of representations, including prefundamental representations.

扭曲的量子 affine 代数与量子顶点代数的 equivariant ϕ-协调模

Twisted quantum affine algebras and equivariant $ϕ$-coordinated modules for quantum vertex algebras

本文研究量子 affine 代数与量子顶点代数的自然联系，提出 ϕ-协调准模的 ϳ-adic 构造，并构建了 ϳ-adic 量子顶点代数 V_L[[ℏ]]^η 及其与扭曲量子 affine 代数的关联。

Twisted quantum affinizations and quantization of extended affine Lie algebras

In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $μ$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $μ$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. When $μ=\mathrm{id}$ and $\mathfrak g$ in affine type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ is the quantum toroidal al…

Parafermionic bases of standard modules for twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_{l+1}^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$

Using the bases of principal subspaces for twisted affine Lie algebras except $A_{2l}^{(2)}$ by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight $kΛ_0$ and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in arXiv:math/0102113.

Torus structure on graphs and twisted partition functions for minimal and affine models

Using the Ocneanu quantum geometry of ADE diagrams (and of other diagrams belonging to higher Coxeter-Dynkin systems), we discuss the classification of twisted partition functions for affine and minimal models in conformal field theory and study several examples associated with the WZW, Virasoro and W_{3} cases.

Combinatorial bases of principal subspaces of modules for twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$

We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $kΛ_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.

将量子 affine 顶点代数的变形 $ϕ$-协调模与正交扭曲 $h$-Yangians 关联

Associating deformed $ϕ$-coordinated modules for the quantum affine vertex algebra with orthogonal twisted $h$-Yangians

本文通过引入变形 $ϕ$-协调模的概念，将量子 affine 顶点代数与正交扭曲 $h$-Yangians 关联，并展示其应用。

Weyl modules for the twisted loop algebras

The notion of a Weyl module, previously defined for the untwisted affine algebras, is extended here to the twisted affine algebras. We describe an identification of the Weyl modules for the twisted affine algebras with suitably chosen Weyl modules for the untwisted affine algebras. This identification allows us to use known results in the untwisted case to compute the dimensions and characters of the Weyl modules for the twisted algebras.

Vertex operators for twisted quantum affine algebras

We construct explicitly the $q$-vertex operators (intertwining operators) for the level one modules $V(Λ_i)$ of the classical quantum affine algebras of twisted types using interacting bosons, where $i=0, 1$ for $A_{2n-1}^{(2)}$, $i=0$ for $D_4^{(3)}$, $i=0, n$ for $D_{n+1}^{(2)}$, and $i=n$ for $A_{2n}^{(2)}$. A perfect crystal graph for $D_4^{(3)}$ is constructed as a by-product.

Drinfeld realization of quantum twisted affine algebras via braid group

The Drinfled realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig's twisted braid group action.

扭量子仿射代数表示的交缠算子

Intertwiners of representations of twisted quantum affine algebras

本文利用q-特征标方法，给出了所有类型扭量子仿射代数第一基本表示R-矩阵的显式计算公式。

仿射旗上的扭曲惠特克范畴与混合量子群的表示范畴

Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

该研究证明仿射旗簇上的扭曲惠特克范畴与混合量子群表示范畴等价，且一般情形下量子范畴O和仿射旗上的扭曲惠特克D-模范畴等价，丰富了量子群与几何表示论的联系。

Vertex-algebraic structure of principal subspaces of the basic modules for twisted Affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}, D_n^{(2)}, E_6^{(2)}$

We obtain a presentation of principal subspaces of basic modules for the twisted affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}$, $D_n^{(2)}$ and $E_6^{(2)}$. Using this presentation, we construct exact sequences among these principal subspaces, and use these exact sequences to obtain recursions satisfied by graded dimensions of the principal subspaces. Solving these recursions, we obtain the graded dimensions of the principal subspaces.

Explicit description of twisted Wakimoto realizations of affine Lie algebras

In a vertex algebraic framework, we present an explicit description of the twisted Wakimoto realizations of the affine Lie algebras in correspondence with an arbitrary finite order automorphism and a compatible integral gradation of a complex simple Lie algebra. This yields generalized free field realizations of the twisted and untwisted affine Lie algebras in any gradation. The free field form of the twisted Sugawara formula and examples are also exhibited.

Vertex algebraic construction of modules for twisted affine Lie algebras of type $A_{2l}^{(2)}$

Let $\tilde{\mathfrak{g}}$ be the affine Lie algebra of type $A_{2l}^{(2)}$. The integrable highest weight $\tilde{\mathfrak{g}}$-module $L(kΛ_0)$ called the standard $\tilde{\mathfrak{g}}$-module is realized by a tensor product of the twisted module $V_L^T$ for the lattice vertex operator algebra $V_L$. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in arXiv:math/0102113.

有限长Lakshmibai-Seshadri路径与扭曲量子齐性代数的零级极值权重模的关联

Semi-infinite Lakshmibai--Seshadri paths and level-zero extremal weight modules over twisted quantum affine algebras

本文通过引入无限长Lakshmibai-Seshadri路径，建立了扭曲量子齐性代数零级极值权重模块与晶体基之间的同构关系。

Representations of Twisted Toroidal Lie Algebras from Twisted Modules over Vertex Algebras

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.

类型 $A_{2n-1}^{(2)}$ 量子仿射代数的 R-矩阵呈现

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。

量子仿射顶点代数的扭曲张量积与余积

Twisted tensor products of quantum affine vertex algebras and coproducts

本文针对可对称化 Kac-Moody 李代数对应的量子仿射顶点代数，构造并证明了连接不同层级扭曲张量积的 $\hbar$-进量子顶点代数余积同态及其余结合性。

Twisted semigroup algebras

We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field k. If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then Spec k[S] is an affine toric variety over k, and we refer to the twists of k[S] as quantum affine toric varieties. We show that every quantum affine toric varieties has a "dense quantum torus", in the sense that it has a localization isomorphic to a quantum torus. We study quantum affine toric varieties and show that many geometric regularity properties survive the deformation process.

Twisted quantum affine algebras and solutions to the Yang-Baxter equation

We construct spectral parameter dependent R-matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral parameter dependent quantum Yang-Baxter equation.

GKLO表示：类型$\mathsf{AI}$的扭曲Yangian及仿射Grassmannian对称商量子化

GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian

本文构造了类型AI扭曲Yangian的GKLO表示，定义其截断，并证明截断的扭曲Yangian量子化仿射Grassmannian横截切片商的概形。

平移扭曲量子仿射代数的表示

Representations of shifted twisted quantum affine algebras

本文引入并研究平移扭曲量子仿射代数，通过平移Cartan-Drinfeld流给出其三角分解和分类，证明有理性和融合积，并分类有限维单模。

Twisted derivations of Hopf algebras

In the paper we introduce the notion of twisted derivation of a bialgebra. Twisted derivations appear as infinitesimal symmetries of the category of representations. More precisely they are infinitesimal versions of twisted automorphisms of bialgebras. Twisted derivations naturally form a Lie algebra (the tangent algebra of the group of twisted automorphisms). Moreover this Lie algebra fits into a crossed module (tangent to the crossed module of twisted automorphisms). Here we calculate this crossed module for universal enveloping algebras and for the Sweedler's Hopf algebra.

Twisted modules for quantum vertex algebras

We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras.

Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras

In this paper, we introduce twisted and folded AR-quivers of type $A_{2n+1}$, $D_{n+1}$, $E_6$ and $D_4$ associated to (triply) twisted Coxeter elements. Using the quivers of type $A_{2n+1}$ and $D_{n+1}$, we describe the denominator formulas and Dorey's rule for quantum affine algebras $U'_q(B^{(1)}_{n+1})$ and $U'_q(C^{(1)}_{n})$, which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for $U'_q(B^{(1)}_{n+1})$ (resp. $U'_q(C^{(1)}_{n})$) using certain statistics on any folded AR-quiver of type $A_{2n+1}$ (resp…

扭曲$q$-Yangian与仿射$\imath$量子群之间的同构：AI型

Isomorphism between twisted $q$-Yangians and affine $\imath$quantum groups: type AI

本研究利用高斯分解，建立了AI型对称对对应的扭曲$q$-Yangian与仿射$\imath$量子群的显式同构，并由此得到该类仿射$\imath$量子群的PBW型基。

Quadratic Relations of the Deformed $W$-Algebra for the Twisted Affine Lie Algebra of Type $A_{2N}^{(2)}$

We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.

Automorphism Groups and Twisted Modules for Lattice Vertex Operator Algebras

We give a complete description of the full automorphism group of a lattice vertex operator algebra, determine the twisted Zhu's algebra for the automorphism lifted from the -1 isometry of the lattice and classify the corresponding irreducible twisted modules through the twisted Zhu algebra.

Twisting of affine algebraic groups, II

We use \cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $\dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U(\g)$ as algebras, where $\g={\rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function …

Twisted modules for Toroidal vertex algebras

This is a paper in a series systematically to study toroidal vertex algebras. Previously, a theory of toroidal vertex algebras and modules was developed and toroidal vertex algebras were explicitly associated to toroidal Lie algebras. In this paper, we study twisted modules for toroidal vertex algebras. More specifically, we introduce a notion of twisted module for a general toroidal vertex algebra with a finite order automorphism and we give a general construction of toroidal vertex algebras and twisted modules. We then use this construction to establish a natural association of toroidal vert…

Three realizations of quantum affine algebra $U_q(A_2^{(2)})$

In this article we establish explicit isomorphisms between three realizations of quantum twisted affine algebra $U_q(A_2^{(2)})$: the Drinfeld ("current") realization, the Chevalley realization and the so-called $RLL$ realization, investigated by Faddeev, Reshetikhin and Takhtajan.

Principal subspaces of basic modules for twisted affine Lie algebras, $q$-series multisums, and Nandi's identities

We provide an observation relating several known and conjectured $q$-series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$-series identities. The first family provides quadruple sum representations for Nandi's identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of level 4 integrable, highest-weight modules of $\mathrm{D}_4^{(3)}$.

Representations of twisted q-Yangians

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…

A Twisted Quantum Toroidal Algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A_1 is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.

Twisted Heisenberg-Virasoro vertex operator algebra

In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor …

Drinfeld 呈现下的仿射 ı 量子群与扭 Yangian

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。

New twisted quantum current algebras

We introduce a twisted quantum affine algebra associated to each simply laced finite dimensional simple Lie algebra. This new algebra is a Hopf algebra with a Drinfeld-type comultiplication. We obtain this algebra by considering its vertex representation. The vertex representation quantizes the twisted vertex operators of Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We also introduce a twisted quantum loop algebra for the Kac-Moody case and give its level one representation.

L-R-smash products and L-R-twisted tensor products of algebras

We introduce a common generalization of the L-R-smash product and twisted tensor product of algebras, under the name L-R-twisted tensor product of algebras. We investigate some properties of this new construction, for instance we prove a result of the type "invariance under twisting" and we show that under certain circumstances L-R-twisted tensor products of algebras may be iterated.

Twisted tensor products of nonlocal vertex algebras

In this paper we introduce and study a twisted tensor product construction of nonlocal vertex algebras. Among the main results, we establish a universal property and give a characterization of a twisted tensor product. Furthermore, we give a construction of modules for a twisted tensor product. We also show that smash products studied by one of us before can be realized as twisted tensor products.

类型 $A_{2l-1}^{(2)}$ 和 $D_{l+1}^{(2)}$ 的扭仿射李代数标准模的组合基：矩形最高权

Combinatorial bases of standard modules of twisted affine Lie algebras in types $A_{2l-1}^{(2)}$ and $D_{l+1}^{(2)}$: rectangular highest weights

本文利用顶点代数技术为两类扭仿射李代数的矩形最高权标准模构造组合基，并计算特征标公式，进而得到两个新的组合恒等式族。

Twisted ${\mathfrak{sl}}(3,\C)\sptilde$-modules and combinatorial identities

The main result of this paper is a combinatorial description of a basis of standard level 1 module for the twisted affine Lie algebra $A_2^{(2)}.$ This description also gives two new combinatorial identities of Göllnitz (or Rogers--Ramanujan) type. Methods used through the paper are mainly developed by J. Lepowsky, R. L. Wilson, A. Meurman and M. Primc, and the crucial role in constructions plays a vertex operator algebra approach to standard representations of affine Lie algebras.

通过半导出Ringel-Hall代数构造扭量子环代数

Twisted quantum loop algebras via semi-derived Ringel-Hall algebras

利用更一般加权射影直线的半导出Ringel-Hall代数，实现了与赋值星形图相关的扭量子环代数，包括Drinfeld新展示中的扭量子仿射代数。

Twisted Quantum Affine Superalgebra $U_q[gl(m|n)^{(2)}]$ and New $U_q[osp(m|n)]$ Invariant R-matrices

The minimal irreducible representations of $U_q[gl(m|n)]$, i.e. those irreducible representations that are also irreducible under $U_q[osp(m|n)]$ are investigated and shown to be affinizable to give irreducible representations of the twisted quantum affine superalgebra $U_q[gl(m|n)^{(2)}]$. The $U_q[osp(m|n)]$ invariant R-matrices corresponding to the tensor product of any two minimal representations are constructed, thus extending our twisted tensor product graph method to the supersymmetric case. These give new solutions to the spectral-dependent graded Yang-Baxter equation arising from $U_q…

Twisted Fock module of toroidal algebra via DAHA and vertex operators

We construct the twisted Fock module of quantum toroidal $\mathfrak{gl}_1$ algebra with a slope $n'/n$ using vertex operators of quantum affine $\mathfrak{gl}_n$. The proof is based on the $q$-wedge construction of an integrable level-one $U_q(\widehat{\mathfrak{gl}}_n)$-module and the representation theory of double affine Hecke algebra. The results are consistent with Gorsky-Neguţ conjecture (Kononov-Smirnov theorem) on stable envelopes for Hilbert schemes of points in the plane and can be viewed as a manifestation of $(\mathfrak{gl}_1,\mathfrak{gl}_n)$-duality.

The Operator Algebra and Twisted KZ Equations of WZW Orbifolds

We obtain the operator algebra of each twisted sector of all WZW orbifolds, including the general twisted current algebra and the algebra of the twisted currents with the twisted affine primary fields. Surprisingly, the twisted right and left mover current algebras are not a priori copies of each other. Using the operator algebra we also derive world-sheet differential equations for the twisted affine primary fields of all WZW orbifolds. Finally we include ground state properties to obtain the twisted Knizhnik-Zamolodchikov equations of the WZW permutation orbifolds and the inner-automorphic W…