Quantum Lie algebras via modified Reflection Equation Algebra

GL型的分级扭反射矩阵

Graded twisted reflection matrices of GL-type

本文分类了GL型R矩阵的Z_2-分级扭反射方程的常数解，基于四指标局域性原理，确定了奇异与可逆解，并关联了分级Satake图与电李代数。

Extended Affine Lie Superalgebras

We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras.

Symplectic reflection algebras and affine Lie algebras

The goal of this paper is to present some results and (more importantly) state a number of conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory of affine Lie algebras (namely, decompositions of the restriction of the basic representation to finite dimensional and affine subalgebras). These conjectures arose from the insight due to R. Bezrukavnikov and A. Okounkov on the link between quantum connections for Hilbert schemes of resolutions of Kleinian singularities and representatio…

Reflection matrices from Hadamard-type Temperley-Lieb R-matrices

We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-matrices obtained from the newly defined general rank- n Hadamard type representations of the Temperley-Lieb algebra $TL_N(\sqrt n)$. They are characterized by a universal set of algebraic equations in a specific canonical basis uniquely defined from the "Master matrix" associated to the chosen realization of Temperley-Lieb algebra

Braided modules and reflection equations

We introduce a representation theory of q-Lie algebras defined earlier in \cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in particular, those based on the so-called reflection equations. We also investigate the truncated tensor product of braided modules.

q-Casimir and q-cut-and-join operators related to Reflection Equation Algebras

In this paper we are dealing with the Reflection Equation algebra ${\cal M}(R)$, associated with a $GL_N$ type Hecke symmetry $R$. In this algebra we define the $q$-analogs of the partial derivatives $\partial_j^i$ in generators $m_i^j$ of ${\cal M}(R)$. The product $\hat L = MD$ of two matrices $M=\|m_i^j\|$ and $D=\|\partial_i^j\|$ turns out to be a generating matrix of a modified Reflection Equation algebra $\hat{\cal L}(R)$ which is similar to the universal enveloping algebra $U(gl_N)$ in many aspects. Central elements of the modified Reflection Equation algebra give rise to $q$-Casimir op…

Extended Affine Root Supersystems

The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular, suggests a systematic study of the root system apart from its connection with the Lie algebra. Although there have been a lot of researches in this regard on Lie algebra level, such an approach has not been considered on Lie superalgebra level. In this work, we introduce and study extended affine root supersystems which are a generalization of both affine refl…

Extended affinization of Invariant Affine Reflection Algebras

Invariant affine reflection algebras are the last and the most general known extension of affine Kac-Moody Lie algebras, introduced in recent years. We develop a method known as "affinization" to the class of invariant affine reflection algebras, and show that starting with an algebra from this class together with a certain finite order automorphism, and applying the so called "loop construction", we obtain again an invariant affine reflection algebra. This can be considered as an important step towards the realization of invariant affine reflection algebras.

Topics in isotopic pairs and their representations. III. Bunches of Lie algebras and modified classical Yang-Baxter equation

This third part of the series is a brief comment to certain aspects of the theory of classical $r$-matrix and bihamiltonian formalism, which motivations lie in constructions of the previous two parts.

Reflection Equation, Twist, and Equivariant Quantization

We prove that the reflection equation (RE) algebra $\La_R$ associated with a finite dimensional representation of a quasitriangular Hopf algebra $\Ha$ is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that $\La_R$ is a module algebra over the twisted tensor square \twist{$\Ha$}{$\Ha$} and the double $\D(\Ha)$. We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.

李超代数 $\mathfrak{gl}_{m|n}$ 的双 Yangian 与反射代数

Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{m|n}$

本文研究李超代数 $\mathfrak{gl}_{m|n}$ 的双 Yangian，证明其 Poincaré-Birkhoff-Witt 定理、构造量子缩并系数形式的中心元，并据此给出 level 0 反射代数的生成元—关系表述。

Quantization of $r-Z$-quasi-Poisson manifolds and related modified classical dynamical $r$-matrices

Le $X$ be a $C^\infty$-manifold and $\g$ be a finite dimensional Lie algebra acting freely on $X$. Let $r \in \ve^2(\g)$ be such that $Z=[r,r] \in \ve^3(\g)^\g$. In this paper we prove that every quasi-Poisson $(\g,Z)$-manifold can be quantized. This is a generalization of the existence of a twist quantization of coboundary Lie bialgebras (\cite{EH}) in the case $X=G$ (where $G$ is the simply connected Lie group corresponding to $\g$). We deduce our result from a generalized formality theorem. In the case Z=0, we get a new proof of the existence of (equivariant) formality theorem and so (equiv…

反射方程代数上的广义 Harish-Chandra 态射

Generalized Harish-Chandra morphism on Reflection Equation algebras

本文将广义 Harish-Chandra 态射从包络代数 U(gl(N)) 推广到反射方程代数，通过引入与斜可逆 Hecke 对称对应的生成矩阵特征值，并利用该态射构造权系统的量子类比。

Quantum Symmetric Pairs and the Reflection Equation

It is shown that central elements in G. Letzter's quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter's approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the unde…

四阶BV结构在超范畴中及修正项链李双代数

Quartic BV structures in supercategories and modified necklace Lie bialgebras

本文提出修正项链李双代数，通过插入箭头对关联箭图表示簇上的四阶泊松/BV结构，并在对称幺半Π-范畴中统一处理，揭示修正与经典运算的对偶关系。

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let $G$ be a simple complex Lie group with Lie algebra $\mf g$ and let $\af$ be the affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of $\N$-graded vertex operator algebras associated to $\mf g$. They are $\af \oplus \af$-modules of dual levels $k, \bar k \notin \Q$ in the sense that $k + \bar k = -2 h^\vee$ where $h^\vee$ is the dual Coxeter number of $\mf g$. Its conformal weight 0 component is the algebra of regular functions on $G$. This family of vertex operator algebras were previously studied by Arkhipov-Gaitsgory and Gorbounov…

Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras

This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron $\Cal M$ of $W$ and an acceptable (corresponding to twisting coefficients) set $P({\Cal M})\subset M$ of vectors orthogonal to faces of $\Cal M$ (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra ${\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$ which is graded by …

Reflection algebra, Yangian symmetry and bound-states in AdS/CFT

We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the boundary Yangian symmetry of an AdS/CFT superstring which ends on a boundary with non-trivial degrees of freedom and which preserves the full bulk Lie symmetry algebra. We also consider the spectrum of bulk and boundary states and some automorphisms of the underlying algebras.

Diagonal reduction algebra and reflection equation

We describe the diagonal reduction algebra D(gl(n)) of the Lie algebra gl(n) in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl(n)).

Tetrahedron Reflection Equation

Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in 2+1 dimensions. This equation can be obtained as an additional to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra.

Reflection algebras for sl(2) and gl(1|1)

We present a generalization the G. Letzter's theory of quantum symmetric pairs of semisimple Lie algebras for the case of quantum affine algebras. We then study solutions of the reflection equation for the quantum affine algebras sl(2) and gl(1|1) and their Yangian limit for singlet (diagonal) and vector (non-diagonal) boundary conditions. We construct the corresponding quantum affine coideal subalgebras that are based on the quantum symmetric pairs, and the (generalized) twisted Yangians.

Center and Lie algebra of outer derivations for algebras of differential operators associated to hyperplane arrangements

We compute the center and the Lie algebra of outer derivations of a familiy of algebras of differential operators associated to hyperplane arrangements of the affine space A 3. The results are completed for 4-braid arrangements and for reflection arrangements associated to the wreath product of a cyclic group with the symmetric group S 3. To achieve this we use tools from homological algebra and Lie-Rinehart algebras of differential operators.

Quasi-reflection algebras and cyclotomic associators

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism B_n^1 -> (ZZ/N ZZ)^n rtimes S_n, where B_n^1 is a braid group of type B. The formality isomorphism depends algebraically on a series Psi_KZ, the ``KZ pseudotwist''. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N,k), mapping to Drinfeld's group GT(k), and whose Lie algebra is isomorphic to its associated graded grtm(N,k). We prove that Ihara's subgroup GTK of the Grothendieck-Teichmüller group, defi…

A parameterization of the canonical bases of affine modified quantized enveloping algebras

For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\textbf{g}$ we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of $\dot{\textbf{U}}(\textbf{g})$, where the root category is the $T^2$-orbit category of the derived category of Dynkin or tame quiver. Our method bases on one theorem of Lin, Xiao and Zhang in [9], which gave th…

Modified quantum dimensions and re-normalized link invariants

In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and gen…

Odd reflections in the Yangian associated with ${\frak gl}(m|n)$

The odd reflections are an effective tool in the Lie superalgebra representation theory, as they relate non-conjugate Borel subalgebras. We introduce analogues of the odd reflections for the Yangian ${\rm Y}({\frak gl}_{m|n})$ and use them to produce a transition rule for the parameters of the highest weight modules corresponding to a change of the parity sequence. This leads to a description of the finite-dimensional irreducible representations of the Yangian associated with an arbitrary parity sequence.

Quantum supergroups IV. The modified form

We construct a canonical basis for a class of tensor product modules of a quantum covering group associated to a Kac-Moody Lie superalgebra of anisotropic type, and use these bases to construct a canonical basis for the modified form of a quantum covering group.

Weyl approach to representation theory of reflection equation algebra

The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the R-matrix is arbitrary (not necessary originated from quantum groups). We suggest a universal method of constructing finite dimensional irreducible non-commutative representations in the framework of the Weyl approach well known in the representation theory of classical Lie groups and algebras. With this method a series of irreducible modules is constructed which are parametrized by Young diagrams. The spectrum of centra…

Solutions of the reflection equation for face and vertex models associated with $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$

We present new diagonal solutions of the reflection equation for elliptic solutions of the star-triangle relation. The models considered are related to the affine Lie algebras $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$. We recover all known diagonal solutions associated with these algebras and find how these solutions are related in the elliptic regime. Furthermore, new solutions of the reflection equation follow for the associated vertex models in the trigonometric limit.

准双曲李代数的变形映射

Deformation maps of quasi-twilled Lie algebras

本文通过引入准双曲李代数，统一研究李代数各类算子的上同调与变形理论，提出两种变形映射类型，涵盖修正r矩阵、交叉同态、导子、同态、相对罗塔-巴克斯算子等，并给出其控制代数与上同调，拓展了现有结果并获得新结论。

Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps

We established a method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev's electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig's parametrizations of the totally positive part of $SL_3$ and $SO_5$, wh…

A lecture on Kac--Moody Lie algebras of the arithmetic type

We name an indecomposable symmetrizable generalized Cartan matrix $A$ and the corresponding Kac--Moody Lie algebra ${\goth g} ^\prime (A)$ {\it of the arithmetic type} if for any $β\in Q$ with $(β| β)<0$ there exist $n(β)\in {\Bbb N}$ and an imaginary root $α\in Δ^{im}$ such that $n(β)β\equiv α\mod Ker\ (.|.)$ on $Q$. Here $Q$ is the root lattice. This generalizes "symmetrizable hyperbolic" type of Kac and Moody. We show that generalized Cartan matrices of the arithmetic type are divided in $4$ types: (a) finite, (b) affine, (c) rank two, and (d) arithmetic hyperbolic type. The last type is ve…

A solution of the 3D reflection equation from quantized algebra of functions of type B

Let A_q(g) be the quantized algebra of functions associated with simple Lie algebra g defined by generators obeying the so called RTT relations. We describe the embedding $A_q(B_2) \hookrightarrow A_q(C_2)$ explicitly. As an application, a new solution of the Isaev-Kulish 3D reflection equation is constructed by combining the embedding with the previous solution for A_q(C_2) by the authors.