We begin a study of the representation theory of quantum continuous $\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $\mathcal E$ are labeled by a continuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcal E$ has many properties familiar from the representation theory of $\mathfrak{gl}_\infty$: vector representations, Fock modules, semi-infinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $\mathcal E$ to spherical double affine Hecke algebras $S\ddot H_N$ for all $N$. A key step in this construction is an identification of a natural bases of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the $K$-theory of Hilbert schemes.
We investigate in detail relationships between the set ${\mathfrak B}^\infty$ of all infinite ``biconvex'' sets in the positive root system $Δ_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and the set ${\mathcal W}^\infty$ of all infinite ``reduced word'' of the Weyl group of ${\mathfrak g}$. The study is applied to the classification of ``convex orders'' on $Δ_+$ (cf. \cite{kI}), which are indispensable to construct ``convex bases'' of Poincaré-Birkhoff-Witt type of the upper triangular subalgebra $U_q^+$ of the quantized universal enveloping algebra $U_q({\mathfrak g})$. We…
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $\mathcal{C}_{\mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q'(\mathfrak{g})$-modules. For a suitable infinite sequence $\widehat{w}_0= \cdots s_{i_{-1}}s_{i_0}s_{i_1} \cdots$ of simple reflections, we introduce subcategories $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ of $\mathcal{C}_{\mathfrak{g}}^0$ for all $a \le b \in \mathbb{Z}\sqcup\{ \pm \infty \}$. Associated with a certain chain $\mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(\mathfrak{C}…
A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\infty)$. For a complete duality datum in the Hernandez-Leclerc category $\mathcal{C}^0_{\mathfrak{g}}$ of a quantum affine algebra $U_q'(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in $\mathcal{C}^0_{\mathfrak{g}}$ has an extended crystal structure isomorphic to the extended crystal $\widehat{B}_{\mathfrak{g}}(\inft…
Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathfrak{g}_{\infty}$ defined over $\mathbb{C}_{\infty}$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that for a class of Lie algebras $\mathfrak{g}_{\infty}$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ …
In this paper, we study the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ associated to the Lie algebra $\mathfrak sl_{\infty}$ of infinite rank. We shall prove that the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ admits both a Hopf algebra structure and a triangular decomposition. In particular, it can be realized as the Drinfeld double of it's certain Hopf subalgebras. We will also study a two-parameter twisted Ringel-Hall algebra $H_{r,s}(A_{\infty})$ associated to the category of all finite dimensional representations of the infinite linear quiver $A_{\infty…
Motivated by recent developments of $\infty$-categorical theories related to differential graded (dg for short) Lie algebras, we develop a general framework for locally finite $\infty$-$\mathfrak{g}$-modules over a dg Lie algebra $\mathfrak{g}$. We show that the category of such locally finite $\infty$-$\mathfrak{g}$-modules is almost a model category in the sense of Vallette. As a homotopy theoretical generalization of Loday and Pirashvili's Lie algebra objects in the tensor category of linear maps, we further study weak Loday-Pirashvili modules consisting of $\infty$-morphisms from locally f…
It is shown that if $\mathfrak B(V) $ is connected Nichols algebra of diagonal type with $\dim V>1$, then $\dim (\mathfrak L^-(V)) = \infty$ $($resp. $ \dim (\mathfrak L(V)) = \infty $$)$ $($ resp. $ \dim (\mathfrak B(V)) = \infty $$)$ if and only if $Δ(\mathfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $\mathfrak B(V)$ equal to zero, viz. $\overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_j\in \mathfrak B(V)$, are given.
In this paper, we extend the Goldman-Millson Theorem for $L_\infty$ algebras. We consider two $L_\infty$ algebras $L$ and $\tilde{L}$ endowed with descending, bounded above and complete filtrations compatible with the $L_\infty$ structures and ${U:L \rightarrow \tilde{L}}$ an $\infty$-morphism respecting the filtrations. We prove that in the setting of the linear part of $U$, say $ψ$, being a quasi-isomorphism on the r-1st page of the spectral sequences and ${H^1 ((\mathfrak{F}_{2^q} L)/(\mathfrak{F}_{\mathrm{min}(2^{q+1},r)} L))=0}$ for every $q$ with $2^q < r$ and ${H^i((\mathfrak{F}_1 \tild…
We prove the longstanding physics conjecture that there exists a unique two-parameter $\mathcal{W}_{\infty}$-algebra which is freely generated of type $\mathcal{W}(2,3,\dots)$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type $\mathcal{W}(2,3,\dots, N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$, the structure constants for the principal $\mathcal{W}$-algebras $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ are rational functions of $k$ and $n$, and…
Let $\mathfrak{g}$ be a curved $L_\infty$-algebra endowed with a complete filtration $\mathfrak{F}\mathfrak{g}$. Suppose there exists an integer $r \in \mathbb{N}_0$ for which the curvature $μ_0$ satisfies $μ_0 \in \mathfrak{F}_{2r+1} \mathfrak{g}$ and the spectral sequence yields $E_{r+1}^{p,q} =0$ for $p,q$ with $p+q=2$. We prove that then a Maurer-Cartan element exists. In addition, we show, as a typical application, that for $P$ a possibly inhomogeneous Koszul operad with generating set in arities 1,2 (e.g. $P$=Com,As,BV,Lie,Ger), a $P_\infty$-algebra $A$ is intrinsically formal if its twi…
The positive part $U^+_q$ of $U_q({\widehat {\mathfrak{sl}}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. In 1993 I. Damiani obtained a PBW basis for $U^+_q$, consisting of some elements $\lbrace E_{nδ+α_0}\rbrace_{n=0}^\infty$, $\lbrace E_{nδ+α_1}\rbrace_{n=0}^\infty$, $\lbrace E_{nδ}\rbrace_{n=1}^\infty$ that are defined recursively. Our goal is to describe these elements in closed form. We achieve this goal using Catalan words and a $q$-shuffle algebra.
Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and let $W$ be a finite subgroup of $\mathbb{C}$-algebra automorphisms of the enveloping algebra $U(\mathfrak{g})$. We show that the derived category of $U(\mathfrak{g})^W$-modules determines isomorphism classes of both $\mathfrak{g}$ and $W.$ Our proofs are based on the geometry of the Zassenhaus variety of the reduction modulo $p\gg 0$ of $\mathfrak{g}.$ Specifically, we use non-existence of certain étale coverings of its smooth locus
We consider the Whittaker modules $M_{1}(λ,μ)$ for the Weyl vertex algebra $M$, constructed in arXiv:1811.04649, where it was proved that these modules are irreducible for each finite cyclic orbifold $M^{\Bbb Z_n}$. In this paper, we consider the modules $M_{1}(λ,μ)$ as modules for the ${\Bbb Z}$-orbifold of $M$, denoted by $M^0$. $M^0$ is isomorphic to the vertex algebra $\mathcal W_{1+\infty, c=-1} = \mathcal M(2) \otimes M_1(1)$ which is the tensor product of the Heisenberg vertex algebra $M_1(1)$ and the singlet algebra $\mathcal M(2)$. Furthermore, these modules are also modules of the Li…
Let $\mathfrak{g}$ be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the "canonical basis" of the adjoint representation of~$\mathfrak{g}$. Here, we present a variation of this idea which leads to a new, and quite elementary construction of~$\mathfrak{g}$ itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of $\mathfrak{g}$.
We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of $End_{\bar{U}_{q}(\mathfrak{sl}_{2})}(X^{\otimes n})$, where $X:=\mathcal{X}^{+}_{2}$ is a two dimensional $\bar{U}_{q}(\mathfrak{sl}_{2})$ module.
The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{gl}}(\infty)$, $\widehat{\mathfrak{o}}(\infty)$, $\widehat{\mathfrak{sp}}(\infty)$, are equivalent to subalgebras of the quantum $W_{1+\infty}$ algebras. They have well-known representations on the Fock space of either a Dirac fermion ($\hat A_\infty$), a Majorana fermion ($\hat B_\infty$ and $\hat D_\infty$) or a symplectic boson ($\hat C_\infty$). Explicit formulas for the action of the quantum $W_{1+\infty}$ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of …
We prove {\rm (i)} Nichols algebra $\mathfrak B(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and $\mathfrak B(V)$ is an arithmetic root system, then $\mathfrak B(V) = F \oplus \mathfrak L(V);$ and {\rm (iii)} if $Δ(\mathfrak B(V))$ is an arithmetic root system and there does not exist any $m$-infinity element with $p_{uu} \not= 1$ for any $u \in D(V)$, then $\dim (\mathfrak B(V) ) = \infty$ if and only if there exists $V'$, which is twisting equivalent to $V$, such…
A problem of defining the quantum analogues for semi-classical twists in $U(\mathfrak{g})[[t]]$ is considered. First, we study specialization at $q=1$ of singular coboundary twists defined in $U_{q}(\mathfrak{g})[[t]]$ for $\mathfrak{g}$ being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when $\mathfrak{g}=\mathfrak{sl}_{3}$ and obtain $q-$deformation of the semi-classical twist introduced by Connes and Moscovici in noncommutative geometry.
There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n) \hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) = \text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ is a natural generalization of the parafermion algebra of $\mathfrak{sl}_2$. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,3,\dots)$. In this paper, we consider an analogous structure of orthogonal type, namely $\mathcal{D}^k(n) …
In the present paper, we give a definition of the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ of the circle $S^1\colon =\mathbb{R}/\mathbb{Z}$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ associated to the rational circle $S^1_\mathbb{Q}\colon= \mathbb{Q}/\mathbb{Z}$ as a direct limit of $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(n))$'s, where the order is given by divisibility of positive integers. The quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ …
The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor $\mathfrak n$ of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for $\mathfrak{sl}(n)$, the rank 2 semi-simple Lie algebras and $\mathfrak{sp}(2n)$. In the present work, we generalize these constructions to the Lie algebras $\mathfrak{so}(2n+1)$. The orthogonal semistandard Young tableaux were defined by M. Kashiwara and T. Nakashima, they form a basis for the shape algebra of $\mathfrak{so}(2n+1)$. Defining the notion of …
In this note, we prove that, for a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K$ of characteristic 0 which contains $\mathbb C$, the Chevalley--Eilenberg complex $\mathrm U(\mathfrak g)\otimes \wedge(\mathfrak g)$, which is in a natural way a deformation quantization of the Koszul complex of $\mathrm S(\mathfrak g)$, is $A_\infty$-quasi-isomorphic to the deformation quantization of the $A_\infty$-bimodule $K=\mathbb K$ provided by the Formality Theorem in presence of two branes (CFFR).
For a finite dimensional Lie algebra $\mathfrak{g}$, the Duflo map $S\mathfrak{g}\rightarrow U\mathfrak{g}$ defines an isomorphism of $\mathfrak{g}$-modules. On $\mathfrak{g}$-invariant elements it gives an isomorphism of algebras. Moreover, it induces an isomorphism of algebras on the level of Lie algebra cohomology $H(\mathfrak{g},S\mathfrak{g})\rightarrow H(\mathfrak{g}, U\mathfrak{g})$. However, as shown by J. Alm and S. Merkulov, it cannot be extended in a universal way to an $A_\infty$-isomorphism between the corresponding Chevalley-Eilenberg complexes. In this paper, we give an elementa…
In third paper of the series we construct a large family of representations of the quantum toroidal $\gl_1$ algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand-Zetlin type basis for a class of irreducible lowest weight $\gl_\infty$-modules.
