Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, and \(Y_{\hbar }(\mathfrak{g})\), \(U_{q}(L\mathfrak{g})\) the corresponding Yangian and quantum loop algebra, with deformation parameters related by \(q=e^{\pi \iota \hbar }\). When \(\hbar \) is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor \(\Gamma \) from the category of finite-dimensional representations of \(Y_{\hbar }(\mathfrak{g})\) to those of \(U_{q}(L \mathfrak{g})\). The functor \(\Gamma \) is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of \(\operatorname{Rep}_{\operatorname{fd}}(Y_{\hbar }(\mathfrak{g}))\) defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on \(\Gamma \) and show that, if \(|q|\neq 1\), it yields an equivalence of meromorphic braided tensor categories, when \(Y_{\hbar }(\mathfrak{g})\) and \(U_{q}(L\mathfrak{g})\) are endowed with the deformed Drinfeld coproducts and the commutative part of their universal \(R\)-matrices. This proves in particular the Kohno–Drinfeld theorem for the abelian \(q\)KZ equations defined by \(Y_{\hbar }(\mathfrak{g})\). The tensor structure arises from the abelian \(q\)KZ equations defined by an appropriate regularisation of the commutative part of the \(R\)-matrix of \(Y_{\hbar }(\mathfrak{g})\).     

Yangians and universal enveloping algebras

Starting from the universal enveloping algebras u(gl(n)), n=1,2,... we construct an algebra A, which gives a realization of the Yangian Y(gl(m)) and is a proper way to define the universal enveloping algebra for infinite-dimensional classical Lie algebras.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 142–150, 1987.     

Twisted Yangians and Mickelsson algebras I

We introduce an analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.     

Extremal loop weight modules and tensor products for quantum toroidal algebras

We define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld “coproduct.” This allows us to recover the vector representations recently introduced by Feigin–Jimbo–Miwa–Mukhin [7 Feigin, B., Jimbo, M., Miwa, T., Mukhin, E. (2013). Representations of quantum toroidal 𝔤𝔩_n. J. Algebra 380:78–108.[Crossref], [Web of Science ®] , [Google Scholar]] and constructed by the author [21 Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials. 2nd ed. Oxford: Oxford Math. Monographs, 1979. [Google Scholar]] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.     

Cayley-dickson algebras and alternative loop algebras

In this paper we establish a relationship between alternative loop algebras and Cayley-Dickson algebras: any Cayley-Dickson algebra is a quotient algebra of an alternative loop algebra. Thus we give a new way of representing a Cayley-Dickson algebra. As an application, a result of Luiz G. X. de Barros is generalized.     

Affine Yangians and deformed double current algebras in type A

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W1+∞-algebras. We prove that they admit a PBW-type basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We construct a Schur-Weyl functor between these algebras and rational Cherednik algebras.     

Many-valued quantum algebras

We deal with algebras of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety. Supported by the Research and Development Council of the Czech Government via the project MSM6198959214.     

Connected quantized Weyl algebras and quantum cluster algebras

For an algebraically closed field $\mathbb{K}$, we investigate a class of noncommutative $\mathbb{K}$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots ,x_n\}$ such that each pair satisfies a relation of the form $x_i{x}_j=q_{ij}{x}_j{x}_i+r_{ij}$, where $q_{ij}\in {\mathbb{K}}^{?}$ and $r_{ij}\in \mathbb{K}$, with, in some sense, sufficiently many pairs for which $r_{ij}\ne 0$. For such an algebra it turns out that there is a single parameter q such that each $q_{ij}=q^{\pm 1}$. Assuming that $q\ne \pm 1$, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let $n\ge 3$ be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type $A_{n?1}$ as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver $P_{n+1}^{(1)}$ identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in $n+2$ variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of $A_{n?1}$ and $P_{n+1}^{(1)}$ are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element.     

Hall algebras,hereditary algebras and quantum groups

4-VIII-1994 & 4-X-1994     

On sernisimple alternative loop algebras

Let QL be the loop algebra of an R.A. loop L over the rational field Q. Assume that the non-trivial commutator in L is not a square of a central element. In [1]: it is shown that QL determines L. In this paper we characterize all fields K, with char K ≠ 2, such that KL determines L for all R..A. 2-loops L with the above property.     

LatticeW algebras and quantum groups

We present Feigin's construction [Lectures given in Landau Institute] of latticeW algebras and give some simple results: lattice Virasoro andW ₃ algebras. For the simplest caseg=sl(2), we introduce the wholeU _q(2)) quantum group on this lattice. We find the simplest two-dimensional module as well as the exchange relations and define the lattice Virasoro algebra as the algebra of invariants ofU _q(sl(2)). Another generalization is connected with the lattice integrals of motion as the invariants of the quantum affine groupU _q(ñ₊). We show that Volkov's scheme leads to a system of difference equations for a function of non-commutative variables.Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 132–147, July, 1994.     

Commutative quantum operator algebras

A key notion bridging the gap between quantum operator algebras [26] and vertex operator algebras [4, 9] is the definition of the commutativity of a pair of quantum operators (see Section 2). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. In [26] we give a definition of a commutative quantum operator algebra. We show in [26] that a vertex operator algebra gives rise to a special case of a CQOA. The main purpose of the current paper is to further develop the foundations for a complete mathematical theory of CQOAs. We give proofs of most of the relevant results announced in [26], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators.     

Representations of quantum algebras

Inventiones mathematicae -     

Vertex algebras and the formal loop space

We construct a certain algebro-geometric version L(X)\mathcal{L}(X) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme L⁰(X)\mathcal{L}^{0}(X) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on L(X)\mathcal{L}(X) supported in L⁰(X)\mathcal{L}^{0}(X) . We also show that L(X)\mathcal{L}(X) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that all linear constructions applied to the free loop space produce vertex algebras.