arXiv subjects
Publications and source records attributed to Nicolas Guay.
We extend to the super Yangian of the special linear Lie superalgebra $\mathfrak{sl}_{m|n}$ and its affine version certain results related to Schur-Weyl duality. We do the same for the deformed double current superalgebra of $\mathfrak{sl}_{m|n}$, which is introduced here for the first time.
In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra ${\mathfrak U}_q {\mathfrak p}_n$ of type $P$. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of ${\mathfrak U}_q {\mathfrak p}_n$. We explain how to relate modules over ${\mathfrak U}_q {\mathfrak p}_n$ to modules over ${\mathfrak p}_n$, the Lie superalgebra of type $P$, and we prove that the category of tensor modules over ${\mathfrak U}_q {\mathfrak p}_n$ is not semisimple.
We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ attached to the Lie superalgebra ${\mathfrak{p}}_n$ of type $P$. The superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${\mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplectic $q$-Brauer algebra and prove that it is the centralizer of the $\mathfrak{U}_q {\mathfrak{p}}_n$-module structure on ${\mathbb C}(n|n)^{\otimes l}$. We end by proposing a definition for a new periplectic $q$-Schur superalgebra.
Given an affine Kac-Moody algebra, we explain how to construct a coproduct for its associated Yangian. In order to prove that this coproduct is an algebra homomorphism, we obtain, in the first half of this paper, a minimalistic presentation of the Yangian when the Kac-Moody algebra is, more generally, symmetrizable.
Using vertex operators, we build representations of the Yangian of a simply laced Kac-Moody algebra and of its double. As a corollary, we prove the PBW property for simply laced affine Yangians.
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.
We continue the study of finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types B, C and D, with focus on those of types BI, CII and DI. After establishing that, for all twisted Yangians of these types, the highest weight of such a module necessarily satisfies a certain set of relations, we classify the finite-dimensional irreducible representations of twisted Yangians for the pairs $(\mathfrak{so}_N,\mathfrak{so}_{N-2} \oplus \mathfrak{so}_2)$ and $(\mathfrak{so}_{2n+1},\mathfrak{so}_{2n})$.
We initiate a theory of highest weight representations for twisted Yangians of types B, C, D and we classify the finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types CI, DIII and BCD0.
We prove the equivalence of two presentations of deformed double current algebras associated to a complex simple Lie algebra, the first one obtained via a degeneration of affine Yangians while the other one naturally appeared in the construction of the elliptic Casimir connection. We also construct a specific central element of these algebras and, in type A, show that they contain a very large center for certain values of their parameters.
We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan's classification. These algebras can be regarded as coideal subalgebras of the extended Yangian for orthogonal or symplectic Lie algebras. They can also be presented as quotients of a reflection algebra by additional symmetry relations. We prove an analogue of the Poincare-Birkoff-Witt Theorem, determine their centres and study also extended reflection algebras.
We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras ${\mathsf {BC}}_{r,s}(q)$. The superalgebra ${\mathsf {BC}}_{r,s}(q)$ is a quantum deformation of the walled Brauer-Clifford superalgebra ${\mathsf {BC}}_{r,s}$ and a super version of the quantum walled Brauer algebra. We prove that ${\mathsf {BC}}_{r,s}(q)$ is the centralizer superalgebra of the action of ${\mathfrak U}_{q}({\mathfrak q}(n))$ on the mixed tensor space $\mathbf{V}_{q}^{r,s}=\mathbf{V}_{q}^{\otimes r} \otimes (\mathbf{V}_q^*)^{\otimes s}$ when $n \ge r+s$, where ${\mathbf V}_{q}=\mathbb{C}(q)^{(n|n)}$ is the natural representation of the quantum enveloping superalgebra ${\mathfrak U}_{q}({\mathfrak q}(n))$ and $\mathbf{V}_q^*$ is its dual space. We also provide a diagrammatic realization of ${\mathsf {BC}}_{r,s}(q)$ as the $(r,s)$-bead tangle algebra ${\mathsf {BT}}_{r,s}(q)$. Finally, we define the notion of $q$-Schur superalgebras of type $\mathsf{Q}$ and establish their basic properties.
We study quantized enveloping algebras called twisted Yangians associated with the symmetric pairs of types CI, BDI and DIII (in Cartan's classification) when the rank is small. We establish isomorphisms between these twisted Yangians and the well known Olshanskii's twisted Yangians of types AI and AII, and also with the Molev-Ragoucy reflection algebras associated with symmetric pairs of type AIII. We also construct isomorphisms with twisted Yangians in Drinfeld's original presentation.
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.
We provide a sufficient condition for the cyclicity of an ordered tensor product $L=V_{a_1}(ω_{b_1})\otimes V_{a_2}(ω_{b_2})\otimes...\otimes V_{a_k}(ω_{b_k})$ of fundamental representations of the Yangian $Y(\mathfrak{g})$. When $\mathfrak{g}$ is a classical simple Lie algebra, we make the cyclicity condition concrete, which leads to an irreducibility criterion for the ordered tensor product $L$. In the case when $\mathfrak{g}=\mathfrak{sl}_{l+1}$, a sufficient and necessary condition for the irreducibility of the ordered tensor product $L$ is obtained. The cyclicity of the ordered tensor product $L$ is closely related to the structure of the local Weyl modules of $Y(\mathfrak{g})$. We show that every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of $Y(\mathfrak{g})$.
We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new involves $\mathbb{C}[u,v]$. We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra $A_1$ instead of the polynomial ring $\mathbb{C}[u,v]$, and, more generally, a rank one rational Cherednik algebra. We study quasi-finite highest weight representations of these Lie algebras.
A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with U_q(sl_2). We study its representation theory, and in particular, its category O.
We study the category O of representations of the rational Cherednik algebra A attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor, from O to the category of H-modules, where H is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between O/O_tor, the quotient of O by the subcategory of A-modules supported on the discriminant and the category of finite-dimensional H-modules. The standard A-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of ``cells'', provided W is a Weyl group and the Hecke algebra H has equal parameters. We prove that the category O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W.