The properties of the set {L} of extended jordanian twists for algebra sl(3) are studied. Starting from the simplest algebraic construction --- the peripheric Hopf algebra U_ P'(0,1)(sl(3)) --- we construct explicitly the complete family of extended twisted algebras {U_ E(θ)(sl(3))} corresponding to the set of 4-dimensional Frobenius subalgebras {L(θ)} in sl(3). It is proved that the extended twisted algebras with different values of the parameter θare connected by a special kind of Reshetikhin twist. We study the relations between the family {U_E(θ)(sl(3))} and the one-dimensional set {U_DJR(λ)(sl(3))} produced by the standard Reshetikhin twist from the Drinfeld--Jimbo quantization U_DJ(sl(3)). These sets of deformations are in one-to-one correspondence: each element of {U_E(θ)(sl(3))} can be obtained by a limiting procedure from the unique point in the set {U_DJR(λ)(sl(3))}.
On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot
Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.
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.
By twisted quantum invariants we mean polynomial invariants of knots in the three-sphere endowed with a representation of the fundamental group into the automorphism group of a Hopf algebra $H$. These are obtained by the Reshetikhin-Turaev construction extended to the $\mathrm{Aut}(H)$-twisted Drinfeld double of $H$, provided $H$ is finite dimensional and $\mathbb{N}^m$-graded. We show that the degree of these polynomials is bounded above by $2g(K)\cdot d(H)$ where $g(K)$ is the Seifert genus of a knot $K$ and $d(H)$ is the top degree of the Hopf algebra. When $H$ is an exterior algebra, our t…
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…
Sapienza Università di Roma（罗马大学）；University of Zagreb Faculty of Science, Department of Mathematics（萨格勒布大学理学院）；Central China Normal University（华中师范大学）
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-Birkhoff-Witt Theorem, determine their centres and also study extended reflection algebras.
We prove that the parallel transport of a flat $n-1$-gerbe on any given target space gives rise to an $n$-dimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimension-independent version of twisted and equivariant Dijkgraaf-Witten models. Finally, we introduce twisted equivariant Dijkgraaf-Witten theories giving us in the 3-2-1-dimensional case a new class of equivariant …
For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the …
We consider the $k$-twisted Nekrasov-Shatashvili limit (NS$_k$ limit) of 5d (K-theoretic) and 6d (elliptic) quiver gauge theory, where one of the multiplicative equivariant parameters is taken to be the $k$-th root of unity. We obtain the extended center of the associated $q$-deformed quiver W-algebras constructed by our formalism [arXiv:1512.08533; arXiv:1608.04651; arXiv:1705.04410], which provides gauge theoretic proof of Bouwknegt-Pilch's statement on the relation to the representation ring of quantum affine algebra.
