Mathematics

The Yangian Y(g) of a simple Lie algebra g can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra U(g[t]) and the coordinate ring of the first congruence subgroup O( G 1 [[ t −1 ]]) . Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe subalgebras form a natural family of commutative subalgebras depending on a group element C of the adjoint group G . The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain. We describe the associated graded of Bethe subalgebras as subalgebras in U(g[t]) and in O( G 1 [[ t −1 ]]) for all semisimple C∈G . We show that associated graded in U(g[t]) of the Bethe subalgebra assigned to the identity of G is the universal Gaudin subalgebra of U(g[t]) obtained from the center of the corresponding affine Kac-Moody algebra at the critical level. This generalizes Talalaev's formula for generators of the universal Gaudin subalgebra to g of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin-Frenkel center at the critical level. Using our general result on associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple C∈G as C goes to an irregular semisimple group element C 0 . We show that this limit is the product of the smaller Bethe subalgebra and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of C 0 in g . This generalizes the Nazarov-Olshansky solution of Vinberg's problem on quantization of shift of argument subalgebras.

In this note we prove an integral identity involving complex powers of generators of quantum group U q (sl(3)) considered as certain positive operators in the setting of positive principal series representations. This identity represents a continuous analog of one of the Lusztig's relations between divided powers of generators of quantum groups, which play an important role in the study of irreducible modules \cite{Lu 1}. We also give definitions of arbitrary functions of U q (sl(3)) generators and give another proofs for some of the known results concerning positive principal series representations of U q (sl(3)) .

In this paper, we give a recursive algorithm to compute the multivariable Zassenhaus formula e X 1 + X 2 +⋯+ X n = e X 1 e X 2 ⋯ e X n ∏ k=2 ∞ e W k and derive an effective recursion formula of W k .

Let $\mathds{k}$ be an algebraically closed field. We give a complete isomorphism classification of non-connected pointed Hopf algebras of dimension 16 with $\operatorname{char}\mathds{k}=2$ that are generated by group-like elements and skew-primitive elements. It turns out that there are infinitely many classes (up to isomorphism) of pointed Hopf algebras of dimension 16. In particular, we obtain infinitely many new examples of non-commutative non-cocommutative finite-dimensional pointed Hopf algebras.

We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we prove that MNSD modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that MNSD tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.

We study parafermion vertex algebras N −3/2 (sl(2)) and N −3/2 (sl(3)) . Using the isomorphism between N −3/2 (sl(3)) and the logarithmic vertex algebra W 0 (2 ) A 2 from [2], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra W(2,3) of central charge c=−10 , and that N −3/2 (sl(3)) is a direct sum of irreducible N −3/2 (sl(2)) -modules. As a byproduct, we prove certain conjectures about the vertex algebra W 0 (p ) A 2 . We also obtain a vertex-algebraic proof of the irreducibility of a family of W(2,3 ) c modules at c=−10 .

We explore the possibility of introducing q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. For the module of 1-forms on the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics satisfying a certain reality condition. Finally, we construct metric connections on a class of projective modules over the quantum 2-sphere.

In this paper we introduce a new quantum algebra which specializes to the 2 -toroidal Lie algebra of type A 1 . We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.

We study analytic properties of " q -deformed real numbers", a notion recently introduced by two of us. A q -deformed positive real number is a power series with integer coefficients in one formal variable~ q . We study the radius of convergence of these power series assuming that $q \in \C.$ Our main conjecture, which can be viewed as a q -analogue of Hurwitz's Irrational Number Theorem, provides a lower bound for these radii, given by the radius of convergence of the q -deformed golden ratio. The conjecture is proved in several particular cases and confirmed by a number of computer experiments. For an interesting sequence of "Pell polynomials", we obtain stronger bounds.

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra M(p) , p≥2 . The first category consists of all finite-length M(p) -modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra W(p) . We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.