Bethe subalgebras of $$U_q(\widehat{\mathfrak {gl}}_n)$$ via shuffle algebras

In this article, we construct certain commutative subalgebras of the big shuffle algebra of type \(A^{(1)}_{n-1}\). This can be considered as a generalization of the similar construction for the small shuffle algebra, obtained in Feigin et al. (J Math Phys 50(9):42, 2009). We present a Bethe algebra realization of these subalgebras. The latter identifies them with the Bethe subalgebras of \(U_q(\widehat{\mathfrak {gl}}_n)\).     

$U({\mathfrak {so}}(7,{\mathbb C}))$的向量表示的范畴化

The aim of this paper is to categorify the n-th tensor power of the vector representation of U (so(7, C)). The main tools are certain singular blocks and projective functors of the BGG category of the complex Lie algebra gl n .     

$$|End \mathfrak{A}| = |Con \mathfrak{A}| = |End \mathfrak{A}| = 2^{|\mathfrak{A}|}$$ for any uncountable 1-unary algebra $$\mathfrak{A}$$

Algebra universalis -     

Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

Theoretical and Mathematical Physics - The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously...     

Quantum algebras with representation ring of $$ \mathfrak{s}\mathfrak{l}_2 $$ type

A reducible representation of the Temperley-Lieb algebra is constructed on a tensor product of n-dimensional spaces. As a centralizer of this action, we obtain a quantum algebra (quasi-triangular Hopf algebra) with the representation ring that is equivalent to the representation ring of the Lie algebra. Bibliography: 23 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 167–177.     

Topological invariants from quantum group $$\mathcal {U}_{\xi }\mathfrak {sl}(2|1)$$ at roots of unity

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra \(\mathfrak {sl}(2|1)\). The construction is based on nilpotent irreducible finite dimensional representations of quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2|1)\) where \(\xi \) is a root of unity of odd order. These constructions use the notion of modified trace and relative \( G \)-modular category of previous authors.     

The Abelian category of quotients of $$\mathcal{L}\mathfrak{F}$$ -spaces

We construct the category of quotients of -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.     

Algebraic Bethe ansatz for $$\mathfrak o_{2n+1}$$ -invariant integrable models

Theoretical and Mathematical Physics - We study the class of $$ \mathfrak{o} _{2n+1}$$ -invariant quantum integrable models in the framework of the algebraic Bethe ansatz and propose a construction...     

Factorizations of one-generated $$ \mathfrak{X} $$-local formations

Under study are the factorizations of one-generated \(\mathfrak{X}\)-local formations, where \(\mathfrak{X}\) is a class of simple groups. Some known results on one-generated local and Baer-local formations appear as particular cases.     

Deformations of fundamental group representations and earthquakes on $${\textit{SO}}(n,1)$$ surface groups

In this article we construct a type of deformations of representations \(\pi _1(M)\rightarrow G\) where G is an arbitrary lie group and M is a large class of manifolds including \(\hbox {CAT}(0)\) manifolds. The deformations are defined based on codimension 1 hypersurfaces with certain conditions, and also on disjoint union of such hypersurfaces, i.e. multi-hypersurfaces. We show commutativity of deforming along disjoint hypersurfaces. As application, we consider Anosov surface groups in \({\textit{SO}}(n,1)\) and show that the construction can be extended continuously to measured laminations, thus obtaining earthquake deformations on these surface groups.     

Signature Characters of Highest-Weight Representations of $U_{q}(\mathfrak {gl}_{n})$

We consider \(U_{q}(\mathfrak {gl}_{n})\), the quantum group of type A for |q| = 1, q generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the technique involves combinatorics of the Gelfand-Tsetlin bases. As an application, we obtain information about unitarity of finite-dimensional irreducible representations for arbitrary q: we classify the continuous spectrum of the unitarity locus. We also recover some known results in the classical limit \(q \rightarrow 1\) that were obtained by different means. Finally, we provide several explicit examples of signature characters.     

On a new form of Bethe ansatz equations and separation of variables in the $$\mathfrak{s}\mathfrak{l}_3 $$ Gaudin model

A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum Gaudin model is presented. Dedicated to V.I. Arnold on the occasion of his 70th birthday     

An almost all result on $${q_1q_2\equiv c ({\rm mod}\, q)}$$

In this paper, we consider the congruence equation \({q_1 q_2 \equiv c ({\rm mod}\, q)}\) with \({a < q_1 \leq a+q^{1/2+\epsilon}}\) and \({b < q_2 \leq b+q^{1/2+\epsilon}}\) and show that it has solution for almost all a and b. Then we apply it to a question of Fujii and Kitaoka as well as generalize it to more variables. At the end, we present a new way to attack the above congruence equation question through higher moments.     

Deformation of \mathfrak{sl}\, (2) and \mathfrak{osp}\, (1|2)-modules of symbols

We consider the $\mathfrak{sl}\, (2)$ -module structure on the spaces of symbols of differential operators acting on the spaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this structure and prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem getting the same results.     

Exceptional sequences of 8 line bundles on $$({\mathbb {P}}^1)^3$$ ( P 1 ) 3

Journal of Algebraic Combinatorics - We investigate maximal exceptional sequences of line bundles on $$({\mathbb {P}}^1)^r$$ , i.e., those consisting of $$2^r$$ elements. For $$r=3$$ we show that...     

The projective general linear group $${mathrm {PGL}}(2,2^m)$$ PGL ( 2 , 2 m ) and linear codes of length $$2^m+1$$ 2 m + 1

Designs, Codes and Cryptography - Let $$q=2^m$$ . The projective general linear group $${mathrm {PGL}}(2,q)$$ acts as a 3-transitive permutation group on the set of points of the projective line....     

Laws of the lattice of all $${\mathfrak {X}}$$ -local formations of finite groups

Ricerche di Matematica - Let $${\mathfrak {X}}$$ be a class of simple groups with a completeness property $$\pi ({\mathfrak {X}}) = \mathrm {char} \, {\mathfrak {X}}$$ . A formation is a class of...