Primitive elements of the free groups of the varieties\mathfrak{A}\mathfrak{N}_n

For groups of the formF/N', we find necessary and sufficient conditions for an elementg∈N/N' to belong to the normal closure of an elementh∈F/N'. It is proved that, in contrast to the case of a free metabelian group, for a free group of the variety , there exists an elementh whose normal closure contains a primitive elementg, but the elementsh andg ^±1 are not conjugate. In the groupF( ), two nonconjugate elements are chosen that have equal normal closures. Translated fromMaternaticheskie Zametki, Vol. 61, No. 6, pp. 884–889, June, 1997. Translated by A. I. Shtern     

The automorphism group of the rank 2 Lie algebra free in the variety \mathfrak{A}\mathfrak{N}_2

We study the location of some finite subgroups in the automorphism group of the rank 2 Lie algebra free in the variety $\mathfrak{A}\mathfrak{N}_2 $ .     

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.     

Parametrisierung von Konjugationsklassen in\mathfrak{s}\mathfrak{l}_n

Ohne Zusammenfassung     

On Projective Modules in Category \mathcal{O}_{int} of Quantum \mathfrak{sl}_{2}

The restriction of a Verma module of ${\bf U}(\mathfrak{sl}_3)$ to ${\bf U}(\mathfrak{sl}_2)$ is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of ${\bf U}(\mathfrak{sl}_2)$ . The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category $\mathcal O_{\rm{int}}$ of quantum $\mathfrak{sl}_2$ is given.     

On \mathfrak{F}-reachable subgroups of finite groups

The structure of $\mathfrak{F}$ -reachable subgroups in Θ-Frattini extensions is established.     

Projective and injective objects of the categories dual to \mathfrak{A}\mathfrak{B}

The categories dual to the category of Abelian groups (including the category of compact Abelian groups) are considered. In these categories, structure theorems on injective and projective objects are proved, and some projective coverings are calculated. In the category of compact Abelian groups, the notion of connected hull is introduced; some results on connected hulls are obtained and examples are given. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 168–182.     

Multiply\mathfrak{L} formations of finite groupsformations of finite groups

We study formations of finite groups. Gomel University, Gomel, Belorussia. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 783–797, June, 2000.     

ℓ-Überlagerungen von ℙ1\\{ \mathfrak{p}_1 ,\mathfrak{p}_2 ,\mathfrak{p}_3 \}

Ohne Zusammenfassung     

Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities

Let be a space of quasipolynomials of dimension N=N ₁+⋅⋅⋅+N _n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ _i=0 ^N A _N−i (x)∂ _x ⁱ annihilating V and such that its leading coefficient A ₀ is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ _i=0 ^N u ⁱ A _N−i (∂ _u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

On the definability of endomorphisms of a free group of the variety\mathfrak{A}\mathfrak{M} by a finite set of valuesby a finite set of values

In the paper we find out in what cases any endomorphism of a free metabelian group of rankn is uniquely determined by its values on finitely many elements of the group. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 916–920, December, 1997 Translated by A. I. Shtern     

The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .