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

Ohne Zusammenfassung     

Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

Bounded reductive subalgebras of \mathfrak{s}{\mathfrak{l}_n}

Let \mathfrakg \mathfrak{g} be a reductive Lie algebra and \mathfrakk ì \mathfrakg \mathfrak{k} \subset \mathfrak{g} be a reductive in \mathfrakg \mathfrak{g} subalgebra. A ( \mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is a \mathfrakg \mathfrak{g} -module for which any element m ∈ M is contained in a finite-dimensional \mathfrakk \mathfrak{k} -submodule of M. We say that a ( \mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is bounded if there exists a constant C _M such that the Jordan-H?lder multiplicities of any simple finite-dimensional \mathfrakk \mathfrak{k} -module in every finite-dimensional \mathfrakk \mathfrak{k} -submodule of M are bounded by C _M . In the present paper we describe explicitly all reductive in \mathfraks\mathfrakl_n \mathfrak{s}{\mathfrak{l}_n} subalgebras \mathfrakk \mathfrak{k} which admit a bounded simple infinite-dimensional ( \mathfraks\mathfrakl_n,\mathfrakk \mathfrak{s}{\mathfrak{l}_n},\mathfrak{k} )-module. Our technique is based on symplectic geometry and the notion of spherical variety. We also characterize the irreducible components of the associated varieties of simple bounded ( \mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-modules.     

Combinatorics of character formulas for the lie superalgebra \mathfrak{g}\mathfrak{l}\left( {m,n} \right)

Let \mathfrakg \mathfrak{g} be the Lie superalgebra \mathfrakg\mathfrakl( m,n ) \mathfrak{g}\mathfrak{l}\left( {m,n} \right) . Algorithms for computing the composition factors and multiplicities of Kac modules for \mathfrakg \mathfrak{g} were given by the second author, [12] and by J. Brundan [1]. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph G \mathcal{G} defined using these diagrams. Each vertex of G \mathcal{G} corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If E \mathcal{E} is the subgraph of G \mathcal{G} obtained by deleting all edges of positive weight, then E \mathcal{E} is the graph that describes nonsplit extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.     

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     

Cartan Invariants in the Category of Restricted Representation for gl（m｜n）

Let k be an algebraically closed field of characteristic p 〉 2, and gl（m｜n） be the general linear Lie superalgebra over k. The Cartan invariants for the restricted supermodule category for gl（m｜n） are presented.     

On the singularity of the irreducible components of a Springer fiber in {\mathfrak{s}\mathfrak{l}_n}

Let ${\mathcal{B}_u}Let B_u{\mathcal{B}_u} be the Springer fiber over a nilpotent endomorphism u ? End(\mathbbCⁿ){u\in {\rm End}(\mathbb{C}^n)}. Let J (u) be the Jordan form of u regarded as a partition of n. The irreducible components of B_u{\mathcal{B}_u} are all of the same dimension. They are labelled by Young tableaux of shape J (u). We study the question of the singularity of the components of B_u{\mathcal{B}_u} and show that all the components of B_u{\mathcal{B}_u} are nonsingular if and only if J(u) ? {(l,1,1,?), (l₁,l₂), (l₁,l₂,1), (2,2,2)}{J(u)\in\{(\lambda,1,1,\ldots), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}}.     

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.     

On the mirabolic Lie algebra \mathfrak{p}_n

We consider the Lie algebra \(\mathfrak{g} = \mathfrak{p}_n \) of (n + 1) × (n + 1) matrices with zeros in the last row. This algebra has received the name of mirabolic; it has many remarkable properties and plays an important role in representation theory. In this paper we study open coadjoint orbits for the corresponding Lie group P _n .     

Bosonic Formulas for \widehat{\mathfrak{s}\mathfrak{l}} 2 Coinvariants

We derive bosonic-type formulas for the characters of ₂ coinvariants.     

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.     

Toroidal actions on level 1 modules ofU_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )

Recently Varagnolo and Vasserot established that theq-deformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra with the level (0,1). In the present article we propose a more detailed proof of this fact than the one given by Varagnolo and Vasserot. The quantum toroidal action on the Fock space depends on a certain parameter . We find that with a specific choice of this parameter, the action on the Fock spaces gives rise to the toroidal action on irreducible level-1 highest weight modules of the affine quantum algebra . Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum totoidal algebra gives rise to a structure on irreducible level-1 highest weight -modules.Supported by the JSPS Research Fellowship for Young Scientists.     

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     

Combinatorics of the \widehat{\mathfrak{s}\mathfrak{l}}_2 Coinvariants: Dual Functional Realization and Recursion

We give the fermionic character formulas for the spaces of coinvariants obtained from level k integrable representations of . We establish the functional realization of the spaces dual to the coinvariant spaces. We parameterize functions in the dual spaces by rigged partitions, and prove the recursion relations for the sets of rigged partitions.