The free \mathfrak{m} -lattice on the poset H

Let H={a ₀, a ₁, a ₂, b ₀, b ₁, b ₂} be the poset defined by a ₀<a ₂<a ₁, b ₀<b ₂<b ₁, a ₀<b ₁, and b ₀<a ₁. For an infinite regular cardinal , we describe the free -lattice on H. This continues the work of I. Rival and R. Wille who accomplished the same for =. In subsequent papers, we show how to apply this result to describe the free -lattice on a poset for a large class of posets, called slender posets.     

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.     

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.     

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.     

Primitive systems of elements in the variety\mathfrak{A}\mathfrak{N}_2 and some locally finite varieties of groupsand some locally finite varieties of groups

Part of any basis of a relatively free group in the variety is called a primitive system of elements. We provide a criterion of being primitive for , where is a variety of Abelian groups satisfying x^m=1, and a variety generated by a finite group. Let be a variety of nilpotent groups of class ≤c. It is proved that, for the group , the property of being primitive for an element g is stronger than the condition of being unimodular on a vector composed of values of Fox derivatives in the ring . The group is not residually finite whenever a system of elements is primitive. Supported by RFFR grant No. 96-01-01948. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 687–699, November–December, 1998.     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

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.     

Harish-Chandra S-homomorphism and\mathfrak{G}-modules generated by a semiprimitive element

We define Harish-Chandra S-homomorphism which generalizes the classical Harish-Chandra homomorphism and study its properties. For -modules , generated by semiprimitive elements we prove the existence of composition sequences.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1031–1037, August, 1990.     

Symmetries of Systems of the Hyperbolic Riccati Type

Let be a Kac–Moody algebra, U(x,y) be a function defined in , and a be a constant element of . We prove that the equation U _xy = [[U,a],U _x] has two symmetry hierarchies connected by a gauge transformation. In particular, the well-known Konno equation appears in the case of the algebra . The corresponding symmetry hierarchies contain the nonlinear Schrödinger and the Heisenberg magnet equations.     

Supports of (\mathfrak{g},\mathfrak{k})-modules of finite type

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}$ be a reductive subalgebra in $\mathfrak{g}$ . We say that a $\mathfrak{g}$ -module M is a $(\mathfrak{g},\mathfrak{k})$ -module if M, considered as a $\mathfrak{k}$ -module, is a direct sum of finite-dimensional $\mathfrak{k}$ -modules. We say that a $(\mathfrak{g},\mathfrak{k})$ -module M is of finite type if all $\mathfrak{k}$ -isotopic components of M are finite-dimensional. In this paper we prove that any simple $(\mathfrak{g},\mathfrak{k})$ -module of finite type is holonomic. A simple $\mathfrak{g}$ -module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the ??directions of growth of M.?? We also prove that for a given pair $(\mathfrak{g},\mathfrak{k})$ the set of possible invariants is finite.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Hilbert Functions, Residual Intersections, and Residually S2 Ideals

Let R be a homogeneous ring over an infinite field, IR a homogeneous ideal, and I an ideal generated by s forms of degrees d ₁,...,d _s so that codim( :I)s. We give broad conditions for when the Hilbert function of R/ or of R/( :I) is determined by I and the degrees d ₁,...,d _s . These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S ₂ ideals. We prove that the residually S ₂ property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.     

Nonsymmetric interpolation macdonald polynomials and \mathfrak{g}\mathfrak{l}_n basic hypergeometric series

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for series. *Supported by the ANR project MARS (BLAN06-2 134516). **Supported by the NSF grant DMS-0401387. ***Supported by the Australian Research Council.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

Semisimple Lie Algebras of Differential Operators

Starting from the commutation relations in a complex semisimple Lie algebra , one may obtain a space of vector fields on Euclidean space such that and are isomorphic when is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of is a Borel subalgebra . Furthermore, one may adjoin to the vector fields in multiplication operators to obtain an -parameter family of distinct presentations of as spaces of differential operators, where is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of can be presented in this way. All of this is carried out explicitly for arbitrary . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

\mathfrak{E} of topological spaces II

In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic.