Diagonal reduction algebra and the reflection equation

We describe the diagonal reduction algebra D(gl _n ) of the Lie algebra gl _n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl _n ).     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Maximal subalgebras of the general linear Lie algebra containing Cartan subalgebras

Let gln(R) be the general linear Lie algebra of all n × n matrices over a unital commutative ring R with 2 invertible, dn(R) be the Cartan subalgebra of gln(R) of all diagonal matrices. The maximal subalgebras of gln(R) that contain dn(R) are classified completely.     

Lie algebra of formal vector fields which are extended by formal g-valued functions

In this paper, we consider the infinite-dimensional Lie algebra W_n⋉g⊗O _n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of the Weyl algebra of (gl _n ⊕ g) by the (2n+1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies of the Lie algebra W_n⋉g⊗O _n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to characteristic classes of tangent and G-bundles. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 205–230.     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

Construction of Graded Covariant GL(m/n) Modules Using Tableaux

Irreducible covariant tensor modules for the Lie supergroups GL(m/n) and the Lie superalgebras gl(m/n) and sl(m/n) are obtained through the use of Young tableaux techniques. The starting point is the graded permutation action, first introduced by Dondi and Jarvis, on V ^l . The isomorphism between this group of actions and the symmetric group S _l enables the graded generalization of the Young symmetrizers, and hence of the column relations and Garnir relations, to be made. Consequently, corresponding to each partition of l an irreducible GL(m/n) module may be obtained as a submodule of V ^l . A basis for the module labeled by the partition is provided by GL(m/n)–standard tableaux of shape defined by Berele and Regev. The reduction of an arbitrary tableau to standard form is accomplished through the use of graded column relations and graded Garnir relations. The standardization procedure is algorithmic and allows matrix representations of the Lie superalgebras gl(m/n) and sl(m/n) to be constructed explicitly over the field of rational numbers. All the various steps of the standardization algorithm are exemplified, as well as the explicit construction of matrices representing particular elements of gl(m/n) and sl(m/n).     

The number of distinct latin squares as a group-theoretical constant

It is shown that the number l_n of all distinct Latin squares of the nth order appears as a structure constant of the algebra defined on the Magic squares of the same order. The algebra is isomorphic to the algebra of double cosets of the symmetric group of degree n² with respect to the intransitive subgroup of all substitutions in the n sets of transitivity, each set being of cardinality n. The representation theory makes it possible then to express l_n in terms of eigenvalues of a certain element of the algebra.     

EXCEPTIONAL SEQUENCES FOR QUIVERS OF DYNKIN TYPE

Let kQ be the path algebra of a quiver Q without oriented cycles with n vertices. An indecomposable kQ-module without self-extensions is called exceptional. The braid group B _n with n ? 1 generators acts naturally on the set of complete exceptional sequences. Crawley-Boevey (Proceedings of ICRA VI, Carleton-Ottawa, 1992) and Ringel (Contemp. Math. 1994, 171, 339–352) have pointed out that this action is transitive. The number of complete exceptional sequences for kQ representation finite will be computed here and it is shown to be independent of the orientation of the arrows of the quiver Q. The factor group of the braid group which acts freely on the set of complete exceptional sequences can be regarded as a subgroup of the symmetric group S _{? n} , where ? _n is the number of complete exceptional sequences of the algebra kQ. This group is known for certain special types of quivers. Some other interesting relations of the acting group will be given.     

A modified BLM approach to quantum affine {\mathfrak{gl}_n}

We introduce a spanning set of Beilinson–Lusztig–MacPherson type, {A(j, r)} _A,j , for affine quantum Schur algebras S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and construct a linearly independent set {A(j)} _A,j for an associated algebra [^(K)]_\vartriangle(n){{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)} . We then establish explicitly some multiplication formulas of simple generators E^\vartriangle_h,h+1(0){E^\vartriangle_{h,h+1}}(\mathbf{0}) by an arbitrary element A(j) in [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}} via the corresponding formulas in S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}} , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall algebras \mathfrak H_\vartriangle(n){{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}} associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for \mathfrak H_\vartriangle(n){{\boldsymbol{\mathfrak H}}_\vartriangle}(n) established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n) . Using these relations, we then show that the subspace \mathfrak A_\vartriangle(n){{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)} of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} spanned by {A(j)} _A,j contains the quantum enveloping algebra U_\vartriangle(n){{{\mathbf U}_\vartriangle}(n)} of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n,r){\boldsymbol{\mathcal S}(n,r)} , the resulting subspace \mathfrak A_\vartriangle(n){{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}} is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of \mathfrakgl_n{\mathfrak{gl}_n} . We conjecture that \mathfrak A_\vartriangle(n){{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}} is a subalgebra of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} .     

Representation theory of the vertex algebraW 1+∞

In our paper [KR] we began a systematic study of representations of the universal central extension of the Lie algebra of differential operators on the circle. This study was continued in the paper [FKRW] in the framework of vertex algebra theory. It was shown that the associated to simple vertex algebraW _1+∞,N with positive integral central chargeN is isomorphic to the classical vertex algebraW(gl _N), which led to a classification of modules overW _1+∞,N . In the present paper we study the remaing nontrivial case, that of a negative central charge-N. The basic tool is the decomposition ofN pairs of free charged bosons with respect togl _N and the commuting withgl _N Lie algebra of infinite matricesĝl. To Alexander Alexandrovich Kirillov on his 60-th birthday Supported in part by NSF grant DMS-9103792.     

Finite degree: algebras in general and semigroups in particular

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations on A. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p _n -sequence), and every finite commutative semigroup has finite degree. We give an example of a five-element unary semigroup that has infinite degree. We also give examples to show that finite degree is not preserved in general under taking subalgebras, homomorphic images, direct products or subdirect factors.     

Gelfand�CTsetlin algebras and cohomology rings of Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL _n . We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand–Tsetlin subalgebra of U(gl _n ) and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of U(gl _n ).     

Coloured peak algebras and Hopf algebras

For G a finite abelian group, we study the properties of general equivalence relations on G _n = G ⁿ ⋊ _n , the wreath product of G with the symmetric group _n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of G _n as well as graded connected Hopf subalgebras of ⨁ _{n≥ o} G _n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects. 2000 Mathematics Subject Classification Primary: 16S99; Secondary: 05E05, 05E10, 16S34, 16W30, 20B30, 20E22Bergeron is partially supported by NSERC and CRC, CanadaHohlweg is partially supported by CRC     

Localization algebras and deformations of Koszul algebras

We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular, the center of A acts by characters on the deformed standard modules, providing a “localization map”. We construct a universal graded deformation of A and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category O{\mathcal{O}} for \mathfrakgl_n{\mathfrak{gl}_n} is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the “category O{\mathcal{O}}” of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.     

Codeterminants and <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We study codeterminants in the q-Schur algebra S _q (n,r) and prove that the standard ones form a basis of S _q (n,r), using a quantized version of the Désarménien matrix. We find elements of the form F _S 1_λ E _T in Lusztig’s modified enveloping algebra of gl(n), which, up to powers of q, map to the basis of standard codeterminants, where F _S ∈U ^– and E _T ∈U ⁺ are explicitly given products of root vectors, depending on Young tableaux S and T.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

Duality for Knizhnik–Zamolodchikov and Dynamical Equations

We consider the Knizhnik–Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of (gl _k ,gl _n ) duality. We show that the KZ and dynamical equations naturally exchange under the duality.