On the Various Realizations of the Basic Representation of An−1(1) and the Combinatorics of Partitions

The infinite dimensional Lie algebra l _n = A _n–1 ⁽¹⁾ can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l _n corresponding to all partitions of n can be described very simply by using combinatorial constructions.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

On <Emphasis Type="Italic">T</Emphasis> -spaces and relations in relatively free,lie nilpotent,associative algebras

The purpose of this work is to obtain the commutator relations and Frobenius relations in a relatively free algebra F ^(l) specified by the identity [x ₁ , . . . , x _l ] = 0 over a field of characteristic p > 0. These relations for l > 3 are analogous to the relations in the algebra F ⁽³⁾ and are applied to the T-spaces in the algebra F ^(l). In order to study the relations in F ^(l) in more detail, we construct a model algebra analogous to the Grassmann algebra.     

The Birman-Murakami-Wenzl algebras of type E n

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E _n for n = 6; 7; 8 are shown to be semisimple and free over the integral domain \mathbbZ[ d^±1,l^±1,m ]

Euler characteristic of certain affine flag varieties

The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic elementn _t istc^l wherel is the rank of the associated finite type Lie algebra. The author's research is supported in part by a National Science Foundation postdoctoral fellowship.     

Random I-colorable graphs

In this article we investigate properties of the class of all l-colorable graphs on n vertices, where l = l(n) may depend on n. Let G^l_n denote a uniformly chosen element of this class, i.e., a random l-colorable graph. For a random graph G^l_n we study in particular the property of being uniquely l-colorable. We show that not only does there exist a threshold function l = l(n) for this property, but this threshold corresponds to the chromatic number of a random graph. We also prove similar results for the class of all l-colorable graphs on n vertices with m = m(n) edges.     

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D _n is shown to be semisimple and free of rank (2 ⁿ  + 1)n!! ? (2 ^n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D _n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ^±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D _n is a subalgebra of the BMW algebra of the same type.     

The bar spectral sequence converging toh *(SO(2n+1))

We study the bar spectral sequence converging toh _*(SO(2n+1)), whereh is an algebra theory overBP. The differentials are determined completely ifh=P(l) andn<2 ^l . These results will be used in a future paper on the MoravaK-theories ofSO(2n+1), with no restriction onn. As another application, we determineBP _*(Spin(7)) including much of its algebra structure.AMS Subject Classification: 57T10, 57T30, 55N22     

Specht Modules and Simple Modules for Centralizer Algebras of the Symmetric Group

Let Σ_n be the symmetric group on n letters. For l ≤ n identify Σ_l with a subgroup of Σ_n in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣ_n^Σ_l that is analogous to James's theory of permutation modules, Specht modules, and simple modules over kΣ_n. We make a conjecture about how to construct all simple kΣ_n^Σ_l-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.     

Edge disjoint cycles in graphs

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ? 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ? 2δ², then for any 3 ? l₁ ? l₂ ? ? ? l_k ? n, 1 = k = [(δ - 1)/2], such graph has K edge disjoint cycles with lengths l₁, l₂…l_k, respectively. In particular, when l₁ = l₂ = ? = l_k = n and k = [(δ - 1)/2], the graph contains [(δ - 1)/2] edge disjoint hamiltonian cycles.     

An improvement and an extension of the Elzinga & Hearn's algorithm to the 1-center problem in ℝn withl 2b-normswithl 2b-norms

Summary In this paper we deal with the 1-center problem in ℝⁿ when the distance is measured by anyl _2b-norm. This type of norm generalizes the Euclidean norm (l ₂-norm) and can be used to estimate road distances or travel times in Locational Analysis, and to measure dissimilarities between data in Cluster Analysis. The problem is to find the smallestb-ellipsoid containing a given finite setA of points in ℝⁿ, which generalizes the one of finding the smallest sphere containingA (1-center problem with thel ₂-norm). We show that this problem has a unique optimal solution. For thel ₂-norm, we use the Elzinga-Hearn algorithm. New starting rules are proposed to initialize and to improve the algorithm. On the other hand, the Elzinga-Hearn algorithm is extended to solve the problem withl _2b-norms. Computational results are given for six differentl _2b-norms, when these new starting rules are used in order to show which is the best starting rule. Problems of up to 5.000 points in ℝⁿ,n=2,4,6,8 and 10, are solved in a few seconds.     

Invariant Measures in Free MV-Algebras

MV-algebras can be viewed either as the Lindenbaum algebras of ?ukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free _n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] ⁿ . The maximal spectrum of Free _n is canonically homeomorphic to [0, 1] ⁿ , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] ⁿ which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure.     

Wiener��s Lemma for Infinite Matrices II

In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of B(l^q_w){\mathcal{B}}(\ell^{q}_{w}), the algebra of all bounded linear operators on the weighted sequence space l^q_w\ell^{q}_{w}, for any 1≤q<∞ and any discrete Muckenhoupt A _q -weight w.     

Polynomial Identities of RA and RA2 Loop Algebras

Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z _F over F. For RA2 loops of type M(Dih(A), ^?1,g ₀), we prove that the loop algebra is in the variety generated by the algebra  ₃ which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, ^?1,g ₀), where G is a non-Abelian group of exponent 4 having exactly 2 squares.     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

Levels of quaternion algebras

The level of a ring R with 1 ≠ 0 is the smallest positive integer s such that −1 can be written as a sum of s squares in R, provided −1 is a sum of squares at all. D. W. Lewis showed that any value of type 2 ⁿ or 2 ⁿ + 1 can be realized as level of a quaternion algebra, and he asked whether there exist quaternion algebras whose levels are not of that form. Using function fields of quadratic forms, we construct such examples. Received: 23 March 2007, Revised: 30 October 2007     

Stability of Numerical Invariants in Free Groups

Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l ₁ and l ₂ be cyclic field extensions of k of degree d with norms n _{l 1/k} and n _{l 2/k} . Minà?'s approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n _{l 1/k}  ? n _{l 2/k} of degree d. Let K = k(X ₁,…, X _{d ²} ). We compute polynomials which are similarity factors of a form of the kind N ? (n _{l 2/k}  ? _k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n _{l 1/k}  ? n _{l 2/k} to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n _{l 1/k} .     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

The Hodge Structure on a Filtered Boolean Algebra

Let (B _n) be the order complex of the Boolean algebra and let B(n, k) be the part of (B _n) where all chains have a gap at most k between each set. We give an action of the symmetric group S _l on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The S _n action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary.We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.Finally we present some data on the homology of B(n, k).     

Ellis Group and the Topological Center of the Flow Generated by the Map {n \mapsto {\lambda^{n}}^{k}}

For each natural number k and each irrational member λ of the unit circle, it is proved that the shift-orbit closure X _f of the function f(n) = l^nk{f(n) = {\lambda^{n}}^{k}} is homeomorphic to a k-torus. Using this homeomorphism, we investigate the Ellis group and its topological center of the dynamical system (X _f , U), where U is the shift operator on l^￥(\mathbbZ){l^{\infty}(\mathbb{Z})}. Finally, it is shown that the topological center of the spectrum of the Weyl algebra is the image of \mathbbZ{\mathbb{Z}} in the spectrum.