A duality theorem for hope module algebras over dedekind rings

Let L be a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p≧3, which has been proved to be generalized restricted in the sense of [Shul, Shu2]. For a generalized restricted L-module M, the homological support variety ‖L‖_M is defined to be that of the primitive p-envelope P{L). A realization L of P(L) is given in Der(&;(m : n)). Furthermore, a class of generalized restricted highest weight L-modules lift to Dist(Tx)V(p)-module structures and their support varieties can be computed by using algebraic group techniques developed in [LN].     

On commutative group algebras of mixed groups

Suppose F is a perfect field of characteristic p 0 and G is an abelian group whose torsion subgroup Gt is p-primary. If Gt is totally projective of countable length, it is shown that G is a direct factor of the group of normalized units V(G) of the group algebra F(G) and that V(G)/G is a totally projective p-group. The proof of this result is based on a new characterization of the class of totally projective p-groups of countable length. Li addition, the same result regarding V(G) is obtained if G has countable torsion-free rank and Gt is totally projective of length less than ω₁ + ω₀ . Finally, these results are applied to the question of whether the existence of an F-i pomorphism F(G) ? F(H) for some group H implies that G?H.     

Congruence Identities and their Local Versions

If an algebra A with a nullary operation c satisfies a congruence identity p=q, then it also satisfies [c]_p=[c]_q, the so-called identity at c. The converse is valid if A is weakly regular and the terms p, q do not contain the operation of relational product. We study these problems for 3-permutability and the so-called permutability at c.     

Niveau hermitien de Certaines algèbres de quaternions

Let D [d] =(a,b/F) a quaternion divisior algebra over a field F of characteristic ? 2. Denote 1, i, j , k the basis of D, such that i²[d] n, j²[d] b, ij [d] -ji [d] k and A :D → D the involution given by i [d] -i, j [d] j (and k [d] k). In [LE] D. LEWIS asks the following question :Does there exist a quadratic Pfister form [S p. 721 [d] such that the hermitian form [d] [d] D is isotropic over (D, [d]) but not hyperbolic &; In this note, we show that the answer of this question is negative, so that the hermitien level [§I], when it is finite, of (D, A) is a power of two. This result holds for quaternion algebras with standard involution [LE].     

Local constant ofInd K L 1

Henniart has computed the local constant ε (Ind _K ^L 1) for an extensionL overK of local fields of odd degree in [H]. In this paper, we show that his formula is a consequence of results of Serre [S4] and of Deligne [D2]. Further we compute the local constant for an extension of even degree, assuming the residual characteristic is not equal to 2.     

Méthodes quantiques en topologie algébrique

Using quantum methods, we introduce here the notion of “neo-algebra” which generalizes the notion of a commutative differential graded algebra. Under some mild finiteness conditions, we can associate functorially to a space a neo-algebra over the finite field F_p: its quasi-isomorphism's class determines the p-adic-homotopy type of X. As a matter of fact, from this data, we can describe in a simple way Steenrod operations in the cohomology of X, as well as the p primary part of its homotopy groups. This point of view extends to finite characteristics the well-known rational homotopy theory of D. Quillen [9] and D. Sullivan [11]. It is deeply related to previous works of P. May, I. Kriz [5] and M.A. Mandell [7], [8].     

Symmetry and nonsymmetry for locally compact groups

The class [S] of locally compact groups G is considered, for which the algebra L¹(G) is symmetric (=Hermitian). It is shown that [S] is stable under semidirect compact extensions, i.e., H ? [S] and K compact implies K ×_sH? [S]. For connected solvable Lie groups inductive conditions for symmetry are given. A construction for nonsymmetric Banach algebras is given which shows that there exists exactly one connected and simply solvable Lie group of dimension ?4 which is not in [S]. This example shows that $\text{G}\text{Z}\text{?}$ [S]. Z the center of G, in general does not imply G ?[S]. It is shown that nevertheless for discrete groups and a (possibly) stronger form of symmetry this implication holds, implying a new and shorter proof of the fact that [S] contains all discrete nilpotent groups.     

Approximations and Mittag-Leffler conditions the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [16], and it does not provide for approximations when R has cardinality ≤ ?₀, [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits.In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [22].     

Narrow Positively Graded Lie Algebras

The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n₁ and n₂ from our list, that are positive parts of the affine Kac–Moody algebras A₁⁽¹⁾ and A₂⁽²⁾, respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].

     

量子代数模的张量积与不可分解模

令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子代数.k是特征为零的代数闭域,A→k(v(?)ξ)是环同态.U_k=U(?)_Ak,u_k是U_k的无穷小量子代数.令ξ是1的p次本原根.本文证明了:若有限维可积U_k模M,V中至少有一个是内射模,或者M,V中有一个模作为u_k模是平凡的,则有U_k模同构M(?)V≌V(?)M.我们还证明了:若有限维可积U_k模V作为u_k模是不可分解的,有限维可积U_k模M是不可分解的,且M|_(uk)是平凡的,则V(?)M是不可分解U_k模.令V和M是有限维可积U_k模,作为u_k模是同构的且具有单基座,本文证明V和M作为U_k模也是同构的.由此得到:不可分解内射u_k模提升为U_k模是唯一的.     

Inconsistency of GPK + AFA

M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X₁ introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X₁ in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties. Mathematics Subject Classification: 03E70, 03E10.     

On the respective terms of the derived and the polycentral series of a free lie algabra and an ideal

Let F be a free Lie algebra of rank> 1 and S be an ideal of F. Denote by F_m and F_{n l},…,n_k the terms of the lower central and the polycentral series of F. The aim of this paper is to provide a sufficient condition for the quotient algebra F_{n l},…,n_k/S_{n l},…,n_k to be infinitely generated. The case F_m/S_m was studied in [6] for free groups and in [ 2 ] for free Lie algebras. In this paper the following main theorem is proved : If F = F2 = S, k > 1 and n_i > 1 for i=l,…, k, then F_{n l}…,n_k/S_{n l} is infinitely generated.     

Algebra retracts and Poincaré-series

Let RA be a local inclusion of noetherian local rings. Assume that R is an algebra retract of A with the local retraction mapping p:AR. Let M be an R-module of finite type. Considering M as A-module via p we get P _M ^A =P _R ^A P _M ^R (Th. 1), where P _N ^S denotes the Poincaréseries of an S-module N. This result is used to give a simple proof of Th. 1 in [2], Also an application to fibre products of rings is given (Th. 2), generalizing slightly a result due to A. Dress and H. Krämer, see [1].     

Galois representations on holomorphic differentials

Using the same methods as in [S2] and [S3] we derive the Brauer trees for the cyclic pblocks for G₂(2^k),G₂(3^k) and the noncyclic pblocks for these groups, where p ≠ 2,3. The results are analogous to those obtained for G₂(q), where q is not divisible by 2 or 3. These results were first announced in [Sl].     

Identities of representations of nilpotent lie algebras

Lret N be a nilpotent of class 2 Lie algebra with one-dimensional centre C = Kc over an infinite field K and let p : N → End_k:(V) be a representation of N in a vector space V such that p(c) is invertible in End_k(V). We find a basis for the identities of the representation p. As consequences we obtain a basis for all the weak polynomial identities of the pair (M₂:(K), s1₂(K)) over an infinite field K of characteristic 2 and describe the identities of the regular representation of Lie algebras related with the Weyl algebra and its tensor powers.     

Koszul Differential Graded Algebras and BGG Correspondence II

The concept of Koszul differential graded （DG for short） algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module Ak is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] （see also Proposition 2.2）, Ak is not compact if H（A） is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H（A） is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.     

Exact sequences for the galois group

When F C→ C is a monoidal functor between suitable symmetric closed categories and H is a finite commutative and cocbmmuta-ive Hopf algebra in C, we obtain, generalizing results of S. Qaeneppel and M. Beattie, two exact sequences involving the Picard, Galois and Brauer groups defined in [21], [11]. Finally we describe the relation between this exact sequences.     

FINITE REPLACEMENT AND FINITE HILBERT-STYLE AXIOMATIZABILITY

We define a property for varieties V, the f.r.p. (finite replacement property). If it applies to a finitely based V then V is strongly finitely based in the sense of [14], see Theorem 2. Moreover, we obtain finite axiomatizability results for certain propositional logics associated with V, in its generality comparable to well-known finite base results from equational logic. Theorem 3 states that each variety generated by a 2-element algebra has the f.r.p. Essentially this implies finite axiomatizability of a 2-valued logic in any finite language.