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.     

Extension groups between simple Mackey functors

In order to better understand the structure of indecomposable projective Mackey functors, we study extension groups of degree 1 between simple Mackey functors. We explicitly determine these groups between simple functors indexed by distinct normal subgroups. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups. We then focus on the case where the simple Mackey functors are indexed by the same subgroup. In this case, the corresponding extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine all extension groups between simple Mackey functors for a p-group and for a group that has a normal p-Sylow subgroup. Finally, we compute higher extension groups between simple Mackey functors for a group that has a p-Sylow subgroup of order p.     

The Projective Class Ring of Basic and Split Hopf Algebras

We describe the structure of the complexification of the projective class ring of a basic and split Hopf algebra using a positive integer determined by the composition series of the projective cover of the trivial module. If q is a root of unity of order n, the projective class algebra of u_q ⁺(sl₂) is the product of n–1 copies of the dual numbers over C and two copies of C.     

Enveloping algebras that are principal ideal rings

Let L be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of L is a principal ideal ring if and only if L is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.     

Weakly open sets in the unit ball of some Banach spaces and the centralizer

We show that every Banach space X whose centralizer is infinite-dimensional satisfies that every non-empty weakly open set in BY has diameter 2, where (N-fold symmetric projective tensor product of X, endowed with the symmetric projective norm), for every natural number N. We provide examples where the above conclusion holds that includes some spaces of operators and infinite-dimensional C^∗-algebras. We also prove that every non-empty weak^∗ open set in the unit ball of the space of N-homogeneous and integral polynomials on X has diameter two, for every natural number N, whenever the Cunningham algebra of X is infinite-dimensional. Here we consider the space of N-homogeneous integral polynomials as the dual of the space (N-fold symmetric injective tensor product of X, endowed with the symmetric injective norm). For instance, every infinite-dimensional L₁(μ) satisfies that its Cunningham algebra is infinite-dimensional. We obtain the same result for every non-reflexive L-embedded space, and so for every predual of an infinite-dimensional von Neumann algebra.     

RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p ^MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.     

Cofiniteness conditions, projective covers and the logarithmic tensor product theory

We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C₁-cofinite in the sense of Li. 2. There exists a positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra A_N(V) is finite dimensional. This result shows that the category of grading-restricted generalized V-modules is a finite abelian category over C. Using the existence of projective covers, we prove that if such a vertex operator algebra V also satisfies Condition 3 that irreducible V-modules are R-graded and C₁-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under operations ?_P(z) for z∈C^×. We also prove that other conditions for applying the logarithmic tensor product theory developed by Lepowsky, Zhang and the author hold. Consequently, for such V, this category has a natural structure of braided tensor category. In particular, when V is of positive energy and C₂-cofinite, Conditions 1-3 are satisfied and thus all the conclusions hold.     

An elementary construction of tilting complexes

Let A be an artin algebra and e∈A an idempotent with add(eA_A)=add(D(_AAe)). Then a projective resolution of Ae_eAe gives rise to tilting complexes for A, where P(l)^• is of term length l+1. In particular, if A is self-injective, then is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes {T(2l)^•}_l?1 for A, where T(2l)^• is of term length 2l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes {T(l)^•}_l?1 for A, where T(l)^• is of term length l+1.     

A Central Extension Theorem-for Hopf Algebras

The construction used in Schur's central extension theorem is generalized by proving that given a cocommutative Hopf algebra H, there is a cocornmutative central extension B of H such that any projective representation of H lifts to an ordinary representation of B. The extension B is the crossed product of H with the finite dual of the group algebra kZ₁, where Z₁denotes the group of normal cocycles on H with values in the ground field k.     

Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ?¹(S) as a Banach right module over ?¹(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c₀(S). For such semigroups S, we also investigate the projectivity of c₀(S). We prove that for many semigroups S for which the Banach algebra ?¹(S) is non-amenable, the ?¹(S)-module ?¹(S) is not injective. The main result about the projectivity of c₀(S) states that for a weakly cancellative inverse semigroup S, c₀(S) is projective if and only if S is finite.     

Projective unitary positive-energy representations of diff (S1)

Let $\text{D}$ be the group of orientation-preserving diffeomorphisms of the circle S¹. Then $\text{D}$ is Fréchet Lie group with Lie algebra (δ_∞)_R the smooth real vector fields on S¹. Let δ_R be the subalgebra of real vector fields with finite Fourier series. It is proved that every infinitesimally unitary projective positive-energy representation of δ_R integrates to a continuous projective unitary representation of $\text{D}$. This result was conjectured by V. Kac.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

The degree of an integral point in Ps minus a hypersurface

Let K be a number field of degree m with ring of integers R and absolute discriminant d_K. Given a hypersurface Z_K of degree d in the projective space P_K^us over K with Zariski closure Z in P_R^s, we give an explicit function of m, d_K, s,d, a Hermitian metric on R^s+1 ⊗_z C, and a projective height of Z defined in [1], 4.1, such that there exists an integral point in P_R^s Z of degree bounded by this function.     

Constructing cotorsion pairs over generalized path algebras

We introduce two adjoint pairs (e _i ^λ , ( ) _i ) and (( ) _i , e _i ^ρ ) and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path algebra and exhibit projective and injective objects of the category M _p which is a generalization of monomorphisms category.     

Symmetry algebra of the AdS4×ℂℙ3 superstring

By direct calculation in the classical theory, we derive the central extension of the off-shell symmetry algebra for a string propagating in AdS ₄ ×?? ³ . It turns out to be the same as in the case of the AdS ₅ ×S ⁵ string. We consider the choice of the κ-symmetry gauge in detail and also explain how this gauge can be chosen without breaking the bosonic symmetries.     

Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type A 1

The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let Λ _t be the Yoneda algebra of a reconstruction algebra of type A ₁ over a field k. In this paper, a minimal projective bimodule resolution of Λ _t is constructed, and the k-dimensions of all Hochschild homology and cohomology groups of Λ _t are calculated explicitly.     

Δ-Filtrations and Projective Resolutions for the Auslander–Dlab–Ringel Algebra

The ADR algebra R _A of an Artin algebra A is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the Δ-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of R _A -modules. As an application, we give a counterexample to a claim by Auslander–Platzeck–Todorov, concerning projective resolutions over the ADR algebra.     

A Class of Finite-Dimensional Representations of U_q（sl_2）

In order to study a class of finite-dimensional representations of Uq（sl2）, we deal with the quotient algebra Uq （m, n, b） of quantum group Uq（sl2） with relations Kr=1, Emr=b, Fnr=0 in this paper, where q is a root of unity. The algebra Uq（m, n, b） is decomposed into a direct sum of indecomposable （left） ideals. The structures of indecomposable projective representations and their blocks are determined.     

Stable n-pointed trees of projective lines

Stable n-pointed trees arise in a natural way if one tries to find moduli for totally degenerate curves: Let C be a totally degenerate stable curve of genus g ≥ 2 over a field k. This means that C is a connected projective curve of arithmetic genus g satisfyingo

1. (a) every irreducible component of C is a rational curve over κ.
2. (b) every singular point of C is a κ-rational ordinary double point.
3. (c) every nonsingular component L of C meets C−L in at least three points. It is always possible to find g singular points P₁,..., P_g on C such that the blow up C of C at P₁,..., P_g is a connected projective curve with the following properties:o
   1. (i) every irreducible component of C is isomorphic to P_k¹
   2. (ii) the components of C intersect in ordinary κ-rational double points
   3. (iii) the intersection graph of C is a tree.

The morphism φ : C → C is an isomorphism outside 2g regular points Q₁, Q₁′, Q_g, Q_g′ and identifies Q_i with Q_j′. is uniquely determined by the g pairs of regular κ-rational points (Q_i, Q_i′). A curve C satisfying (i)-(iii) together with n κ-rational regular points on it is called a n-pointed tree of projective lines. C is stable if on every component there are at least three points which are either singular or marked. The object of this paper is the classification of stable n-pointed trees. We prove in particular the existence of a fine moduli space B_n of stable n-pointed trees. The discussion above shows that there is a surjective map πB_2g → D_g of B_2g onto the closed subscheme D_g of the coarse moduli scheme M_g of stable curves of genus g corresponding to the totally degenerate curves. By the universal property of M_g, π is a (finite) morphism. π factors through B_2g = B_2g mod the action of the group of pair preserving permutations of 2g elements (a group of order 2^gg, isomorphic to a wreath product of S_g and ℤ/2ℤThe induced morphism π: B_2g → D_g is an isomorphism on the open subscheme of irreducible curves in D_g, but in general there may be nonequivalent choices of g singular points on a totally degenerated curve for the above construction, so π has nontrivial fibres. In particular, π is not the quotient map for a group action on B_2g. This leads to the idea of constructing a Teichmüller space for totally degenerate curves whose irreducible components are isomorphic to B_2g and on which a discontinuous group acts such that the quotient is precisely D_g; π will then be the restriction of this quotient map to a single irreducible component. This theory will be developped in a subsequent paper.In this paper we only consider stable n-pointed trees and their moduli theory. In 4 1 we introduce the abstract cross ratio of four points (not necessarily on the same projective line) and show that for a field κ the κ-valued points in the projective variety B_n of cross ratios are in 1 − 1 correspondence with the isomorphy classes of stable n-pointed trees of projective lines over κ. We also describe the structure of the subvarieties B(T, ψ) of stable n-pointed trees with fixed combinatorial type.We generalize our notion in 4 2 to stable n-pointed trees of projective lines over an arbitrary noetherian base scheme S and show how the cross ratios for the fibres fit together to morphisms on S. This section is closely related to [Kn], but it is more elementary since we deal with a special case.4 3 contains the main result of the paper: the canonical projection B_{n + 1} → B_n is the universal family of stable n-pointed trees. As a by-product of the proof we find that B_n is a smooth projective scheme of relative dimension 2n - 3 over ℤ. We also compare B_n to the fibre product B_n−1 × _Bn-2 B_{n − 1} and investigate the singularities of the latter.In 4 4 we prove that the Picard group of B_n is free of rank 2ⁿ⁻1−(n+1)−n(n−3)/2.We also give a method to compute the Betti numbers of the complex manifold B_n(ℂ).In 4 5 we compare B_n to the quotient Q_n: = ℙ_ssⁿ/PGL₂ of semi-stable points in ℙ₁ⁿ for the action of fractional linear transformations in every component. This orbit space has been studied in greater detail by several authors, see [GIT], [MS], [G]. It turns out that B_n is a blow-up of Q_n, and we describe the blow-up in several steps where at each stage the obtained space is interpreted as a solution to a certain moduli problem.