Complements to the Almost Complete Tilting A (m)-Modules

Let A be a finite dimensional hereditary algebra over an algebraically closed field and A ^(m) be the mth replicated algebra of A. We prove that if T is a faithful almost complete tilting A ^(m)-module with pd _{A ^(m)} T ≤ m, then T has exactly m + 1 indecomposable nonisomorphic complements with projective dimensions at most m. Moreover, we give an explicit distribution of the complements to T.     

Actions of the dipper-donkin quantization GL 2 on the clifford algebra C(l,3)

Following the method already developed for studying the actions of GL_q (2,C) on the Clifford algebra C(l,3) and its quantum invariants [1], we study the action on C(l, 3) of the quantum GL ₂ constructed by Dipper and Donkin [2]. We are able of proving that there exits only two non-equivalent cases of actions with nontrivial “perturbation” [1]. The spaces of invariants are trivial in both cases.

We also prove that each irreducible finite dimensional algebra representation of the quantum GL ₂ q^m ≠1, is one dimensional.

By studying the cases with zero “perturbation” we find that the cases with nonzero “perturbation” are the only ones with maximal possible dimension for the operator algebra ?.     

Rank-one nonincreasing mappings on triangular matrices and some related preserver problems

Let T_n (F) be the algebra of all n×n upper triangular matrices over an arbitrary field F. We first characterize those rank-one nonincreasing mappings ψ: T_n (F)→T_m (F)n?m such that ψ(I_n ) is of rank n. We next deduce from this result certain types of singular rank-one r-potent preservers and nonzero r-potent preservers on T_n (F). Characterizations of certain classes of homomorphisms and semi-homomorphisms on T_n (F) are also given.     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

The Loewy Length of the Descent Algebra of Type <Emphasis Type="Italic">D</Emphasis>

In this article the Loewy length of the descent algebra of D _2m + 1 is shown to be m + 2, for m ≥ 2, by providing an upper bound that agrees with the lower bound in Bonnafé and Pfeiffer (2006). The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D _2m + 1 is at most m + 1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group W is identified with an algebra associated to the reflection arrangement of W.     

Toeplitz operators on the Fock space: Radial component effects

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT _a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT _a such that spT _a =ess-spT _a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.     

L ∞ Structures on Spaces with Three One-Dimensional Components

Abstract

L _∞ structures have been a subject of recent interest in physics, where they occur in closed string theory and in gauge theory. This paper provides a class of easily constructible examples of L _n and L _∞ structures on graded vector spaces with three one-dimensional components. In particular, it demonstrates a way to classify all possible L _n and L _∞ structures on V?=?V _m ?⊕?V _m+1?⊕?V _m+2 when each of the three components is one-dimensional. Included are necessary and sufficient conditions under which a space with an L ₃ structure is a differential graded Lie algebra. It is also shown that some of these differential graded Lie algebras possess a nontrivial L _n structure for higher n.     

On the number of homogeneous models of a given power

It is shown that, For each complete theoryT, the nomberh _T(m) of homogeneous models ofT of powerm is a non-increasing function of uncountabel cardinalsm Moreover, ifh _T(ℵ₀)≦ℵ₀, then the functionh _T is also non-increasing ℵ₀ to ℵ₁. This work was supported in part by NSF contracts GP 4257 and GP5913.     

Marcinkiewicz multiplier theorem for the Weyl functional calculus

Let {X _j , Y _j , T : 1 ≤ j ≤ n} be a basis satisfying the commutation relation for the Heisenberg Lie algebra . Then we obtain a multi-parameter Marcinkiewicz multiplier theorem for the operator defined by m(X ₁,..., X _n , Y ₁,..., Y _n , T).     

Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

Differential operators and BV structures in noncommutative geometry

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the commutative algebra are replaced by \mathbbDer(A){\mathbb{D}{\rm er}(A)}, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A){\mathcal{F}(A)}, a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)){\mathcal{D}(\mathcal{F}(A))} of differential operators is filtered and gr D(F(A)){\mathcal{D}(\mathcal{F}(A))}, the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr D(F(A)){\mathcal{D}(\mathcal{F}(A))} is closely related to the double Poisson structure on T_A \mathbbDer(A){T_{A} \mathbb{D}{\rm er}(A)} introduced by Van den Bergh. Specifically, we prove that gr D(F(A)) @ F(T_A(\mathbbDer(A)),{\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),} provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space F(A){\mathcal{F}(A)} is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras. In this language, D(F(A)){\mathcal{D}(\mathcal{F}(A))} becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper, we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on \mathbbDer(A){\mathbb{D}{\rm er}(A)} gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator, that makes F(T_A(\mathbbDer(A))){\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))} a BV wheelgebra. In the final section, we explain how the wheeled differential operators D(F(A)){\mathcal{D}(\mathcal{F}(A))} produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1.     

Preconditioners for block Toeplitz systems based on circulant preconditioners

We study the numerical solution of a block system T _m,n x=b by preconditioned conjugate gradient methods where T _m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q ₁(T _m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q ₂(T _m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.     

Specht property for the 2-graded identities of the Jordan algebra of a bilinear form

Let K be a field of characteristic zero and V a vector space of dimension m>1 with a nondegenerate symmetric bilinear form f:V×V→K. The Jordan algebra B_m = K⊕V of the form f is a ?₂-graded algebra with this decomposition. We prove that the ideal of all the ?₂-graded identities of B_m satisfies the Specht property and we compute the ?₂-graded cocharacter sequence of B_m.     

Rees algebras with minimal multiplicity

Let (R,m) be a local ring. Let S_M denote the Rees algebra S=R[m^rt] localized at its unique maximal homogeneous ideal M=(m,m^rt). Let TN denote the extended Rees algebra T= R[m^rt, t^-1] localized at its unique maximal homogeneous idea N= (t^?1,m,m^r). Multiplicity formulas are developedfor S_M and T_N. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that S_M and T_N are Cohen-Macaulay with minimal multiplicity     

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x ₁, x ₂, . . . , x _n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x ₁, x ₂, . . . , x _n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.