Representation dimension of exterior algebras

We determine the representation dimension of exterior algebras. This provides the first known examples of representation dimension > 3. We deduce that the Loewy length of the group algebra over F₂ of a finite group is strictly bounded below by the 2-rank of the group (a conjecture of Benson). A key tool is the use of the concept of dimension of a triangulated category.     

Representation dimension of extensions of hereditary algebras

We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of End_H(M)^op, then the representation dimension of is less than or equal to 3 whenever one of the following conditions holds: (i) H is of finite representation type; (ii) H is tame and M is a direct sum of regular and preprojective modules; (iii) M has no self-extensions.     

Quotients of dimension effect algebras

We show that the quotient of a dimension effect algebra by its dimension equivalence relation is a unital bounded lattice-ordered positive partial abelian monoid that satisfies a version of the Riesz decomposition property. For a dimension effect algebra of finite type, the quotient is a centrally orthocomplete Stone–Heyting MV-effect algebra; moreover, an orthocomplete effect algebra in which equality is a dimension equivalence relation is the same thing as a complete Stone–Heyting MV-effect algebra.     

Serre dimension of monoid algebras

Let R be a commutative Noetherian ring of dimension d, M a commutative cancellative torsion-free monoid of rank r and P a finitely generated projective R[M]-module of rank t. Assume M is Φ-simplicial seminormal. If \(M\in \mathcal {C}({\Phi })\), then Serre dim R[M]≤d. If r≤3, then Serre dim R[int(M)]≤d. If \(M\subset \mathbb {Z}_{+}^{2}\) is a normal monoid of rank 2, then Serre dim R[M]≤d. Assume M is c-divisible, d=1 and t≥3. Then P?∧ ^t P⊕R[M] ^t?1. Assume R is a uni-branched affine algebra over an algebraically closed field and d=1. Then P?∧ ^t P⊕R[M] ^t?1.     

Representation dimension and tilting

Let Λ be a finite dimensional k-algebra over an algebraically closed field k and let _ΛT be a splitting tilting module of projective dimension at most 1. Let Γ=End_ΛT. If the representation dimension of Λ is at most 3 then the main result asserts that the representation dimension of Γ does not exceed that of Λ.     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

效应代数的表示理论

The aim of this paper is to characterize representable and weak representable effect algebras and establish a representation theory of effect algebras. An effect algebra E is said to be representable if there exists a Hilbert space H and a monomorphism π from E into the Hilbert space effect algebra ε(H) and it is said to be weakly representable if there exists an injective morphism from E into some ε(H). It is proved that an effect algebra E with the nonempty state space S(E) is representable if and only if x, y ∈ E, f(x)+f(y) ≤ 1 implies x⊕y is defined; it is weakly representable if and only if the state space S(E) separates the points of E. Some operational properties of representable effect algebras are established, and some applications of the obtained results are listed.     

Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.     

On the dimension of graded algebras

To each graded algebra R with a finite number of generators we associate the series T(R, z) = d_nzⁿ, where d_n is the dimension of the homogeneous component of R. It is proved that if the dimensions d_n have polynomial growth, then the Krull dimension of R cannot exceed the order of the pole of the series T(R, z) for z=1 by more than 1.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 209–216, August, 1973.     

Finitistic dimension of standardly stratified algebras

We prove that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.     

Finitistic dimension of properly stratified algebras

We prove that the finitistic dimension of a properly stratified algebra having a simple preserving duality and for which every tilting module is cotilting, equals twice the projective dimension of the characteristic tilting module. As a corollary, we get that the global dimension of a quasi-hereditary algebra with duality equals twice the projective dimension of the characteristic tilting module. As another corollary, we obtain an affirmative answer to the conjecture of Erdmann and Parker. Finally, we calculate the finitistic dimension of the blocks of certain parabolic generalizations of the category .     

Essential dimension of central simple algebras

Let p be a prime integer, 1≤s≤r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed _p (\( Al{{g}_{{{{p}^r},{{p}^s}}}} \)) of the class \( Al{{g}_{{{{p}^r},{{p}^s}}}} \) of central simple algebras of degree p ^r and exponent dividing p ^s . In particular, we show that ed(Alg _8,2)=ed₂(Alg _8,2)=8 and ed _p (\( Al{{g}_{{{{p}^2},p}}} \))=p ²+p for p odd.     

Pointed hopf algebras of dimension 32

We give a complete classification of the 32-dimensional pointed Hopf algebras over an algebraically closed field k with chark k ≠ 2. It turns out that there are infinite families of isomorphism classes of pointed Hopf algebras of dimension 32. In [AS1], [BDG] and [Ge] are given families of counterexamples for the tenth Kaplansky conjecture. Up to now, 32 is the lowest dimension where Kaplansky conjecture fails.