Artinian Hopf algebras are finite dimensional

We prove that an artinian Hopf algebra over a field is finite dimensional. This answers a question of Bergen.

     

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

Amalgamation in finite dimensional cylindric algebras

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA _n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. February 22, 1999.     

Locally finite dimensional lie algebras and their derivation algebras

The derivation algebras of all locally finite dimensional locally simple Lie algebras over a field of characteristic 0 are determined. Every locally finite dimensional Lie algebra of countable dimension is a subalgebra of the outer derivation algebra outder (ℒ) for every Lie algebra ℒ, which is the direct limit of diagonally embedded classical Lie algebras. These outer derivation algebras have dimension ℒ and are never locally finite dimensional. Dedicated to Prof. H. Petersson on the occasion of his 60th birthday     

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. Let T(Λ)=Λ?D(Λ) be the trivial extension of Λ by its minimal injective cogenerator D(Λ). We characterize, in terms of quivers and relations, the algebras Λ^′ such that T(Λ)?T(Λ^′).     

Endomorphisms and null subalgebras of finite dimensional algebras

In [5] it was shown that a finite dimensional binary algebra without nilpotents over an algebraically closed field of characteristic zero has only the trivial derivation and, consequently, a finite automorphism group. It is shown here that an m-ary algebra without nilpotents has only finitely many endomorphisms, with no assumptions on the characteristic of the base field. An upper bound on the dimension of the algebraic set of endomorphisms of a finite dimensional m-ary algebra, depending on the dimension of a maximal null subalgebra, is obtained if the base field is the complex field. The null subalgebras arising from a single derivation are examined for the case of an algebraically closed base field of characteristic zero.     

Relative invariants for representations of finite dimensional algebras

 Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ ^M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ ^M (β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d ^M _β on the variety of modules of dimension vector β, such that d ^M _β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d ^M _β with M indecomposable. Received: 31 July 2001     

On extending real functions to finite dimensional algebras

In a finite dimensional real algebra, we define functional calculus for real functions. The analytic hypothesis is replaced by a differentiable hypothesis. High order differentibility is essentially necessary and sufficient to our discussions. Moreover, we give some examples. Partially supported by NSF of China The authors gratefully acknowledge the support of K. C. Wong Education Foundation, Hong Kong