Isomorphism Classes of Algebras with Radical Cube Zero II

We study to classify, up to isomorphism, algebras Λ over a field k such that the radical cubed is zero and Λ modulo the radical is a product of copies of k. The number of local quasi-Frobenius k-algebras with the condition is shown to be not less than the cardinality of k. In particular, the canonical forms of those algebras of dimension 5 are presented and their isomorphism classes are completely determined under some conditions on k.      

Composition algebras and outer automorphisms of algebraic groups via triality

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form n over a field k of characteristic not two, and a category arising from an action of the projective similarity group of n on certain pairs of automorphisms of the group scheme PGO⁺(n) defined over k. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.     

The Module Decomposition of Super-Symmetric Powers of Matrices

ABSTRACT

Let M be the k  ×  m matrices over ?. The GL ( k ) ×  GL ( m ) decompositions of the symmetric and of the exterior powers of M are described by two classical theorems. We describe a theorem for Lie superalgebras, which implies both of these classical theorems as special cases. The constructions of both the exterior and the symmetric algebras are generalized to a class of algebras defined by partitions. That superalgebra theorem is further generalized to these algebras.     

Mutation of Auslander Generators for Selfinjective Algebras

Let Λ be an artin algebra with representation dimension equal to three and M an Auslander generator of Λ. We show how, under certain assumptions, we can mutate M to get a new Auslander generator whose endomorphism ring is derived equivalent to the endomorphism ring of M. We apply our results to selfinjective algebras with radical cube zero of infinite representation type, where we construct an infinite set of Auslander generators.     

The Coradical Filtration for Drinfeld Double Ringel-Hall Algebras

Abstract

Let 𝒟(Λ) be the Drinfeld double Ringel-Hall algebra with Λ being any finite dimensional hereditary algebra over a finite field k. We determine the coradical filtration for 𝒟(Λ). As an application, we describe the group of Hopf algebra automorphisms of the Drinfeld double Ringel composition algebra of Λ.     

Rank one quotients of vector groups

ABSTRACT

In this paper we study the possible torsion in even-dimensional higher class groups Cl _2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 _F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl _2n (Λ) can only occur for primes p dividing prime ideals ? of 𝒪 _F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.     

Automorphic equivalence in the varieties of representations of Lie algebras

Abstract

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.     

POLYNOMIALS RELATED TO HALL NUMBERS

Let Λ be a finite dimensional algebra of finite representation type over a finite field k. For any modules A, B and Pin mod Λ with P projective, we prove that there exists a polynomial ? ^B _(P)Λ over Z whose evaluation at |E| for any conservative finite field extension E of Λ is the sum of Hall numbers F ^{B E} _{C ^E A ^E} where C ^E runs through isoclasses in mod Λ ^E and P ^E is the projective cover of C ^E . As a consequence of this result and its dual version, Hall polynomials ? ^E _CA exist when C or A is semisimple. As applications of the main result, we obtain the existence of Hall polynomials for Nakayama algebras and some selfinjective algebras.

     

On Jordan-Nilalgebras of Index 3

We study commutative algebras that satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a new bound of the index of nilpotency. We prove that every commutative nilalgebra of nilindex 3 generated by k elements over a field of characteristic ≠ 2, 3 is nilpotent of index less than or equal to k + 5.     

Static Subcategories of the Module Category of a Finite-Dimensional Hereditary Algebra

Let k be a field, Λ a finite-dimensional hereditary k-algebra, and modΛ the category of all finite-dimensional Λ-modules. We are going to characterize the representation type of Λ (tame or wild) in terms of the possible subcategories statM of all M-static modules, where M is an indecomposable Λ-module.     

A Generalization of a Two-Parameter Quantization for the Group GL 2(k)

We generalize a well-known two-parameter quantization for the group GL ₂(k) (over an arbitrary field k). Specifically, a certain class of Hopf algebras is constructed containing that quantization. The algebras are constructed given an arbitrary coalgebra and an arbitrary pair of its commuting anti-isomorphisms, and are defined by quadratic relations. They are densely linked to the compact quantum groups introduced by Woronowicz. We give examples of Hopf algebras that can be rowed up to the two-parameter quantization for GL ₂(k).     

Ramified partition algebras

For each natural number n, poset T, and |T|–tuple of scalars Q, we introduce the ramified partition algebra P _n ^(T) (Q), which is a physically motivated and natural generalization of the partition algebra [24, 25] (the partition algebra coincides with case |T|=1). For fixed n and T these algebras, like the partition algebra, have a basis independent of Q. We investigate their representation theory in case ${{T=\underline{{2}}:=({1,2},\leq)}}$. We show that ${{P_n^{(\underline{{2}})$ (Q) is quasi–hereditary over field k when Q ₁ Q ₂ is invertible in k and k is such that certain finite group algebras over k are semisimple (e.g. when k is algebraically closed, characteristic zero). Under these conditions we determine an index set for simple modules of ${{P_n^{(\underline{{2}})$ (Q), and construct standard modules with this index set. We show that there are unboundedly many choices of Q such that ${{P_n^{(\underline{{2}})$ (Q) is not semisimple for sufficiently large n, but that it is generically semisimple for all n. We construct tensor space representations of certain non–semisimple specializations of ${{P_n^{(\underline{{2}})$ (Q), and show how to use these to build clock model transfer matrices [24] in arbitrary physical dimensions. Sadly Ahmed died before this work was completed. His memory lives on.     

Classification of Hopf algebras of dimension 18

This paper contributes to the classification problems of finite dimensional Hopf algebras H over an algebraically closed field k of characteristic zero. It is shown that for a non-semisimple Hopf algebra H of dimension 18 either H or H* is pointed.     

Basic Hopf algebras and quantum groups

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers given in terms of a finite group G and sequence of elements of G in the following way can occur. The quiver has vertices and arrows , where the set is closed under conjugation with elements in G and for each g in G, the sequences W and are the same up to a permutation. We show how is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R]. Received August 15, 1994; in final form May 16, 1997     

Endoprimal algebras

An algebra A is endoprimal if, for all the only maps from A ^k to A which preserve the endomorphisms of A are its term functions. One method for finding finite endoprimal algebras is via the theory of natural dualities since an endodualisable algebra is necessarily endoprimal. General results on endoprimality and endodualisability are proved and then applied to the varieties of sets, vector spaces, distributive lattices, Boolean algebras, Stone algebras, Heyting algebras, semilattices and abelian groups. In many classes the finite endoprimal algebras turn out to be endodualisable. We show that this fails in general by proving that , regarded as either a bounded semilattice or upper-bounded semilattice is dualisable, endoprimal but not endodualisable. Received May 16, 1997; accepted in final form November 6, 1997.     

Intermediate Growth of Solvable Lie Superalgebras

Finitely generated solvable Lie algebras have an intermediate growth between polynomial and exponential. Recently the second author suggested the scale to measure such an intermediate growth of Lie algebras. The growth was specified for solvable Lie algebras F(A ^q , k) with a finite number of generators k, and which are free with respect to a fixed solubility length q. Later, an application of generating functions allowed us to obtain more precise asymptotic. These results were obtained in the generality of polynilpotent Lie algebras. Now we consider the case of Lie superalgebras; we announce that main results and describe the methods. Our goal is to compute the growth for F(A ^q , m, k), the free solvable Lie superalgebra of length q with m even and k odd generators. The proof is based upon a precise formula of the generating function for this algebra obtained earlier. The result is obtained in the generality of free polynilpotent Lie superalgebras. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Sharply transitive linear algebraic groups

We study Zariski-closed linear groupsG GL _n (k) over fieldsk of characteristic 0 which act sharply transitively on the non-zero vectors ofk ⁿ . For square-freen, orn15, or ifk has cohomological dimension 1 we obtain a complete classification (i.e. a reduction to questions about associative division algebras). The main tools are representation theory of Lie algebras over algebraically closed and non-closed fields, and results about simple associative algebras in order to control the interplay between linear Lie algebras and the associative algebras generated by them. The relation to nearfields and left-symmetric division algebras is also discussed.     

On the Hopf–Schur group of a field

Let k be any field. We consider the Hopf–Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf–Schur group can be much larger than the Schur group of k.