Forms of pointed Hopf algebras

Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p ² and p ³, with p odd. Received: 22 November 1998     

Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

In this paper, we investigate the tensor structure of the category of finite- dimensional weight modules over the Hopf–Ore extensions kG(χ^?1,a,0) of group algebras kG. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that k is an algebraically closed field of characteristic zero, and the orders of χ and χ(a) are the same.     

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G ⁿ , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.     

Isotopy and invariants of Albert algebras

Let k be a field with characteristic different from 2 and 3. Let B be a central simple algebra of degree 3 over a quadratic extension K/k, which admits involutions of second kind. In this paper, we prove that if the Albert algebras and have same and invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with , and . We conclude with a construction of Albert division algebras, which are pure second Tits' constructions. Received: December 9, 1997.     

Trace Invariants of Finite-Dimensional Algebras

With every finite-dimensional algebra A over any field k we associate an 8-tuple of linear or bilinear forms on A, all of which are defined in terms of traces. For every groupoid 𝒞 formed by a class of k-algebras of fixed finite dimension, this passage is functorial and, when composed with any map that is constant on isoclasses, gives rise to an abundance of maps f: 𝒞 → I such that the fibres of f form a block decomposition of 𝒞. We study this decomposition for specific choices of 𝒞 and f, thereby putting established results from diverse algebraic theories into a unifying perspective, but also gaining new insight into classical groupoids of algebras such as associative unital algebras, division algebras, or composition algebras over general ground fields.     

Varieties of solvable index-two alternative algebras over a field of characteristic three

The subvarieties of the variety Alt₂ of solvable index-two alternative algebras over an arbitrary field of characteristic 3 are studied. The main types of such varieties are singled out in the language of identities, and inclusions between these types are established. The main results is the following.Theorem.The topological rank of the variety Alt₂ of solvable index-two alternative algebras over an arbitrary field of characteristic 3 is equal to five. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 556–566, October, 1999.     

Etude de la categorie des algebres de Hopf commutatives connexes sur un corps

Let k be a perfect field of characteristic p0; the categoryH of connected abelian Hopf algebras over k is abelian and locally noetherian. Technics of locally noetherian categories are used here to obtain Krull and homological dimensions ofH (which are respectively 1 and 2), and a decomposition ofH in a product of categories. First we have, whereH ^– is the category of Grassman algebras, andH ⁺ consists of Hopf algebras which are zero in odd degrees; then we prove thatH ⁺ itself is a product of isomorphic categoriesH _n, n^*, and we give an equivalence betweenH _n and a category of modules. This is compared to some results of algebraic geometry about Greenberg modules.     

Examples of Sweedler Cohomology in Positive Characteristic

There have been few examples of computations of Sweedler cohomology, or its generalization in low degrees known as lazy cohomology, for Hopf algebras of positive characteristic. In this paper we first provide a detailed calculation of the Sweedler cohomology of the algebra of functions on (?/2) ^r , in all degrees, over a field of characteristic 2. Here the result is strikingly different from the characteristic zero analog.

Then we show that there is a variant in characteristic p of the result obtained by Kassel and the author in characteristic zero, which provides a near-complete calculation of the second lazy cohomology group in the case of function algebras over a finite group; in positive characteristic, the statement is, rather surprisingly, simpler.     

On the structure of a relatively free Grassmann algebra

We investigate the multiplicative and T-space structure of the relatively free algebra F ⁽³⁾ with a unity corresponding to the identity [[x ₁ , x ₂], x ₃] = 0 over an infinite field of characteristic p > 0. The highest emphasis is placed on unitary closed T-spaces over a field of characteristic p > 2. We construct a diagram containing all basic T-spaces of the algebra F ⁽³⁾, which form infinite chains of the inclusions. One of the main results is the decomposition of quotient T-spaces connected with F ⁽³⁾ into a direct sum of simple components. Also, the studied T-spaces are commutative subalgebras of F ⁽³⁾; thus, the structure of F(3) and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics of the case p = 2. In the Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic.     

Growth in Poisson algebras

A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x ₁, y ₁}, {x ₂, y ₂}, . . . , {x _m , y _m }} = 0 and {x ₁, y ₁} · {x ₂, y ₂} · . . . · {x _m , y _m } = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.     

Matrix relation algebras

Square matrices over a relation algebra are relation algebras in a natural way. We show that for fixed n, these algebras can be characterized as reducts of some richer kind of algebra. Hence for fixed n, the class of n × n matrix relation algebras has a first–order characterization. As a consequence, homomorphic images and proper extensions of matrix relation algebras are isomorphic to matrix relation algebras. Received July 18, 2001; accepted in final form April 24, 2002.     

Hochschild cohomology of algebras of quaternionic type: The family Q(2ℬ)<Subscript>1</Subscript>(<Emphasis Type="Italic">k,s, a,c</Emphasis>) over a field of characteristic different from 2

In a previous joint paper of the author with A.I. Generalov and S.O. Ivanov, the Hochschild cohomology algebra of quaternionic-type algebras from the family Q(2ℬ)₁ over an algebraically closed field of characteristic 2 was calculated. In this paper, the Hochschild cohomology groups of algebras from this family over an algebraically closed field of characteristic different from 2 are calculated. As a corollary, the additive structure of the Hochschild cohomology of algebras of type Q(2 $ A $ A ) over a field of characteristic not 2 is described.     

Similarity to symmetric matrices over fields which are not formally real

We show that a matrix is similar to a symmetric matrix over a field of characteristic 2 if and only if the minimum polynomial of the matrix is not the product of distinct irreducible polynomials whose splitting fields are inseparable extensions. When the field is not of characteristic 2, a known theorem is generalized by considering k, the number of elementary divisors of odd degree of the n × n A: If ?1 is a sum of 2^v squares and n differs from a multiple of 2 ^{v + 1} by at most ±k, then A is similar to a symmetric matrix.     

On the notion of essential dimension for algebraic groups

We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our groupG isS _n , these objects are field extensions; ifG=O _n , they are quadratic forms; ifG=PGL _n , they are division algebras (all of degreen); ifG=G ₂, they are octonion algebras; ifG=F ₄, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, relating essential dimension to Galois cohomology.Partially supported by NSA grant MDA904-9610022 and NSF grant DMS-9801675     

AUTOMORPHISMS OF EXCEPTIONAL SIMPLE LIE ALGEBRAS

The scheme describing automorphisms of exceptional simple Lie algebras possessing geometrical realization over an algebraically closed field of characteristic p < 7 is proposed. In particular, automorphisms of Melikyan algebras g(m ₁, m ₂) (p = 5) and Skryabin algebras Y(m ₁, m ₂, m ₃) (p = 3) are found.     

Four-Dimensional Real Third-Power Associative Division Algebras

We study third-power associative division algebras A over a field 𝕂 of characteristic different from 2. Those algebras having dimension ≤2 are commutative. When 𝕂 is the field ? of real numbers, those algebras having dimension 4 are power-commutative in each of the following two cases:

1. A contains a central element;

2. A satisfies the additional identity (x, x³, x) = 0.

     

The p-Modular Descent Algebra of the Symmetric Group

The descent algebra of the symmetric group, over a field ofnon-zero characteristic p, is studied. A homomorphism into thealgebra of generalised p-modular characters of the symmetricgroup is defined. This is then used to determine the radical,and its nilpotency index. It also allows the irreducible representationsof the descent algebra to be described. 1991 Mathematics SubjectClassification 20F32.     

On Auslander-Reiten Quivers of Enveloping Algebras of Restricted Lie Algebras

Let (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≥ 3, X ∈ L* a linear form. In this article we study the Auslander-Reiten quivers of certain blocks of the reduced enveloping algebra u(L,x). In particular, it is shown that the enveloping algebras of supersolvable Lie algebras do not possess AR-components of Euclidean type.