Projective Extensions of Fields

Every field K admits proper projective extensions, that is,Galois extensions where the Galois group is a non-trivial projectivegroup, unless K is separably closed or K is a pythagorean formallyreal field without cyclic extensions of odd degree. As a consequence,it turns out that almost all absolute Galois groups decomposeas proper semidirect products. We show that each local field has a unique maximal projectiveextension, and that the same holds for each global field ofpositive characteristic. In characteristic 0, we prove thatLeopoldt's conjecture for all totally real number fields isequivalent to the statement that, for all totally real numberfields, all projective extensions are cyclotomic. So the realizabilityof any non-procyclic projective group as Galois group over Qproduces counterexamples to the Leopoldt conjecture.     

The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.     

Algebras whose right nucleus is a central simple algebra

We generalize Amitsur's construction of central simple algebras over a field F which are split by field extensions possessing a derivation with field of constants F to nonassociative algebras: for every central division algebra D over a field F of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is D and whose left and middle nucleus are a field extension K of F splitting D, where F is algebraically closed in K.We then give a short direct proof that every p-algebra of degree m, which has a purely inseparable splitting field K of degree m and exponent one, is a differential extension of K and cyclic. We obtain finite-dimensional division algebras over a field F of characteristic $p>0$ whose right nucleus is a division p-algebra.     

The Cartan matrix of the Schur algebra S(2, r)

Let k be an infinite field of prime characteristic and let r be a positive integer. Using admissible decompositions, we determine explicitly the entries of the decomposition matrix of the Schur algebra S(2, r) over k and prove that any two blocks with the same number of simple modules have the same decomposition matrix and hence the same Cartan matrix.     

A proof of the strong no loop conjecture

The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field.     

The Schur group of an abelian number field

We characterize the maximum r-local index of a Schur algebra over an abelian number field K in terms of global information determined by the field K for an arbitrary rational prime, r. This completes and unifies previous results of Janusz in [G.J. Janusz, The Schur group of an algebraic number field, Ann. of Math. (2) 103 (1976) 253-281] and Pendergrass in [J.W. Pendergrass, The 2-part of the Schur group, J. Algebra 41 (1976) 422-438].     

Factor Algebras of Free Algebras: On a Problem of G. Bergman

Let A_n = K x₁,...,x_n be a free associative algebra over a fieldK. In this paper, examples are given of elements u A_n, n 3,such that the factor algebra of A_n over the ideal generatedby u is isomorphic to A_n–1, and yet u is not a primitiveelement of A_n (that is, it cannot be taken to x₁ by an automorphismof A_n). If the characteristic of the ground field K is 0, thisyields a negative answer to a question of G. Bergman. On theother hand, by a result of Drensky and Yu, there is no suchexample for n = 2. It should be noted that a similar questionfor polynomial algebras, known as the embedding conjecture ofAbhyankar and Sathaye, is a major open problem in affine algebraicgeometry. 2000 Mathematics Subject Classification 16S10, 16W20(primary); 14A05, 13B25 (secondary).     

The Composition Algebra and the Composition Monoid of the Kronecker Quiver

The Kronecker quiver K is considered, and the relations forthe specialisation at q = 0 of the generic composition algebraare given, as well as those for Reineke's composition monoid.As a corollary, it is deduced that the composition monoid isa proper factor of the specialisation of the composition algebra.A normal form is also obtained for the varieties occurring inthe composition monoid in terms of Schur roots.     

On τ-tilting finiteness of the Schur algebra

We determined the τ-tilting finiteness of Schur algebras over an algebraically closed field of arbitrary characteristic, except for a few small cases.     

Bosonic and fermionic representations of Lie algebra central extensions

Given any representation of an arbitrary Lie algebra g over a field K of characteristic 0, we construct representations of a central extension of g on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H²(g;K). We illustrate these techniques with several concrete examples.     

Embedding Properties of Metabelian Lie Algebras and Metabelian Discrete Groups

We show that for every natural number m a finitely generatedmetabelian group G embeds in a quotient of a metabelian groupof type FP_m. Furthermore, if m 4, the group G can be embeddedin a metabelian group of type FP_m. For L a finitely generatedmetabelian Lie algebra over a field K and a natural number mwe show that, provided the characteristic p of K is 0 or p >m, then L can be embedded in a metabelian Lie algebra of typeFP_m. This result is the best possible as for 0 < p m everymetabelian Lie algebra over K of type FP_m is finite dimensionalas a vector space.     

Bimodule resolution of a group algebra

Relationship between the bimodule resolution of the group algebra of a finite group G over a commutative ring and the usual projective resolution of the trivial G-module is studied. In particular, an analog of Happel’s lemma is proved; this lemma has been established earlier for finite-dimensional algebras over fields. As an example of application of the results, the bimodule resolution is constructed for the integer group ring of the dihedral group of order 4m. Bibliography: 10 titles.     

Integral Schur algebras forGL 2

The structure of Schur algebrasS(2,r) over the integral domainZ is intensively studied from the quasi-hereditary algebra point of view. We introduce certain new bases forS(2,r) and show that the Schur algebraS(2,r) modulo any ideal in the defining sequence is still such a Schur algebra of lower degree inr. A Wedderburn-Artin decomposition ofS _K (2,r) over a fieldK of characteristic 0 is described. Finally, we investigate the extension groups between two Weyl modules and classify the indecomposable Weyl-filtered modules for the Schur algebrasS _Zp(2,r) withr<p ² . Research supported by ARC Large Grant L20.24210     

An Invariant for Fraction Fields of McConnell Algebras

In their seminal work [1] on the fields of fractions of theenveloping algebra of an algebraic Lie algebra, Gel'fand andKirillov formulate the following conjecture. Assume that g isa finite-dimensional algebraic Lie algebra over a field of characteristiczero. Then D(g) is a Weyl skew-field over a purely transcendentalextension of the base field. They showed that neither the conjecture nor its negation holdsfor all non-algebraic algebras. In [2], A. Joseph gave a particularlyeasy non-algebraic counterexample devised by L. Makar-Limanov:this is a non-algebraic 5-dimensional solvable Lie algebra,providing a counterexample despite the fact that the centreis one-dimensional. Besides, he raised a question of generalizationof this method for any completely solvable Lie algebra. On the other hand, consider A(V, , ), the McConnell algebrafor the triple (V, , ) as defined in [4, 14.8.4] and below.McConnell in [3] described the completely prime quotients ofthe enveloping algebra of a solvable Lie algebra in terms ofA(V, , ), and found a complete set of invariants to separatethem. In [2], A. Joseph raised the question whether the fieldsof fractions of these McConnell algebras remain non-isomorphic.The purpose of this note is to extend the work of L. Makar-Limanovreported in [2, Section 6], and so provide an integer-valuedinvariant which, for McConnell algebras defined over Z, saysprecisely when this skew-field is isomorphic to a Weyl skew-field:this number has simply to be positive. This result thereforegives a large supply of skew-fields which ‘resemble’a Weyl skew-field very nearly, but nevertheless are not isomorphicto it. 1991 Mathematics Subject Classification 17B35.     

Plactic algebra of rank 3

The structure of the algebra K[M] of the plactic monoid M of rank 3 over a field K is studied. The minimal prime ideals of K[M] are described. There are only two such ideals and each of them is a principal ideal determined by a homogeneous congruence on M. Moreover, in case K is uncountable and algebraically closed, the left and right primitive spectrum and the corresponding irreducible representations of the algebra K[M] are described. All these representations are monomial. As an application, a new proof of the semiprimitivity of K[M] is given.     

The Linear Coordinate Preserving Problem

We prove that every K-endomorphism of a rank 2 polynomial algebra over an algebraically closed field K of positive characteristic taking all linear coordinates to coordinates is an automorphism. We give a new characterization of coordinates of K[t][x, y], where K is an algebraically closed field of any characteristic. We also explore the close connection between coordinates and permutation polynomials of finite fields.     

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let Z be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g^∗ with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G₂.     

An explicit example of a noncrossed product division algebra

The paper presents an explicit example of a noncrossed product division algebra of index and exponent 8 over the field ?(s)(t). It is an iterated twisted function field in two variables D(x, σ)(y, τ ) over a quaternion division algebra D which is defined over the number field ?(√3,√?7). The automorphisms σ and τ are computed by solving relative norm equations in extensions of number fields. The example is explicit in the sense that its structure constants are known. Moreover, it is pointed out that the same arguments also yield another example, this time over the field ?((s))((t)), given by an iterated twisted Laurent series ring D((x, σ))((y, τ )) over the same quaternion division algebra D. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)