It is well known that the second homology group of any Kac–Moody Lie algebra and the Virasoro algebra is trivial. This is equivalent to saying that any Kac–Moody Lie algebra (or the Virasoro algebra) is its own universal covering algebra; see, for example, [1] and [4]. Recently, Loday and Pirashvili in [12] proved that the second Leibniz homology group of the Virasoro algebra is also trivial. The Leibniz homology is a noncommutative analogue of Lie algebra homology developed by J-L. Loday (see [11] and [12]). It can be extended for a larger class of algebras called Leibniz algebras.

In this paper, we study the second Leibniz homology group HL2(g(A)) of a Kac–Moody Lie algebra g(A). We shall show that HL2(g(A)) = 0 for any non-affine Kac–Moody algebra g(A), but HL2(g(A)) ≠ 0 for any affine Kac–Moody algebra g(A). In the latter case, we use Wilson's idea [15] to carry out HL2(g(A)) explicitly in terms of certain subspaces of Kähler differentials over the Laurent polynomial ring. More precisely, we have the following result.

In this paper, we give an elementary proof of a curious identity of elliptic functions. It is very similar to a beautiful proof given by Coates of a different identity. The result was strongly motivated by Wildeshaus' generalisation of Zagier's conjecture.

Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the Euler–Poincaré characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [Ascr ], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [Ascr ] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts.

Our formula for the equivariant Euler characteristic over [Ascr ] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper.

Let p, q be distinct odd primes, and let a, b be positive integers. In this paper we prove that if S(pa, qb) is a Storer difference set with the parameters ν = paqb, k = (ν−1)/4 and λ =(ν−5)/16, then we have a = b = 1, p =(ρ3r+ρ−3r−1)/3 and q = ρ3r+ρ−3r+1, where ρ = 2+√3, ρ− = 2 − √3, and r is a positive integer.

Let p be a prime number, and let G be a p-group which is not elementary abelian. For every mod-p cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξp = 0. This gives upper bounds for nilpotency degrees of such classes of G and of nilpotent mod-p cohomology classes of finite groups.

We define a Henstock-type integral on measure spaces with metric topologies, and give an example of a function which is integrable but whose absolute value is not.

Two new families of commutative semifields of order q4 with middle nucleus GF(q2), where q is an odd prime power, are defined. One family occurs when q ≡ 3 (mod 4) and the other when q ≡ 1 (mod 4).

We prove that a finite solvable group whose order is divisible by a prime p has at least 2√p−1 conjugacy classes.

In this paper we obtain the precise structure of right nonsingular semiperfect right CS-rings. It is shown that such rings are direct sums of certain specific block matrix rings.

Keedwell has shown that none of the groups of order less than 5 has a weak uniquely completable set. We prove that a weak uniquely completable set exists in a latin square based on a finite group if and only if the group is of order greater than 5.

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <[mid ]X[mid ], the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory.

We show that the norms of the sequence of spectral projections of certain non-self-adjoint ordinary differential operators cannot be polynomially bounded. The proof involves the investigation of ‘tame’ bi-orthogonal systems in Banach spaces.

We consider the equation

a1x1 + ··· + anxn = 1, (1.1)

where the coefficients a1, …, an are non-zero elements of a field K, to be solved for (x1, …, xn) in a finitely generated multiplicative subgroup G of (K*)n. Evertse [2] and van der Poorten and Schlickewei [6] showed, independently, for the most important case in which K is a number field and G is a group of S-units in K, that (1.1) has only finitely many solutions (x1, …, xn) for which no subsum in (1.1) vanishes. Such solutions are called non-degenerate.

In 1990, Schlickewei [7] was the first to obtain an explicit upper bound for the number of non-degenerate solutions of (1.1) when G is a group of S-units of K. His result went through various successive improvements, and the best result to date is the bound (235n2)n3s by Evertse [3, Theorem 3], where s is the cardinality of S. Very recently, Evertse, Schlickewei and Schmidt [4, 5] obtained a remarkable result. They showed, again for K a number field, that the number of non-degenerate solutions of (1.1) with (x1, …, xn) in a finitely generated subgroup of (K*)n of rank r is at most c(n)r+2, with c(n) = exp((6n)4n). The importance of this result lies with its uniformity with respect to the rank r of the group and its independence of the field K. The proofs of the above-mentioned results are all quite difficult and depend on deep tools from Schmidt's Subspace Theorem and diophantine approximation. In this paper we consider equation (1.1) in the rational function field K = k(t) where k is an algebraically closed field of characteristic 0. Of course, results of this type will follow rather easily from the above-mentioned results by a specialization argument. However, our object is to show that our approach to the function field case is quite elementary and certainly very different, being a simple and direct consequence of the powerful abc-theorem.

Before stating our result, we first define proportional solutions of (1.1). We say that (x1, …, xn) and (x′1, …, x′n) are proportional if xi, /, x′i ∈ k, 1 [les ] i [les ] n. This determines equivalence classes of solutions.

Let v be a henselian valuation of a field K. In this paper it is proved that any finite extension (K′, v′) of (K, v) is tame if and only if there exists α ≠ 0 in K′ such that v′(α) = v(TrK′/K(α)) using elementary results of valuation theory. A special case of this result, when the characteristic of the residue field of v is p > 0 and (K′, v′)/(K, v) is an extension of degree p, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math. 404 (1990) 1–38).

We provide a new lower bound for the greatest prime factor of the norm of algebraic numbers of the form axm+byn. The improvement concerns the dependence on the number field containing a, b, x and y.

Let F be a field, and let Σn be the symmetric group on n letters. In this paper we address the following question: given two irreducible FΣn-modules D1 and D2 of dimensions greater than 1, can it happen that D1 [otimes ] D2 is irreducible? The answer is known to be ‘no’ if char F = 0 [12] (see also [2] for some generalizations). So we assume from now on that F has positive characteristic p. The following conjecture was made in [4].

CONJECTURE. Let D1and D2be two irreducible FΣn-modules of dimensions greater than 1. Then D1 [otimes ] D2 is irreducible if and only if p = 2, n = 2 + 4l for some positive integer l; one of the modules corresponds to the partition (2l + 2, 2l) and the other corresponds to a partition of the form (n − 2j − 1, 2j + 1), 0 [les ] j < l. Moreover, in the exceptional cases, one has

formula here

The main result of this paper is the following theorem, which establishes a big part of the conjecture.

It is shown that a rational function of degree [ges ] 2 admits an invariant line field with respect to some measure μ, which is an equilibrium state of a Hölder continuous potential whose topological pressure is greater than its supremum, only in very special cases when the Julia set is either a geometric circle or an interval, or totally disconnected and contained in a real-analytic curve.

We prove uniform and tangential approximation theorems for superharmonic functions in abstract harmonic spaces. Our tangential approximation theorem differs from traditional ones by the absence of the so-called long islands condition. The result is new also in the case of the classical potential theory.

In this note we wish to prove a purely characteristic p > 0 variant of the Kodaira–Akizuki–Nakano vanishing for smooth complete intersections of dimension at least two in projective space. This has some interesting applications; in particular, we show that all Frobenius pull-backs of the tangent bundle of any complete intersection of general type and of dimension at least three in Pn are stable. We also show (see Remark 3.4) that a small modification of the techniques of [5] and a theorem of Mehta and Ramanathan (see [3]) together allow us to extend this stability result to smooth projective hypersurfaces of degree d, where (n + 1)/2 < d < n + 1 (that is, to some Fano hypersurfaces).

It is well known that behaviour of stability under Frobenius pull-backs is a subtle problem of the theory of vector bundles in characteristic p > 0, and hence this result is not without interest. We end with an obvious conjectural form of our variant for a general class of varieties.

Let f[ratio ]C→Ĉ be a function which is either transcendental meromorphic or rational with degree at least 2. We discuss the uniform perfectness of the attracting or parabolic cycle of stable domains of f(z), and include a proof that the Julia set of a meromorphic function of finite type is uniformly perfect.