Throughout this paper, $G$ is a connected semisimple algebraic group defined over an algebraically closed field ${\mathbb k}$ of characteristic zero, and ${\frak g}$ is its Lie algebra.

The main goals of the paper are to establish a priori bounds for the prime power divisors of the discriminants of spinor regular positive definite primitive integral ternary quadratic lattices, and to describe a procedure for determining all such lattices.

It is proved that if $F$ is an infinite field with characteristic different from $2$, whose theory is supersimple, and $C$ is an elliptic or hyperelliptic curve over $F$ with generic ‘modulus’, then $C$ has a generic $F$-rational point. The notion of generity here is in the sense of the supersimple field $F$.

Let $A > 0$ be an integer. The equation $x^5 - y^5 = Az^5$ was first studied by Dirichlet and Lebesgue. Lebesgue conjectured in 1843 that if $A$ has no prime divisors of the form $10k+1$, the equation has no solutions except the visible ones. Partial results were obtained by Lebesgue and by Terjanian in 1987. The purpose of the paper is to prove Lebesgue's conjecture. The main tool used is the method known as the elliptic Chabauty method.

The paper contains a final identification theorem for the ‘generic’ $K^*$-groups of finite Morley rank.

One-parameter overpartition-theoretic analogues are given of two classical families of partition identities: Andrews' combinatorial generalization of the Gollnitz–Gordon identities and a theorem of Andrews and Santos on partitions with attached odd parts. Geometric counterparts arising from multiple $q$-series identities are also discussed. These involve representations of overpartitions in terms of generalized Frobenius partitions.

Let $p \,{\geq}\, 3$ be a prime number, $F$ be a number field with $\zeta_p \notin F^{\times}$, and $K = F(\zeta_p)$. In a previous paper, the author proved, under some assumption on $p$ and $F$, that an unramified cyclic extension $N/F$ of degree $p$ has a normal integral basis if and only if the pushed-up extension $NK/K$ has a normal integral basis. This addendum shows that the assertion holds without the above-mentioned assumption.

If $K$ is an algebraic function field of one variable over an algebraically closed field $k$ and $F$ is a finite extension of $K$, then any element $a$ of $K$ can be written as a norm of some $b$ in $F$ by Tsen's theorem. All zeros and poles of $a$ lead to zeros and poles of $b$, but in general additional zeros and poles occur. The paper shows how this number of additional zeros and poles of $b$ can be restricted in terms of the genus of $K$, respectively $F$. If $k$ is the field of all complex numbers, then we use Abel's theorem concerning the existence of meromorphic functions on a compact Riemann surface. From this, the general case of characteristic 0 can be derived by means of principles from model theory, since the theory of algebraically closed fields is model-complete. Some of these results also carry over to the case of characteristic $p>0$ using standard arguments from valuation theory.

Let $X$ be a subset in $[-1,1]^{n_0}\,{\subset}\,\Real^{n_0}$ defined by the formula \[ X=\{ {\bf x}_0\,{\mid}\,Q_1{\bf x}_1 Q_2{\bf x}_2 \cdots Q_{\nu}{\bf x}_{\nu} (({\bf x}_0,{\bf x}_1,\,{\ldots}\,,{\bf x}_{\nu}) \in X_{\nu})\}, \] where $Q_i \in \{ \exists, \forall \}$, $Q_i \neq Q_{i{+}1}$, ${\bf x}_i \in \Real^{n_i}$, and $X_{\nu}$ may be either an open or a closed set in $[-1,1]^{n_0+ \cdots +n_{\nu}}{\!}$, being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of $X$ is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving $X_{\nu}$.

In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of $X_{\nu}$ are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.

The paper deals with locally finite groups $G$ having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. The following theorem gives a very detailed description of such groups.

Let $G$ be a locally finite group having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. Then $G/[G,\phi]$ has finite rank. Furthermore, $[G,\phi]'$ contains a characteristic subgroup $B$ such that the following hold.

(1) $B$ is a product of finitely many subgroups normal in $[G,\phi]$ isomorphic to either PSL$(2,K)$ or SL$(2,K)$ for some infinite locally finite fields $K$ of odd characteristic.

(2) $[G,\phi]'/B$ has finite rank.

A previous conjecture is verified for any normal surface singularity which admits a good ${\mathbb C}^*$-action. This result connects the Seiberg–Witten invariant of the link (associated with a certain ‘canonical’ spin$^c$ structure) with the geometric genus of the singularity, provided that the link is a rational homology sphere.

As an application, a topological interpretation is found of the generalized Batyrev stringy invariant (in the sense of Veys) associated with such a singularity.

The result is partly based on the computation of the Reidemeister–Turaev sign-refined torsion and the Seiberg–Witten invariant (associated with any spin$^c$ structure) of a Seifert 3-manifold with negative orbifold Euler number and genus zero.

The study of the pentagon equation leads to results on the structure and classification of finite quantum groups. It is proved that $L$ is a finite-dimensional Hopf algebra if and only if there exists an invertible matrix $R$, solution of the pentagon equation $R^{12}R^{13}R^{23}=R^{23}R^{12}$, such that $L\cong P(n, R)$; the Hopf algebra structure of $P(n, R)$ is explicitly described using generators and relations. Finally, it is proved that there exists a one-to-one correspondence between the set of types of $n$-dimensional Hopf algebras and the set of orbits of the action ${\rm GL}_n(k)\times (M_n(k)\otimes M_n(k)) \,{\longrightarrow}\, M_n(k)\otimes M_n(k), (u, R)\,{\longrightarrow}\, (u\otimes u)R(u\otimes u)^{-1}$, the representatives of which are invertible solutions of length $n$ for the pentagon equation.

A profinite group $G$ of finite cohomological dimension with (topologically) finitely generated closed normal subgroup $N$ is studied. If $G$ is pro-$p$ and $N$ is either free as a pro-$p$ group or a Poincaré group of dimension 2 or analytic pro-$p$, it is shown that $G/N$ has virtually finite cohomological dimension ${\rm cd}(G)\,{-}\,{\rm cd}(N)$. Some other cases when $G/N$ has virtually finite cohomological dimension are also considered.

If $G$ is profinite, the case of $N$ projective or the profinite completion of the fundamental group of a compact surface is considered.

Let $U^+$ be the plus part of the quantized enveloping algebra of a simple Lie algebra of type $A_n$ and let $\mathcal{B}^*$ be the dual canonical basis of $U^+$. Let $b$, $b'$ be in $\mathcal{B}^*$, and suppose that one of the two elements is a $q$-commuting product of quantum flag minors. It is shown that $b$ and $b'$ are multiplicative if and only if they $q$-commute.

Let $H$ be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field $k$ of characteristic 0. If $H$ has no nontrivial skew-primitive elements, some bounds are found for the dimension of $H_1$, the second term in the coradical filtration of $H$. Using these results, it is shown that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. Also a Hopf algebra of dimension $pq$ where $p$ and $q$ are odd primes with $p < q \leq 1 + 3p$ and $q \leq 13$ is semisimple and thus a group algebra or the dual of a group algebra. Some partial results in the classification problem for dimension 16 are also given.

A theorem of Kac on quiver representations states that the dimension vectors of indecomposable representations are precisely the positive roots of the associated symmetric Kac–Moody Lie algebra. Here this result is generalised to representations respecting an admissible quiver automorphism, and the positive roots of an associated symmetrisable Kac–Moody Lie algebra are obtained.

Also the relationship with species of valued quivers over finite fields is discussed. It is known that the number of isomorphism classes of indecomposable representations of a given dimension vector for a species is a polynomial in the size of the base field. It is shown that these polynomials are non-zero if and only if the dimension vector is a positive root of the corresponding symmetrisable Kac–Moody Lie algebra.

Let $G$ be a finite group and suppose that $G = AB$, where $A$ and $B$ are abelian subgroups. By a theorem of Ito, the derived subgroup $G'$ is known to be abelian. If either of the subgroups $A$ or $B$ is cyclic, then more can be said. The paper shows, for example, that $G'{/}(G' \cap A)$ is isomorphic to a subgroup of $B$ in this case.

Let ${\mathcal H}_{q}(d)$ be the Iwahori–Hecke algebra of the symmetric group, where $q$ is a primitive $l$th root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for $l \geq 4$ the multiplicities in a Specht or dual Specht module filtration of an ${\mathcal H}_q(d)$-module are well defined. A cohomological criterion is given for when an ${\mathcal H}_{q}(d)$-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting $q = 1$, results are obtained for the symmetric group in characteristic $p \geq 5$. These results are false in general for $p = 2$ or $3$.

Let U be a domain, convex in $x$ and symmetric about the $y$-axis, which is contained in a centered and oriented rectangle $S$. It is proved that $H_t{(U^+)}/H_t{(U)}\,{\leq}\, H_t{(S^+)}/H_t{(S)}$ where $H_t$ stands for heat content, that is, the remaining heat in the domain at time $t$ if it initially has uniform temperature 1, with Dirichlet boundary conditions, where $A^+\,{=}\,A\,{\cap}\, \{(x,y)\,{:}\,x\,{>}\,0\}$. It is also shown that the analog of this inequality holds for some other Schrödinger operators.

The concept of Yang–Baxter bases is useful in interpreting Young's constructions for the symmetric group. This concept is extended first to any Weyl group, and then to any Coxeter group.