In this paper we study simple associative algebras with finite $\mathbb{Z}$ -gradings. This is done using a simple algebra ${{F}_{g}}$ that has been constructed in Morita theory from a bilinear form $g:\,U\,\times \,V\,\to \,A$ over a simple algebra $A$ . We show that finite $\mathbb{Z}$ -gradings on ${{F}_{g}}$ are in one to one correspondence with certain decompositions of the pair $\left( U,\,V \right)$ . We also show that any simple algebra $R$ with finite $\mathbb{Z}$ -grading is graded isomorphic to ${{F}_{g}}$ for some bilinear from $g:\,U\,\times \,V\,\to \,A$ , where the grading on ${{F}_{g}}$ is determined by a decomposition of $\left( U,\,V \right)$ and the coordinate algebra $A$ is chosen as a simple ideal of the zero component ${{R}_{0}}$ of $R$ . In order to prove these results we first prove similar results for simple algebras with Peirce gradings.

Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice $\Gamma$ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniformlattice Dirac comb, and its diffraction measure is periodic, with the dual lattice ${{\Gamma }^{*}}$ as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base.

Let $\Phi $ be an algebraically closed field of characteristic zero, $G$ a finite, not necessarily abelian, group. Given a $G$ -grading on the full matrix algebra $A\,=\,{{M}_{n}}\left( \Phi\right)$ , we decompose $A$ as the tensor product of graded subalgebras $A\,=\,B\,\otimes \,C,\,B\,\cong \,{{M}_{p}}\left( \Phi\right)$ being a graded division algebra, while the grading of $C\,\cong \,{{M}_{q}}\left( \Phi\right)$ is determined by that of the vector space ${{\Phi }^{n}}$ . Now the grading of $A$ is recovered from those of $A$ and $B$ using a canonical “induction” procedure.

We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type $C\left( n \right),D\left( m,n \right),D\left( 2,1;\alpha\right)\left( \alpha \in \mathbb{F}\backslash \left\{ 0,-1 \right\} \right),F(4)$ , and $G(3)$ .

We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs.

We prove polytopality of the generalized associahedra introduced in [5].

We classify the subalgebras of the general Lie conformal algebra $\text{g}{{\text{c}}_{N}}$ that act irreducibly on $\mathbb{C}\,{{\left[ \partial\right]}^{N}}$ and that are normalized by the $\text{s}{{\text{l}}_{2}}$ -part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_{n}^{\left( -\sigma ,\sigma\right)}$ , $\sigma \,\in \,C$ . The connection goes both ways—we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.

We begin by reviewing Monstrous Moonshine. The impact of Moonshine on algebra has been profound, but so far it has had little to teach number theory. We introduce (using ‘postcards’) a much larger context in which Monstrous Moonshine naturally sits. This context suggests Moonshine should indeed have consequences for number theory. We provide some humble examples of this: new generalisations of Gauss sums and quadratic reciprocity.

We construct a class of fermions (or bosons) by using a Clifford (or Weyl) algebra to get two families of irreducible representations for the extended affine Lie algebra $\widetilde{\mathfrak{g}{{\mathfrak{l}}_{N}}\left( {{\mathbb{C}}_{q}} \right)}$ of level (1, 0) (or (−1, 0)).

This paper characterizes when a Delone set $X$ in ${{\mathbb{R}}^{n}}$ is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set $X$ , let ${{N}_{X}}\left( T \right)$ count the number of translation-inequivalent patches of radius $T$ in $X$ and let ${{M}_{X}}\left( T \right)$ be the minimum radius such that every closed ball of radius ${{M}_{X}}\left( T \right)$ contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to $X$ being an ideal crystal.

Explicitly, for ${{N}_{X}}\left( T \right)$ , if $R$ is the covering radius of $X$ then either ${{N}_{X}}\left( T \right)$ is bounded or ${{N}_{X}}\left( T \right)\,\ge \,T/2R$ for all $T\,>\,0$ . The constant $1/2R$ in this bound is best possible in all dimensions.

For ${{M}_{X}}\left( T \right)$ , either ${{M}_{X}}\left( T \right)$ is bounded or ${{M}_{X}}\left( T \right)\ge T/3$ for all $T\,>\,0$ . Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set $X$ has ${{M}_{X}}\left( T \right)\,\ge \,c\left( n \right)T$ for all $T\,>\,0$ , for a certain constant $c\left( n \right)$ which depends on the dimension $n$ of $X$ and is $>\,1/3$ when $n\,>\,1$ .

In this paper we study specializations and one-sided bimodules of simple Jordan superalgebras.

We study the representations of extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ where $q$ is $N$ -th primitive root of unity ( $({{\mathbb{C}}_{q}}$ is the quantum torus in two variables). We first prove that $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$ for a suitable number of copies is a quotient of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ . Thus any finite dimensional irreducible module for $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$ lifts to a representation of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ . Conversely, we prove that any finite dimensional irreducible module for $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ comes from above. We then construct modules for the extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)\oplus \mathbb{C}{{d}_{1}}\oplus \mathbb{C}{{d}_{2}}$ which is integrable and has finite dimensional weight spaces.

We embed the moduli space $Q$ of 5 points on the projective line ${{S}_{5}}$ -equivariantly into $\mathbb{P}\left( V \right)$ , where $V$ is the 6-dimensional irreducible module of the symmetric group ${{S}_{5}}$ . This module splits with respect to the icosahedral group ${{A}_{5}}$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of ${{A}_{5}}$ on $Q$ to those on the regular icosahedron.

We consider two dynamical systems associated with a substitution of Pisot type: the usual $\mathbb{Z}$ -action on a sequence space, and the $\mathbb{R}$ -action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the “balanced pairs algorithm” and the “overlap algorithm”) and study the relation between them. In particular, we show that pure discrete spectrum for the $\mathbb{R}$ -action implies pure discrete spectrum for the $\mathbb{Z}$ -action, and obtain a partial result in the other direction. As a corollary, we prove pure discrete spectrum for every $\mathbb{R}$ -action associated with a two-symbol substitution of Pisot type (this is conjectured for an arbitrary number of symbols).

Quantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type ${{\text{A}}_{1}}$ , $\text{C}$ and $\text{BC}$ . We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type $\text{C}$ .