Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has...

a discrete subgroup of PSL(2,R). The modular group is a lattice in PSL(2,R), but it is not cocompact. Kleinian groups are, by definition, discrete subgroups...

points Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure Lattice (module)...

In mathematics, especially in geometry, a double lattice in ℝn is a discrete subgroup of the group of Euclidean motions that consists only of translations...

but the rational numbers, Q, do not. A lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space...

Congruence relation Equivalence class Equivalence relation Lattice (group) Lattice (discrete subgroup) Multiplication table Prime number Up to Abelian variety...

arithmetic lattice in S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbb {R} )} . A lattice in a Lie group is usually defined as a discrete subgroup with...

algorithms for factoring and finding discrete logarithms in quantum computing are instances of the hidden subgroup problem for finite abelian groups, while...

subgroup; in that case, if it extends in n directions it is a lattice. Discrete subgroups of Dih(R2 ) which contain translations in one direction are of...

derivative Darboux derivative Lie groupoid Lie algebroid Lattice (group) Lattice (discrete subgroup) Frieze group Wallpaper group Space group Crystallographic...

some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the...

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...

homomorphism is just the fiber over the identity in H and is a discrete normal subgroup of G. The kernel K is closed in G if and only if G is Hausdorff...

group WK of the maximal compact subgroup K. If we fix a fundamental chamber for the Weyl group of K, then the discrete series representation are in 1:1...

have infinite order. The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here...

If 0 is removed from this structure it remains a lattice but ceases to be complete. The subgroups of any given group under inclusion. (While the infimum...

theorem states that every group G {\displaystyle G} is isomorphic to a subgroup of the symmetric group on (the underlying set of) G {\displaystyle G} ...

space, and its closed subgroup L^ dual to L turns out to be a lattice in V^. Therefore, L^ is the natural candidate for dual lattice, in a different vector...

\mathbf {b} _{d}\}} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d ≤ n {\displaystyle d\leq n} , the LLL algorithm...

reflections, inversions and rotoinversions – i.e., the finite subgroups of O(n); (2) infinite lattice groups, which include only translations; and (3) infinite...

the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co0 = 2 · Co1 of automorphisms of the Leech lattice, and shows that it has two complex irreducible...

example, SL(n,Z) is an arithmetic subgroup of SL(n,Q). For a Lie group G, a lattice in G means a discrete subgroup Γ of G such that the manifold G/Γ has...

Bethe lattices also occur as the discrete subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups. As such, they are also lattices in the...

finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL ⁡ ( n , K ) {\displaystyle \operatorname...

equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation. Analogously, an operator A on functions...

The Hidden Subgroup Problem - Review and Open Problems, arXiv:quant-ph/0411037 Regev, Oded (January 2004). "Quantum Computation and Lattice Problems"....

mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson...

modular lattices are the lattice of submodules of a module (hence modular), the lattice of two-sided ideals of a ring, and the lattice of normal subgroups of...

θ. Not surprisingly, SO(2) and its subgroups are all abelian; addition of rotation angles commutes. For discrete cyclic groups Cn, elements Cnk = R(2πk/n)...

mathematics, the term lattice group is used for two distinct notions: a lattice (group), a discrete subgroup of Rn and its generalizations a lattice ordered group...