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A highest weight module is a module generated by a weight vector v, subject to k λ · v = d λ v for all weights μ, and e i · v = 0 for all i. Similarly, a quantum group can have a lowest weight representation and lowest weight module, i.e. a module generated by a weight vector v, subject to k λ · v = d λ v for all weights λ, and f i ...
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in ...
A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ([x, y]) = 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra [g,g] and hence descends to a weight on the abelian Lie algebra g/[g,g]. Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple ...
The Bravais lattice concept is used to formally define a crystalline arrangement and its (finite) frontiers. A crystal is made up of one or more atoms, called the basis or motif, at each lattice point. The basis may consist of atoms, molecules, or polymer strings of solid matter, and the lattice provides the locations of the basis.
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
A lattice is positive definite if the norm of all nonzero elements is positive. The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (a i, a j), where the elements a i form a basis for the lattice. An integral lattice is unimodular if its determinant is 1 or −1.
Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by α, β, γ [1]. A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal.
Let be a locally compact group and a discrete subgroup (this means that there exists a neighbourhood of the identity element of such that = {}).Then is called a lattice in if in addition there exists a Borel measure on the quotient space / which is finite (i.e. (/) < +) and -invariant (meaning that for any and any open subset / the equality () = is satisfied).