Definitions
- A lattice is a free abelian group of finite rank with an integral symmetric bilinear form (·,·).
- The dimension of a lattice is the same as its rank (as a Z-module).
- A lattice is positive definite if (a, a) is always positive for non-zero a.
- The discriminant of a lattice is the determinant of the matrix with entries (ai, aj), where the elements ai form a basis for the lattice.
- A lattice is unimodular if its discriminant is 1 or −1.
- A unimodular lattice is even or type II if (a, a) is always even, otherwise odd or type I.
- Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is.
- The signature of a lattice is the signature of the form on the vector space.
Read more about this topic: Unimodular Lattice
Famous quotes containing the word definitions:
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)
“The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
—Pamela Patrick Novotny (20th century)