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Every semisimple Lie algebra g is classified by its Dynkin diagram. A Dynkin diagram is a graph with a few different kinds of possible edges. The connected components of the ...

The geometry of the Lie group consisting of real matrices of the form [1 x y; 0 1 z; 0 0 1], i.e., the Heisenberg group.

A theorem proved by É. Cartan in 1913 which classifies the irreducible representations of complex semisimple Lie algebras.

Let N be a nilpotent, connected, simply connected Lie group, and let D be a discrete subgroup of N with compact right quotient space. Then N/D is called a nilmanifold.

Two lines that lie on top of one another are called coincident lines.

The covering of a plane curve with disks of radius epsilon whose centers lie on the curve.

Two curves are tangent externally at a point P if they lie on opposite sides of their common tangent at P

Two curves are tangent internally at a point P if they lie on the same side of their common tangent at P

Two lines or plane congruent geometric figures which lie on top of each other are said to be coincident.

Given a set S of points in the plane, an enclosing circle is a circle for which all points S lie in its interior.