Linear algebraic groupIn mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).
Topological orderIn physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.
Orbifold notationIn geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.
Core (group theory)In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group. For a group G, the normal core or normal interior of a subgroup H is the largest normal subgroup of G that is contained in H (or equivalently, the intersection of the conjugates of H). More generally, the core of H with respect to a subset S ⊆ G is the intersection of the conjugates of H under S, i.e.
Dehn twistIn geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold). Suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a tubular neighborhood of c. Then A is an annulus, homeomorphic to the Cartesian product of a circle and a unit interval I: Give A coordinates (s, t) where s is a complex number of the form with and t ∈ [0, 1]. Let f be the map from S to itself which is the identity outside of A and inside A we have Then f is a Dehn twist about the curve c.
Global dimensionIn ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings.
Isohedral figureIn geometry, a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B.
Snake lemmaThe snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms. In an (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram: where the rows are exact sequences and 0 is the zero object.
