In mathematics, a Lie group is a group that is also a differentiable manifold, with the property that the group operations are smooth. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.
In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.
An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold.
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions.
In mathematics, the adjoint representation of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is GL(n), its Lie algebra is the vector space of all n-by-n matrices. So in this example, the adjoint representation is the vector space of n-by-n matrices , and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: .
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes.
A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold whose flow preserves the geodesic structure of without necessarily preserving the affine parameter of any geodesic. More intuitively, the flow of the projective maps geodesics smoothly into geodesics without preserving the affine parameter.
The gauge covariant derivative is a variation of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations.
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection.
The Lichnerowicz formula is a fundamental equation in the analysis of spinors on pseudo-Riemannian manifolds. In dimension 4, it forms a piece of Seiberg–Witten theory and other aspects of gauge theory. It is named after noted mathematicians André Lichnerowicz who proved it in 1963, and Roland Weitzenböck. The formula gives a relationship between the Dirac operator and the Laplace–Beltrami operator acting on spinors, in which the scalar curvature appears in a natural way. The result is significant because it provides an interface between results from the study of elliptic partial differential equations, results concerning the scalar curvature, and results on spinors and spin structures.
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function T : V → W that is:
A Representation up to homotopy is a concept in differential geometry that generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. It was introduced by Abad and Crainic.
This is a glossary of algebraic geometry.
In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes.
