Flashcards: Lie groups

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n×n unitary matrices with determinant 1.

In mathematics the Jacobi identity is a property a binary operation can have that determines how the order of evaluation behaves for the given operation.

In mathematics, a Lie algebra (/liː/, not /laɪ/) is a vector space together with a non-associative multiplication called "Lie bracket" .

In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G.

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary.

In mathematics, the adjoint representation (or adjoint action) 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.

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n), the latter is called the compact symplectic group.

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (nongravitational) physical phenomena.

In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C).

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.

In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G → H is a continuous group homomorphism.

In mathematics, the simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan.

In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups).

In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space.

In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space.

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has as an inversion symmetry preserving the Hermitian structure.

In mathematics, the projective unitary group PU(n) is the quotient of the unitary group U(n) by the right multiplication of its center, U(1), embedded as scalars.

In theoretical physics and mathematics, the Wess–Zumino–Witten (WZW) model, also called the Wess–Zumino–Novikov–Witten model, is a simple model of conformal field theory whose solutions are realized by affine Kac–Moody algebras.

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system.

In mathematics, the special linear group of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

This is a list of Lie group topics, by Wikipedia page.

In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way is holomorphic.

In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra.

In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V,Q) on the associated projective space P(V).

In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.

In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers.

In mathematics, the length of an element w in a Weyl group W, denoted by l(w), is the smallest number k so that w is a product of k reflections by simple roots.