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mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of...

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that...

to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\displaystyle...

related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity...

quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of...

mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such...

normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism. Two Lie groups are locally isomorphic...

upper half-space model of hyperbolic 3-space. A lattice in a Lie group is a discrete subgroup such that the Haar measure of the quotient space is finite...

mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R)...

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for...

theorem. The quotient of a Lie group by a closed normal subgroup is a Lie group. The universal cover of a connected Lie group is a Lie group. For example...

kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group. In even dimensions in characteristic...

equivalent to compactness of the quotient space G \ X. Now assume G is a topological group and X a topological space on which it acts by homeomorphisms...

group is a linear space, vectors in the Lie algebra can be canonically identified with vectors in the group. The Lie algebra of the Heisenberg group is...

this group. The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket...

\mathbb {R} } . More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient G/Γ is of finite measure, for the measure on it...

discrete and its quotient modulo the center is a simple group, Sp(2n, F) is considered a simple Lie group. The real rank of the corresponding Lie algebra, and...

general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group PGL(V) = GL(V) / Z(V)...

Meanwhile, for connected compact Lie groups, we have the following result: Theorem: Every connected compact Lie group is the quotient by a finite central subgroup...

quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories...