NationMaster - Encyclopedia: Spin group

In mathematics the spin group Spin(n) is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n). As a Lie group Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cℓ(n). In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... Clifford algebras are a type of associative algebra in mathematics. ...

Accidental isomorphisms

In low dimensions, there are isomorphisms among the classical Lie groups called accidental isomorphisms. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups. Specifically, we have

There are certain vestiges of these isomorphisms left over for n = 7,8 (see Spin(8) for more details). For higher n, these isomorphisms disappear entirely. Notice the equalities Spin(2) = SO(2) and Spin(2,2) = SO(2,2). This is due to the fact that the fundamental groups of these orthogonal groups do not contain a two element cyclic $mathbb{Z}_2$ component. Instead, $pi_1 (SO(2)) = mathbb{Z}$ and $pi_1 (SO(2,2)^+) = mathbb{Z} times mathbb{Z}$ . Consequently any finite coverings of these groups are isomorphic to the groups themselves. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of nÃ—n unitary matrices with unit determinant. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of nÃ—n unitary matrices with unit determinant. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of nÃ—n unitary matrices with unit determinant. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

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See also

Accidental isomorphisms