Conformal Groups in Geometry and Spin Structures by Pierre Anglè (auth.), Pierre Anglès (eds.)

By Pierre Anglè (auth.), Pierre Anglès (eds.)

Conformal teams play a key position in geometry and spin constructions. This ebook offers a self-contained evaluate of this crucial zone of mathematical physics, starting with its origins within the works of Cartan and Chevalley and progressing to contemporary examine in spinors and conformal geometry.

Key subject matters and features:

* Focuses in the beginning at the fundamentals of Clifford algebras

* reviews the areas of spinors for a few even Clifford algebras

* Examines conformal spin geometry, starting with an undemanding research of the conformal crew of the Euclidean plane

* Treats overlaying teams of the conformal staff of a customary pseudo-Euclidean area, together with a bit at the complicated conformal group

* Introduces conformal flat geometry and conformal spinoriality teams, by means of a scientific improvement of riemannian or pseudo-riemannian manifolds having a conformal spin structure

* Discusses hyperlinks among classical spin constructions and conformal spin buildings within the context of conformal connections

* Examines pseudo-unitary spin constructions and pseudo-unitary conformal spin buildings utilizing the Clifford algebra linked to the classical pseudo-unitary space

* considerable workouts with many tricks for solutions

* accomplished bibliography and index

This textual content is appropriate for a direction in mathematical physics on the complex undergraduate and graduate degrees. it is going to additionally profit researchers as a reference text.

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Example text

If q is positive, O0+ SO(n, R). Then G+ 0 = Spin(E, q) is denoted by Spin n and called the classical spinor group of degree n. 2 Proposition (a) Let (E, q) be a quadratic regular n-dimensional complex space or Euclidean real space. The spin group Spin(E, q) is the group consisting of products in the Clifford algebra C(E, q) of an even number of unitary vectors in E. Spin(E, q) is connected and simply arcwise connected and constitutes a twofold covering of SO(E, q). (b) Let (E, q) = Er,s be a standard pseudo-Euclidean space of type (r, s), Spin(Er,s ) = Spin (r, s), the corresponding spin group is the group consisting of products in the Clifford algebra C(Er,s ) of an even number of ai ∈ E such that q(ai ) = 1 and of an even number of bj such that q(bj ) = −1.

Thus cogredient involutions are essentially merely different representations of the same abstract involution. 1 Clifford Algebras Cr,s and Cr,s Let V = Er,s be the standard m-dimensional pseudo-Euclidean space of type (r, s), r + s = m. Let (x|y) = x 1 y 1 + · · · + x r y r − x r+1 y r+1 − · · · − x r+s y r+s be its scalar product, relative to an orthogonal basis of V , namely e = {e1 , . . , en } with q(ei ) = 1, 1 ≤ i ≤ r and q(ej ) = −1, r +1 ≤ j ≤ m. C(V ) = Cr,s denotes its Clifford algebra.

The dimension of M(n, R) is n2 over R. In the same way we obtain the following list:5 GL(n, C) gl(n, C) M(n, C), dimension: 2n2 over R. SL(n, C) sl(n, C) {X ∈ M(n, C), Tr X = 0}, dimension: n2 − 1 over C, 2 2(n − 1) over R. U (n, C) u(n, C) {X ∈ M(n, C),t X = −X} consisting of skew-hermitian matrices, dimension n2 over R. O(n, C) o(n, C) {X ∈ M(n, C),t X = −X}, consisting of complex skewsymmetric matrices, dimension: n(n − 1) over R. SU (n, C) su(n, C) {X ∈ M(n, C),t X = −X, T r(X) = 0} consisting of skew-hermitian matrices with null trace.

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