## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 17, Number 2 (2010), 193-200.

### On the Geometry of the Conformal Group in Spacetime

N. G. Gresnigt and P. F. Renaud

#### Abstract

The study of the conformal group in $R^{p,q}$ usually involves the conformal compactification of $R^{p,q}$. This allows the transformations to be represented by linear transformations in $R^{p+1,q+1}$. So, for example, the conformal group of Minkowski space, $R^{1,3}$ leads to its isomorphism with $SO(2,4)$. This embedding into a higher dimensional space comes at the expense of the geometric properties of the transformations. This is particularly a problem in $R^{1,3}$ where we might well prefer to keep the geometric nature of the various types of transformations in sight. In this note, we show that this linearization procedure can be achieved with no loss of geometric insight, if, instead of using this compactification, we let the conformal transformations act on two copies of the associated Clifford algebra. Although we are mostly concerned with the conformal group of Minkowski space (where the geometry is clearest), generalization to the general case is straightforward.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 2 (2010), 193-200.

**Dates**

First available in Project Euclid: 26 May 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1274896198

**Digital Object Identifier**

doi:10.36045/bbms/1274896198

**Mathematical Reviews number (MathSciNet)**

MR2667386

**Zentralblatt MATH identifier**

1192.22011

**Subjects**

Primary: 22E46: Semisimple Lie groups and their representations

Secondary: 17B15: Representations, analytic theory 22E70: Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10]

**Keywords**

Clifford algebra conformal group Minkowski space

#### Citation

Gresnigt, N. G.; Renaud, P. F. On the Geometry of the Conformal Group in Spacetime. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 2, 193--200. doi:10.36045/bbms/1274896198. https://projecteuclid.org/euclid.bbms/1274896198