Biinvariant

A biinvariant is a mathematical object, typically a function or a measure, that remains unchanged under certain group operations. In the context of Lie groups, a biinvariant metric is a Riemannian metric that is invariant under both left and right translations. This means that if you take a tangent vector at a point on the Lie group and translate it by any group element, the length of the translated vector remains the same. Similarly, if you consider two tangent vectors and translate them by the same group element, their inner product remains unchanged.

Biinvariant metrics are particularly important in the study of Lie groups because they simplify many geometric and analytic problems. For example, the geodesics of a biinvariant metric are straight lines in the group manifold. This property makes it easier to understand the exponential map and the structure of the group.

A key result in the theory of Lie groups is that every simple Lie group admits a unique (up to scalar multiples) biinvariant metric. This metric is often referred to as the Killing-Cartan metric. In more general Lie groups, biinvariant metrics may not exist or may not be unique. However, if a biinvariant metric exists, it can be constructed from a suitable inner product on the Lie algebra. The existence and properties of biinvariants are fundamental to understanding the geometry and representation theory of Lie groups.