∞-Lie theory (higher geometry)
Formal Lie groupoids
Super-Algebra and Super-Geometry
Where a vector specifies a direction and a magnitude , a bivector specifies a plane and a magnitude.
For a vector space, a bivector in is an element of the second exterior power of .
This is canonically identified with an element of degree 2 in the Grassmann algebra .
Clifford algebra and rotations
If is equipped with a non-degenerate inner product then the space of bivectors is also canonically identified with a subspace of the Clifford algebra .
If we write for the CLifford algebra element corresponding to a vector , then this identification is given by the map
(The inverse of this map is called the symbol map.)
Under the commutator in the Clifford algebra bivectors go to bivectors and hence form a Lie algebra. This Lie algebra is the special orthogonal Lie algebra of .
Discussion of Clifford algebra and exterior algebra that amplifies the role of bivectors is notably in the references at Geometric Algebra .
Revised on August 30, 2011 01:42:31
by Urs Schreiber