Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Super-Algebra and Super-Geometry



Where a vector specifies a direction and a magnitude, a bivector specifies a plane and a magnitude.


For VV a kk-vector space, a bivector in VV is a decomposable element b=b 1b 2 2Vb = b_1\wedge b_2 \in \wedge^2 V of the second exterior power of VV. If the dimension of VV is 33 or less every element in Λ 2V\Lambda^2 V is decomposable, hence bivectors form the vector space, namely Λ 2V\Lambda^2 V itself. The plane associated to a nonzero bivector b 1b 2b_1\wedge b_2 is Span{b 1,b 2}Span\{b_1,b_2\}.

Alternatively, a bivector is a class of equivalence of pairs (b 1,b 2)V×V(b_1,b_2)\in V\times V by the smallest equivalence relation for which (b 1,b 2)(b 1+λb 2,b 2)(b_1,b_2)\sim (b_1+\lambda b_2,b_2) for any λk\lambda\in k, (b 1,b 2)(b 1,λb 1+b 2)(b_1,b_2)\sim (b_1,\lambda b_1+b_2) for any λk\lambda\in k, and (b 1,b 2)(b 1μ,μb 2)(b_1,b_2)\sim(\frac{b_1}\mu,\mu b_2) for any μk\{0}\mu\in k\backslash \{0\}. This axiomatics is parallel (and related) to the axiomatic definition of a 2×22\times 2 determinant.

Some authors (including English wikipedia) by a bivector mean any element Σ ib 1ib 2iΛ 2V\Sigma_i b_{1 i}\wedge b_{2 i}\in\Lambda^2 V, regardless the decomposability property (and in particular, regardless the dimension of VV which guarantees decomposability it dimV3dim V\leq 3), while they call the decomposable elements “simple bivectors”. This terminology may be inconvenient in analytic geometry as only the nonzero simple bivectors define a 2-plane and most standard constructions which are used in the Euclidean geometry with bivectors need the restriction, for example the notion of a vector being parallel to a bivector (vector vv is parallel to a (simple) bivector bb iff b=vwb = v\wedge w for some vector ww, or equivalently when vv is in the span of some pair b 1,b 2Vb_1,b_2\in V such that we can write b=b 1b 2b = b_1\wedge b_2).

A bivector is canonically identified with an element of degree 2 in the Grassmann algebra V\wedge^\bullet V.


If VV is equipped with a non-degenerate inner product then the space Λ 2V\Lambda^2 V is also canonically identified with a subspace of the Clifford algebra Cl(V)Cl(V).

If we write γ vCl(V)\gamma_v \in Cl(V) for the Clifford algebra element corresponding to a vector vVv \in V, then this identification is given by the map

vw12(γ vγ wγ wγ v). v \wedge w \mapsto \frac{1}{2}\left(\gamma_{v} \cdot \gamma_w - \gamma_w \cdot \gamma_v\right) \,.

(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 𝔰𝔬(V)\mathfrak{so}(V) of VV.


Discussion of Clifford algebra and exterior algebra that amplifies the role of bivectors is notably in the references at Geometric Algebra .

See also

Revised on June 2, 2017 07:21:46 by Urs Schreiber (