Algebras: multiplication of vectors

The abelian group of vectors in a vector space can also be given a new structure by defining multiplication between vectors to get another vector. The Cartesian inspiration here might be considered to be the vector cross product (AKA vector product or outer product) in 3 dimensions.

  • Algebra: defines a bilinear product distributive over vector addition (no commutativity, associativity, or identity required)
  • Associative algebra: associative product; turns the abelian group of vectors into a ring; can always be naturally extended to include a multiplicative identity
  • Lie algebra: product denoted \({\left[u,v\right]}\); satisfies two other attributes of the cross product, anti-commutativity \({\left[u,v\right]=-\left[v,u\right]}\) and the Jacobi identity \({\left[\left[u,v\right],w\right]+\left[\left[w,u\right],v\right]+\left[\left[v,w\right],u\right]=0}\)

For example, the Cartesian vectors under the cross product are a non-associative Lie algebra, while the real \({n\times n}\) matrices under matrix multiplication are an associative algebra.

Δ Note that “algebra” is sometimes defined to include associativity and/or an identity.
Δ If the scalars of a Lie algebra are a field of characteristic 2, then we no longer have \({\left[u,v\right]=-\left[v,u\right]\Rightarrow\left[v,v\right]=0}\), and the latter is imposed as a separate requirement in the definition.

In a Lie algebra (pronounced “lee”), the product is called the Lie bracket, and the notation \({\left[u,v\right]}\) in place of of \({uv}\) reflects the close relationship between Lie algebras and associative algebras: every associative algebra can be turned into a Lie algebra by defining the Lie bracket to be \({\left[u,v\right]\equiv uv-vu}\). In these cases the Lie bracket is called the Lie commutator. The Poincaré-Birkhoff-Witt theorem provides a converse to this: that every Lie algebra is isomorphic to a subalgebra of an associative algebra called the universal enveloping algebra under the Lie commutator. An abelian algebra has \({uv=vu}\); thus an abelian Lie algebra (AKA commutative Lie algebra) has \({\left[u,v\right]=\left[v,u\right]\Rightarrow\left[u,v\right]=0}\). Note that an associative Lie algebra is not necessarily abelian, but does satisfy \({\left[\left[u,v\right],w\right]=0}\) via the Jacobi identity.

Any algebra over a field is completely determined by specifying scalars called structure coefficients (AKA structure constants), defined in a given basis as follows:

\(\displaystyle e_{\mu}e_{\nu}=c^{\rho}{}_{\mu\nu}e_{\rho} \)

However, different structure coefficients may define isomorphic algebras.

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