Algebra theory MOC

Algebra over a field

An algebra (𝑉, ⋅) over a field 𝕂 is a Vector space 𝑉 over 𝕂 equipped with a bilinear product ( ⋅) :𝑉 ×𝑉 →𝑉, falg i.e. for any 𝑥,𝑦,𝑧 ∈𝑉 and 𝑎,𝑏,𝑐 ∈𝕂

1. (𝑥 +𝑦)𝑧 =𝑥𝑧 +𝑦𝑧
2. 𝑧(𝑥 +𝑦) =𝑧𝑥 +𝑧𝑦
3. (𝑎𝑥)(𝑏𝑦) =(𝑎𝑏)(𝑥𝑦)

This may be generalized to an R-algebra.

Examples

• K-monoid^[In these notes I will try and reserve infix notation for associative algebras, as there is a tendency to assume such things to be associative.]
  • Commutative
    • Complex number
    • Quaternion (non-commutative)
    • Symmetric algebra
  • Non-commutative
    • Matrix algebra over a field
    • Endomorphism ring
    • Tensor algebra
    • Clifford algebra
    • Monoid ring
    • Extension field as a unital associative algebra
• Category ring
• Non-associative
  • Lie algebra

Properties

• The product within an algebra is completely determined by its Structure constants

---

develop | en | SemBr