Interface Algebra<E,F>

All Superinterfaces:
AbelianGroup<E>, AbelianMonoid<E>, Group<E>, Magma<E>, Module<E,F>, Monoid<E>, Ring<E>, Semiring<E>, Set<E>
All Known Subinterfaces:
BanachAlgebra<E,F>, CStarAlgebra<E,F>, GradedAlgebra<E,F>, LieAlgebra<E,S>
All Known Implementing Classes:
MatrixLieAlgebra
public interface Algebra<E,F> extends Module<E,F>, Ring<E>
An algebra over a field is a vector space equipped with a bilinear product.

Mathematical Definition

An algebra A over a field K is a vector space equipped with a binary operation (x, y) ↦ x · y such that for all x, y, z ∈ A and a, b ∈ K:

  • (x + y) · z = x · z + y · z
  • x · (y + z) = x · y + x · z
  • (ax) · (by) = (ab) (x · y)

Examples

  • Square matrices (Mâ‚™(K))
  • Polynomials (K[x])
  • Complex numbers (as ℝ-algebra)
  • Quaternions (as ℝ-algebra)
  • Lie Algebras (with bracket [x,y] as product)
Since:
1.0
Author:
Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

  • Method Details

    • multiply

      default E multiply(E a, E b)
      The bilinear product operation.

      This is the same as the Ring multiplication, but explicitly viewed as the algebra product.

      Specified by:
      multiply in interface Semiring<E>
      Parameters:
      a - the first factor
      b - the second factor
      Returns:
      a × b