Division algebra

Definition

A division algebra is a type of algebraic structure where division is possible, except by zero. It consists of a vector space equipped with a bilinear product that allows for non-zero elements to have multiplicative inverses, maintaining the essential property of associativity or alternative associativity. This structure is important in understanding how certain algebras can extend the properties of familiar number systems and is closely related to the Cayley-Dickson construction, which builds new algebras from existing ones.

5 Must Know Facts For Your Next Test

1. Division algebras must satisfy specific properties, such as having a bilinear product and allowing for division by non-zero elements.
2. The most well-known examples of division algebras are the real numbers, complex numbers, and quaternions.
3. Division algebras can be classified into associative and non-associative types, with the quaternions being a notable non-associative example.
4. The Cayley-Dickson construction generates new division algebras from existing ones, progressively doubling the dimension of the space each time.
5. Every finite-dimensional division algebra over the real numbers is either one-dimensional (the reals) or two-dimensional (the complex numbers), with higher dimensions only possible over other fields.

Review Questions

• Compare and contrast division algebras and fields, highlighting their similarities and differences.
  • Both division algebras and fields allow for division by non-zero elements and have a bilinear product structure. However, division algebras can be either associative or non-associative, while fields are strictly associative. Additionally, fields require that every non-zero element has a multiplicative inverse under the defined operations, which is also true for division algebras but may not involve the same constraints regarding associativity.
• Evaluate the significance of the Cayley-Dickson construction in generating new division algebras from existing ones.
  • The Cayley-Dickson construction is significant because it systematically produces new algebras by taking an existing division algebra and creating a higher-dimensional algebra that maintains some key properties. This process allows for the exploration of non-associative structures, such as octonions, broadening our understanding of algebraic systems. It highlights how complex mathematical structures can evolve from simpler ones while retaining essential features like division.
• Analyze how the properties of division algebras relate to their applications in physics and geometry.
  • The properties of division algebras have profound implications in physics and geometry, particularly in understanding rotations and transformations in higher dimensions. For instance, quaternions are utilized in 3D computer graphics for representing rotations without suffering from gimbal lock. Additionally, octonions find applications in string theory and special relativity due to their unique non-commutative properties. The mathematical framework provided by division algebras thus plays a critical role in various scientific fields by enabling complex modeling and analysis.

Related terms

Field: A set equipped with two operations, addition and multiplication, satisfying certain axioms, including the existence of multiplicative inverses for all non-zero elements.

Associative Algebra: An algebra in which the multiplication operation is associative, meaning that the product does not depend on how the factors are grouped.

Cayley-Dickson Construction: A method for constructing new algebras from existing ones, creating algebras that may not be associative or commutative, leading to structures such as octonions.