4.4 Composition algebras
8 min read•august 21, 2024
Composition algebras are a fascinating subset of non-associative algebras with unique properties. They include real numbers, complex numbers, quaternions, and octonions, each doubling in dimension and losing algebraic properties.
These algebras are characterized by a multiplicative and play crucial roles in physics and mathematics. limits real composition algebras to four types, while the ###-Dickson_Construction_0### provides a method for generating higher-dimensional algebras.
Definition of composition algebras
- Composition algebras form a crucial subset of non-associative algebras, characterized by specific algebraic properties
- These algebras play a significant role in the study of non-associative structures, providing insights into alternative rules
- Understanding composition algebras lays the foundation for exploring more complex non-associative algebraic systems
Key properties
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- Finite-dimensional algebra A over a field F with a non-degenerate quadratic form N
- Multiplicative property of the quadratic form: for all x, y in A
- Existence of an identity element 1 such that
- Bilinear form derived from N:
- Composition algebras are generally non-associative, with octonions being the largest non-associative example
Examples of composition algebras
- Real numbers (R): simplest composition algebra, one-dimensional over R
- Complex numbers (C): two-dimensional composition algebra over R
- Quaternions (H): four-dimensional composition algebra, non-commutative
- Octonions (O): eight-dimensional composition algebra, non-associative and non-commutative
- Split-complex numbers: two-dimensional composition algebra with a different quadratic form
Hurwitz's theorem
- Hurwitz's theorem represents a fundamental result in the theory of composition algebras
- This theorem establishes a crucial connection between composition algebras and normed division algebras
- Understanding Hurwitz's theorem provides insights into the limitations and structure of composition algebras
Statement of theorem
- Every finite-dimensional real composition algebra is isomorphic to one of four algebras:
- Real numbers (R)
- Complex numbers (C)
- Quaternions (H)
- Octonions (O)
- These four algebras are the only normed division algebras over the real numbers
- The dimensions of these algebras follow the pattern 2^n (n = 0, 1, 2, 3)
- No composition algebras exist in dimensions other than 1, 2, 4, and 8
Implications and significance
- Limits the possible dimensions of composition algebras over real numbers
- Establishes a connection between composition algebras and division algebras
- Provides a classification of all finite-dimensional real composition algebras
- Demonstrates the unique properties of octonions as the largest composition algebra
- Influences research in various areas of mathematics and theoretical physics
Cayley-Dickson construction
- Cayley-Dickson construction serves as a fundamental method in non-associative algebra for generating composition algebras
- This process allows for the systematic creation of higher-dimensional algebras from lower-dimensional ones
- Understanding the Cayley-Dickson construction provides insights into the structure and properties of composition algebras
Process overview
- Start with a composition algebra A with involution * and norm N
- Define a new algebra A' with elements (a,b) where a,b are in A
- Addition in A':
- Multiplication in A':
- Conjugation in A':
- Norm in A':
Doubling of algebras
- Real numbers (R) → Complex numbers (C): introduces imaginary unit i
- Complex numbers (C) → Quaternions (H): introduces additional units j and k
- Quaternions (H) → Octonions (O): introduces four new units
- Each doubling step results in:
- Doubling of dimension
- Loss of one algebraic property (commutativity, , alternativity)
- Preservation of the composition algebra structure
Normed division algebras
- Normed division algebras form a special class of composition algebras with unique properties
- These algebras play a crucial role in various areas of mathematics and physics
- Understanding normed division algebras provides insights into the structure of composition algebras
Real numbers
- One-dimensional over itself
- Commutative, associative, and distributive
- Norm given by absolute value:
- Serves as the foundation for all other normed division algebras
Complex numbers
- Two-dimensional normed division algebra over real numbers
- Commutative and associative, but not ordered
- Norm given by modulus squared:
- Introduces the concept of imaginary numbers
Quaternions
- Four-dimensional normed division algebra over real numbers
- Non-commutative but associative
- Norm given by sum of squares:
- Used in 3D rotations and computer graphics
Octonions
- Eight-dimensional normed division algebra over real numbers
- Non-commutative and non-associative, but alternative
- Norm given by sum of squares of all eight components
- Applications in theoretical physics and exceptional Lie groups
Composition algebra identities
- Composition algebra identities form the core algebraic properties that define these structures
- These identities play a crucial role in understanding the behavior of composition algebras
- Mastering these identities provides a foundation for working with composition algebras in various contexts
Multiplicative norm property
- Fundamental identity: for all x, y in the algebra
- Preserves the norm under multiplication
- Generalizes the absolute value property of real and complex numbers
- Holds for all composition algebras, including non-associative ones (octonions)
Polarization identity
- Relates the norm to the inner product:
- Allows for the definition of an inner product space structure
- Useful in deriving other properties of composition algebras
- Connects the geometric and algebraic aspects of composition algebras
Applications in physics
- Composition algebras find significant applications in various areas of physics
- These algebraic structures provide powerful tools for modeling and understanding physical phenomena
- The unique properties of composition algebras make them particularly useful in certain branches of physics
Quantum mechanics
- Complex numbers: fundamental in describing quantum states and operators
- Quaternions: used in some formulations of quantum mechanics (quaternionic quantum mechanics)
- Octonions: explored in attempts to unify quantum mechanics with other fundamental theories
- Spin and angular momentum: described using the properties of composition algebras
Special relativity
- Quaternions: provide a compact representation of Lorentz transformations
- Octonions: investigated for potential connections to higher-dimensional spacetime models
- Minkowski space: can be described using the 2x2 matrices over composition algebras
- : naturally arise from the properties of composition algebras
Generalized composition algebras
- Generalized composition algebras extend the concept beyond the classical examples
- These extensions allow for the exploration of broader algebraic structures
- Understanding generalized composition algebras provides insights into the limits and possibilities of the composition algebra concept
Infinite-dimensional cases
- Hilbert spaces: infinite-dimensional generalizations of Euclidean spaces
- von Neumann algebras: infinite-dimensional analogs of matrix algebras
- Banach algebras: complete normed algebras, some of which have composition-like properties
- Operator algebras: used in quantum field theory, some with composition-like structures
Non-associative extensions
- Flexible algebras: weaker condition than associativity, includes some composition algebras
- Power-associative algebras: every element generates an associative subalgebra
- Alternative algebras: satisfy (xx)y = x(xy) and (yx)x = y(xx) for all elements
- Malcev algebras: non-associative analogs of Lie algebras, related to octonions
Composition algebras vs other algebras
- Comparing composition algebras with other algebraic structures highlights their unique properties
- Understanding these relationships provides a broader context for composition algebras in mathematics
- These comparisons help in identifying the strengths and limitations of composition algebras
Comparison with Lie algebras
- Composition algebras: have a multiplicative norm, not necessarily anticommutative
- Lie algebras: anticommutative, satisfy Jacobi identity
- Derivations of composition algebras often form Lie algebras
- Exceptional Lie algebras (G2, F4, E6, E7, E8) related to octonions
- Both used in theoretical physics, but in different contexts
Relation to Jordan algebras
- Composition algebras: not necessarily commutative, have a multiplicative norm
- algebras: commutative, satisfy Jordan identity
- Exceptional : 3x3 Hermitian matrices over octonions
- Both arise in quantum mechanics: composition algebras for states, Jordan algebras for observables
- Some Jordan algebras can be constructed from composition algebras
Automorphisms of composition algebras
- Automorphisms of composition algebras reveal their symmetry properties
- These transformations play a crucial role in understanding the structure of composition algebras
- Studying automorphisms connects composition algebras to group theory and geometry
Symmetry groups
- Real numbers: trivial automorphism group (only identity)
- Complex numbers: automorphism group Z2 (identity and complex conjugation)
- Quaternions: automorphism group SO(3) (rotations in 3D space)
- Octonions: exceptional Lie group G2
- Automorphisms preserve the algebraic structure and norm of the composition algebra
Exceptional Lie groups
- G2: automorphism group of octonions, smallest exceptional Lie group
- F4: isometry group of exceptional Jordan algebra
- E6, E7, E8: larger exceptional Lie groups related to octonions
- These groups have unique properties not found in classical Lie groups
- Connections to string theory and attempts at unified physical theories
Algebraic structures
- Algebraic structures provide a framework for understanding composition algebras
- These structures help classify and analyze the properties of composition algebras
- Studying these structures reveals the relationships between composition algebras and other mathematical objects
Graded algebras
- Composition algebras can be given a graded structure
- Grading: decomposition into subspaces A = A0 ⊕ A1 ⊕ ... ⊕ An
- Z2-grading of octonions: split into even and odd parts
- Graded structure relates to supersymmetry in physics
- Grading helps in understanding the multiplication rules of composition algebras
Alternative algebras
- Composition algebras are examples of alternative algebras
- Alternative law: (xx)y = x(xy) and (yx)x = y(xx) for all elements
- Octonions: largest finite-dimensional real alternative division algebra
- Alternative algebras generalize associative algebras
- Moufang loops: multiplicative structure of units in alternative division algebras
Composition algebras in geometry
- Composition algebras have significant applications in various areas of geometry
- These algebras provide powerful tools for describing and analyzing geometric structures
- Understanding the geometric aspects of composition algebras enhances their applicability in mathematics and physics
Projective geometry
- Composition algebras used to construct projective planes and spaces
- Real projective plane: RP^2, related to real numbers
- Complex projective plane: CP^2, related to complex numbers
- Quaternionic projective plane: HP^2, related to quaternions
- Octonionic projective plane: OP^2, exceptional geometric object
- Projective lines over composition algebras correspond to spheres of dimensions 1, 2, 4, and 8
Conformal geometry
- Composition algebras provide models for conformal geometries
- Möbius transformations: conformal maps of the Riemann sphere, related to complex numbers
- Conformal maps in higher dimensions related to quaternions and octonions
- Twistor theory: uses quaternionic and octonionic structures in 4D and 8D spaces
- Conformal compactifications of spaces related to composition algebras
Representation theory
- Representation theory studies how abstract algebraic structures can be represented by linear transformations
- For composition algebras, representation theory provides insights into their structure and properties
- Understanding representations of composition algebras connects them to other areas of mathematics and physics
Regular representations
- Left regular representation: algebra acts on itself by left multiplication
- Right regular representation: algebra acts on itself by right multiplication
- For non-associative algebras (octonions), left and right representations differ
- Dimension of regular representation equals dimension of the algebra
- Regular representations reveal the multiplication structure of the algebra
Spin representations
- Spin groups: double covers of special orthogonal groups
- Spin(3) isomorphic to unit quaternions, represents rotations in 3D
- Spin(4) related to pairs of quaternions
- Spin(8) has a special triality property related to octonions
- Spinors: elements of the space on which spin groups act
- Applications in particle physics and quantum field theory
Key Terms to Review (17)
Alternative Algebra: Alternative algebra refers to a type of non-associative algebra where the product of any two elements is associative when either element is repeated. This means that in an alternative algebra, the identity \(x \cdot (x \cdot y) = (x \cdot x) \cdot y\) holds for all elements \(x\) and \(y\). This property creates a unique structure that connects to various mathematical concepts, showcasing its importance in areas like Lie algebras, composition algebras, and Jordan algebras.
Artin–Schreier Theorem: The Artin–Schreier theorem is a fundamental result in field theory that characterizes the structure of certain algebraic extensions of fields, specifically those that can be described as extensions of finite fields. It establishes a crucial link between algebraic and transcendental extensions, demonstrating that every finite separable extension of a field of characteristic p can be obtained by adjoining the root of a polynomial of the form $x^{p^n} - a$, where $a$ is in the base field. This theorem is essential for understanding how composition algebras can be constructed over finite fields.
Associativity: Associativity is a property of certain binary operations that states the grouping of operands does not affect the result of the operation. This means that when performing an operation on three elements, the way in which they are grouped will yield the same outcome, whether it is (a * b) * c or a * (b * c). This property is crucial in various algebraic structures, ensuring consistent results regardless of how calculations are arranged.
Cayley: Cayley refers to the work of the mathematician Arthur Cayley, particularly in the context of composition algebras. Composition algebras are algebraic structures that allow for the definition of a quadratic form, which can be associated with a vector space and exhibit properties such as associativity and distributivity. Cayley's contributions to this area include the establishment of the Cayley numbers, which form a specific type of non-associative algebra that extends the complex numbers.
Cayley-Dickson Construction: The Cayley-Dickson construction is a method used to generate new algebras from existing ones by doubling the dimension of the algebra and defining new multiplication rules. This process creates a sequence of algebras, starting with the real numbers and successively producing complex numbers, quaternions, octonions, and even further extensions. The construction preserves certain algebraic structures, leading to interesting properties such as non-associativity and the development of composition algebras.
Hamilton: In the context of composition algebras, Hamilton refers to the work of Sir William Rowan Hamilton, who introduced the concept of quaternions, which are a specific type of composition algebra. Quaternions extend complex numbers and allow for calculations involving three-dimensional space, providing a powerful tool in both mathematics and physics. Hamilton's work laid the groundwork for understanding higher-dimensional algebras and their applications in various fields.
Hurwitz's Theorem: Hurwitz's Theorem states that a composition algebra over the reals can only exist in dimensions 1, 2, 4, or 8. This theorem highlights the connection between algebraic structures and the geometry of the underlying space, shedding light on the limits of constructing composition algebras. The significance of this theorem extends into various mathematical fields, revealing foundational principles about quadratic forms and their properties in non-associative algebras.
Jordan: In mathematics, Jordan refers to structures and systems that are derived from the work of mathematician Camille Jordan, particularly in the context of algebraic systems. This includes Jordan algebras, which are commutative, non-associative algebras that satisfy specific identities, and Jordan triple systems, which are a generalization of these algebras. Both concepts emphasize the unique properties of elements in these systems and their applications in various areas of mathematics and theoretical physics.
Jordan Algebra: A Jordan algebra is a non-associative algebraic structure characterized by a bilinear product that satisfies the Jordan identity, which states that the product of an element with itself followed by the product of this element with any other element behaves in a specific way. This type of algebra plays a significant role in various mathematical fields, including radical theory, representation theory, and its connections to Lie algebras and alternative algebras.
Multiplication: Multiplication is a mathematical operation that combines two elements to produce a third element, often referred to as the product. In the context of non-associative algebra, multiplication can exhibit unique properties that differentiate it from standard multiplication seen in familiar algebraic structures. This operation is central to various algebraic systems, influencing how elements interact and contribute to the overall structure of rings, algebras, and specific number systems like octonions.
Non-associative operation: A non-associative operation is a binary operation in which the grouping of operands affects the result, meaning that the equation (a * b) * c does not necessarily equal a * (b * c). This property distinguishes non-associative operations from associative ones, where the grouping does not change the outcome. Such operations can lead to different results based on how they are arranged, which is crucial for understanding structures like composition algebras.
Norm: In the context of composition algebras, a norm is a function that assigns a non-negative real number to each element of the algebra, reflecting its 'size' or 'length'. This concept is important as it helps in defining geometric interpretations, establishing identities, and understanding the structure of the algebra. Norms are crucial in assessing properties such as invertibility and the algebra's behavior under multiplication.
Normed division algebra: A normed division algebra is a type of algebraic structure that combines both the properties of a normed vector space and a division algebra, where every non-zero element has a multiplicative inverse. This structure is equipped with a norm that satisfies certain properties, making it possible to define the length of vectors and perform geometric operations. It is particularly important in understanding composition algebras, the construction and properties of octonions, and their applications in advanced theoretical frameworks such as string theory.
Octonion Algebra: Octonion algebra is a type of non-associative algebra that extends the concept of complex numbers and quaternions to eight dimensions. It is characterized by its multiplication being not only non-commutative but also non-associative, which means that the order in which you multiply the elements matters and can lead to different results. Octonions are particularly interesting because they are one of the four normed division algebras, alongside real numbers, complex numbers, and quaternions, highlighting their unique properties in the realm of algebra.
Quadratic forms: Quadratic forms are polynomial expressions of degree two in several variables, typically represented in the form $$Q(x_1, x_2, ext{...}, x_n) = a_{11}x_1^2 + a_{12}x_1x_2 + a_{22}x_2^2 + ... + a_{nn}x_n^2$$, where the coefficients are real numbers. They play a crucial role in various areas such as algebraic geometry and number theory. Quadratic forms can be analyzed using matrices, leading to concepts such as definiteness, which helps classify the nature of the quadratic form based on its eigenvalues.
Quaternion algebra: Quaternion algebra is a type of non-associative algebra that extends the concept of complex numbers to higher dimensions, specifically four dimensions. It is defined over a field, typically the real numbers, and consists of elements called quaternions, which can be represented as combinations of a real part and three imaginary units. This structure allows for multiplication and addition, but the multiplication is not commutative, meaning the order of multiplication matters.
Spinors: Spinors are mathematical objects used in physics and mathematics that describe the quantum states of particles with half-integer spin, such as electrons. They generalize the concept of vectors and provide a framework for understanding how these particles transform under rotations and Lorentz transformations. Spinors play a crucial role in theories of quantum mechanics and relativity, particularly in the context of composition algebras and differential geometry.