Basics Background Information Formally Real Special and Exceptional Summary
Jordan Algebras
Katie Sember, Lindsey Scoppetta, Amanda Clemm
George Washington University Summer Program for Women in Mathematics
July 7, 2011
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Rings
Definition A ring is a commutative group under addition equipped with a second binary operation, namely multiplication, that satisfies: Closure Associativity Distribution over addition
Definition A field is a commutative ring whose nonzero elements form a group under multiplication. Thus every element x of a field has an inverse x−1 that is also in the field.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Rings
Definition A ring is a commutative group under addition equipped with a second binary operation, namely multiplication, that satisfies: Closure Associativity Distribution over addition
Definition A field is a commutative ring whose nonzero elements form a group under multiplication. Thus every element x of a field has an inverse x−1 that is also in the field.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Rings
Definition A ring is a commutative group under addition equipped with a second binary operation, namely multiplication, that satisfies: Closure Associativity Distribution over addition
Definition A field is a commutative ring whose nonzero elements form a group under multiplication. Thus every element x of a field has an inverse x−1 that is also in the field.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Rings
Definition A ring is a commutative group under addition equipped with a second binary operation, namely multiplication, that satisfies: Closure Associativity Distribution over addition
Definition A field is a commutative ring whose nonzero elements form a group under multiplication. Thus every element x of a field has an inverse x−1 that is also in the field.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Rings
Definition A ring is a commutative group under addition equipped with a second binary operation, namely multiplication, that satisfies: Closure Associativity Distribution over addition
Definition A field is a commutative ring whose nonzero elements form a group under multiplication. Thus every element x of a field has an inverse x−1 that is also in the field.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
An Algebra: Review
Let (V, +, ·) be a vector space over a field. Then V is an algebra when it has a binary operation × : V × V → V , called multiplication, and an identity element referred to as the unit, such that the following properties hold: (u + v) × w = (u × w) + (v × w) u × (v + w) = (u × v) + (u × w) (au) × (bv) = (ab)(u × v) There exists an element 1 ∈ V such that 1 × u = u × 1 = u Note: Algebras are not assumed to be associative under multiplication.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
An Algebra: Review
Let (V, +, ·) be a vector space over a field. Then V is an algebra when it has a binary operation × : V × V → V , called multiplication, and an identity element referred to as the unit, such that the following properties hold: (u + v) × w = (u × w) + (v × w) u × (v + w) = (u × v) + (u × w) (au) × (bv) = (ab)(u × v) There exists an element 1 ∈ V such that 1 × u = u × 1 = u Note: Algebras are not assumed to be associative under multiplication.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
An Algebra: Review
Let (V, +, ·) be a vector space over a field. Then V is an algebra when it has a binary operation × : V × V → V , called multiplication, and an identity element referred to as the unit, such that the following properties hold: (u + v) × w = (u × w) + (v × w) u × (v + w) = (u × v) + (u × w) (au) × (bv) = (ab)(u × v) There exists an element 1 ∈ V such that 1 × u = u × 1 = u Note: Algebras are not assumed to be associative under multiplication.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
An Algebra: Review
Let (V, +, ·) be a vector space over a field. Then V is an algebra when it has a binary operation × : V × V → V , called multiplication, and an identity element referred to as the unit, such that the following properties hold: (u + v) × w = (u × w) + (v × w) u × (v + w) = (u × v) + (u × w) (au) × (bv) = (ab)(u × v) There exists an element 1 ∈ V such that 1 × u = u × 1 = u Note: Algebras are not assumed to be associative under multiplication.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Jordan Algebras!
Definition A Jordan algebra is a non-associative algebra A over a field F that satisfies the identities: a × b = b × a a2 × (b × a) = (a2 × b) × a for all a, b ∈ A Note: non-associative refers to for all a, b, c ∈ A, (a × b) × c 6= a × (b × c).
Definition A matrix is self-adjoint if it is equal to its conjugate transpose.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Jordan Algebras!
Definition A Jordan algebra is a non-associative algebra A over a field F that satisfies the identities: a × b = b × a a2 × (b × a) = (a2 × b) × a for all a, b ∈ A Note: non-associative refers to for all a, b, c ∈ A, (a × b) × c 6= a × (b × c).
Definition A matrix is self-adjoint if it is equal to its conjugate transpose.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Jordan Algebras!
Definition A Jordan algebra is a non-associative algebra A over a field F that satisfies the identities: a × b = b × a a2 × (b × a) = (a2 × b) × a for all a, b ∈ A Note: non-associative refers to for all a, b, c ∈ A, (a × b) × c 6= a × (b × c).
Definition A matrix is self-adjoint if it is equal to its conjugate transpose.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Jordan Algebras!
Definition A Jordan algebra is a non-associative algebra A over a field F that satisfies the identities: a × b = b × a a2 × (b × a) = (a2 × b) × a for all a, b ∈ A Note: non-associative refers to for all a, b, c ∈ A, (a × b) × c 6= a × (b × c).
Definition A matrix is self-adjoint if it is equal to its conjugate transpose.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Jordan Algebras!
Definition A Jordan algebra is a non-associative algebra A over a field F that satisfies the identities: a × b = b × a a2 × (b × a) = (a2 × b) × a for all a, b ∈ A Note: non-associative refers to for all a, b, c ∈ A, (a × b) × c 6= a × (b × c).
Definition A matrix is self-adjoint if it is equal to its conjugate transpose.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Important Mathematicians
Professor Pascual Jordan - 1933 applications of Jordan algebra: Quantum Mechanics Number Theory Extremal Black Holes Born: October 18th, 1902 - Hanover, Germany quantum mechanics and field theory May of 1933 - Nazi Party considered "politically unreliable" 1954 - Nobel Prize in Physics given to Max Born Abraham Adrian Albert - Albert algebra
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Question
Why was Jordan led away from associative algebras? Answer: Quantum Mechanics
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Quantum Mechanics
Definition Physicist study observables A system observable is a property of the system state that can be determined by some sequence of physical operations.
Example Submitting the system to electromagnetic fields and reading a value off of a gauge.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Quantum Mechanics
Definition Physicist study observables A system observable is a property of the system state that can be determined by some sequence of physical operations.
Example Submitting the system to electromagnetic fields and reading a value off of a gauge.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Quantum Mechanics
Definition Physicist study observables A system observable is a property of the system state that can be determined by some sequence of physical operations.
Example Submitting the system to electromagnetic fields and reading a value off of a gauge.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Quantum Mechanics
Definition Physicist study observables A system observable is a property of the system state that can be determined by some sequence of physical operations.
Example Submitting the system to electromagnetic fields and reading a value off of a gauge.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Multiplication of Observables
In quantum mechanics, observables are closed under addition and scalar multiplication, but multiplying observables is more complicated.
Commutative Product of Observables 1 2 2 2 1 x ◦ y = 2 ((x + y) − x − y ) = 2 (xy + yx)
Physicists like this definition because it considered partial ordering.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Power Associative
Definition A sub-algebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
Definition An algebra is said to be power associative if the sub-algebra generated by any element is associative.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Power Associative
Definition A sub-algebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
Definition An algebra is said to be power associative if the sub-algebra generated by any element is associative.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Formally Real Jordan Algebras
Definition Formally real Jordan algebra: commutative, power-associative algebra J satisfying 2 2 x1 + ··· + xn = 0 ⇒ x1 = ··· = xn = 0 ∀n
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Infinite Families of Formally Real Jordan Algebras
The Jordan algebra of n × n self-adjoint real matrices. The Jordan algebra of n × n self-adjoint complex matrices. The Jordan algebra of n × n self-adjoint quaternionic matrices.
The Jordan algebra freely generated by Rn with the relations where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type. the exceptional Jordan algebra The Jordan algebra of 3 × 3 self-adjoint octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Infinite Families of Formally Real Jordan Algebras
The Jordan algebra of n × n self-adjoint real matrices. The Jordan algebra of n × n self-adjoint complex matrices. The Jordan algebra of n × n self-adjoint quaternionic matrices.
The Jordan algebra freely generated by Rn with the relations where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type. the exceptional Jordan algebra The Jordan algebra of 3 × 3 self-adjoint octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Infinite Families of Formally Real Jordan Algebras
The Jordan algebra of n × n self-adjoint real matrices. The Jordan algebra of n × n self-adjoint complex matrices. The Jordan algebra of n × n self-adjoint quaternionic matrices.
The Jordan algebra freely generated by Rn with the relations where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type. the exceptional Jordan algebra The Jordan algebra of 3 × 3 self-adjoint octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Infinite Families of Formally Real Jordan Algebras
The Jordan algebra of n × n self-adjoint real matrices. The Jordan algebra of n × n self-adjoint complex matrices. The Jordan algebra of n × n self-adjoint quaternionic matrices.
The Jordan algebra freely generated by Rn with the relations where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type. the exceptional Jordan algebra The Jordan algebra of 3 × 3 self-adjoint octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Infinite Families of Formally Real Jordan Algebras
The Jordan algebra of n × n self-adjoint real matrices. The Jordan algebra of n × n self-adjoint complex matrices. The Jordan algebra of n × n self-adjoint quaternionic matrices.
The Jordan algebra freely generated by Rn with the relations where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type. the exceptional Jordan algebra The Jordan algebra of 3 × 3 self-adjoint octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Infinite Families of Formally Real Jordan Algebras
The Jordan algebra of n × n self-adjoint real matrices. The Jordan algebra of n × n self-adjoint complex matrices. The Jordan algebra of n × n self-adjoint quaternionic matrices.
The Jordan algebra freely generated by Rn with the relations where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type. the exceptional Jordan algebra The Jordan algebra of 3 × 3 self-adjoint octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Associative Algebra
Let U be an associative algebra over a field F that does not have characteristic two. Aside from the associative multiplication of the elements of U, defined a · b, we will define two new operations:
[a, b] = a · b − b · a 1 ab = 2 (a · b + b · a)
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Associative Algebra
Let U be an associative algebra over a field F that does not have characteristic two. Aside from the associative multiplication of the elements of U, defined a · b, we will define two new operations:
[a, b] = a · b − b · a 1 ab = 2 (a · b + b · a)
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Two New Algebras: U −, U +
If we keep the vector space structure of this associative algebra U that we have defined, but replace its multiplication by [a, b], the result is a new algebra, U −.
The multiplication implemented by U −, [a, b], is called the commutator of elements in U.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
U − with the Commutator
Upon investigation it can be seen that U − satisfies the Jacobi Identity,
[A, [B,C]] + [B, [C,A]] + [C, [A, B]] = 0 where A, B, C ∈ U,
and is a Lie algebra.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
P oincare´ − Birkhoff − W itt T heorem
The P oincare´ − Birkhoff − W itt theorem states that every Lie algebra is isomorphic to a sub-algebra of some algebra U −.
Recall A sub-algebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Multiplication of U +
If we take the vector space structure of U and replace its associative multiplication with ab, we attain another new algebra, U +.
Recall 1 ab = 2 (a · b + b · a)
Now let’s consider U + with the multiplication ab. We can see that this multiplication is commutative and satisfies the identities we have defined for a Jordan algebra.
Thus U + is a Jordan algebra.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
P oincare´ − Birkhoff − W itt T heorem
The analogue of the P oincare´ − Birkhoff − W itt theorem is not true for Jordan algebras.
However, if a Jordan Algebra W is isomorphic to a sub-algebra of U+, then W is called a special Jordan algebra.
All other Jordan algebras, those that are not special, are called exceptional Jordan algebras.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Examples of Special and Exceptional Jordan Algebras
Definition A self-adjoint matrix B is a matrix such that B is equal to its conjugate transpose, (B = B∗).
Some Examples: The set of self-adjoint real, complex, or quaternionic matrices with multiplication
(xy+yx) 2 The set of 3x3 self-adjoint matrices over the non-associative octonions, again with multiplication
(xy+yx) 2 , "The" exceptional Jordan algebra.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Examples of Special and Exceptional Jordan Algebras
Definition A self-adjoint matrix B is a matrix such that B is equal to its conjugate transpose, (B = B∗).
Some Examples: The set of self-adjoint real, complex, or quaternionic matrices with multiplication
(xy+yx) 2 The set of 3x3 self-adjoint matrices over the non-associative octonions, again with multiplication
(xy+yx) 2 , "The" exceptional Jordan algebra.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Examples of Special and Exceptional Jordan Algebras
Definition A self-adjoint matrix B is a matrix such that B is equal to its conjugate transpose, (B = B∗).
Some Examples: The set of self-adjoint real, complex, or quaternionic matrices with multiplication
(xy+yx) 2 The set of 3x3 self-adjoint matrices over the non-associative octonions, again with multiplication
(xy+yx) 2 , "The" exceptional Jordan algebra.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Examples of Special and Exceptional Jordan Algebras
Definition A self-adjoint matrix B is a matrix such that B is equal to its conjugate transpose, (B = B∗).
Some Examples: The set of self-adjoint real, complex, or quaternionic matrices with multiplication
(xy+yx) 2 The set of 3x3 self-adjoint matrices over the non-associative octonions, again with multiplication
(xy+yx) 2 , "The" exceptional Jordan algebra.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Review of Key Concepts
A Jordan algebra was created by Pascual Jordan for quantum mechanics and is a power-associate algebra. By investigating the algebra U +, we can observe two different categories of Jordan algebras, special and exceptional. A specific type of Jordan algebras known as the Formally real Jordan algebras can be classified as real, complex, quaternionic, and octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Review of Key Concepts
A Jordan algebra was created by Pascual Jordan for quantum mechanics and is a power-associate algebra. By investigating the algebra U +, we can observe two different categories of Jordan algebras, special and exceptional. A specific type of Jordan algebras known as the Formally real Jordan algebras can be classified as real, complex, quaternionic, and octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras Basics Background Information Formally Real Special and Exceptional Summary
Review of Key Concepts
A Jordan algebra was created by Pascual Jordan for quantum mechanics and is a power-associate algebra. By investigating the algebra U +, we can observe two different categories of Jordan algebras, special and exceptional. A specific type of Jordan algebras known as the Formally real Jordan algebras can be classified as real, complex, quaternionic, and octonionic matrices.
K. Sember, L. Scoppetta, A. Clemm Jordan Algebras