Hurwitz quaternion
Encyclopedia
In mathematics
, a Hurwitz quaternion (or Hurwitz integer) is a quaternion
whose components are either all integer
s or all half-integer
s (halves of an odd integer; a mixture of integers and half-integers is not allowed). The set of all Hurwitz quaternions is
It can be confirmed that H is closed under quaternion multiplication and addition, which makes it a subring
of the ring
of all quaternions H.
A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integer
s. The set of all Lipschitz quaternions
forms a subring of the Hurwitz quaternions H.
As a group
, H is free abelian
with generators {½(1+i+j+k), i, j, k}. It therefore forms a lattice
in R4. This lattice is known as the F4 lattice since it is the root lattice of the semisimple Lie algebra F4. The Lipschitz quaternions L form an index 2 sublattice of H.
The group of units in L is the order 8 quaternion group
Q = {±1, ±i, ±j, ±k}. The group of units in H is a nonabelian group of order 24 known as the binary tetrahedral group. The elements of this group include the 8 elements of Q along with the 16 quaternions {½(±1±i±j±k)} where signs may be taken in any combination. The quaternion group is a normal subgroup
of the binary tetrahedral group U(H). The elements of U(H), which all have norm 1, form the vertices of the 24-cell inscribed in the 3-sphere
.
The Hurwitz quaternions form an order (in the sense of ring theory
) in the division ring
of quaternions with rational
components. It is in fact a maximal order; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an integral quaternion, also form an order. However, this latter order is not a maximal one, and therefore (as it turns out) less suitable for developing a theory of left ideals comparable to that of algebraic number theory
. What Adolf Hurwitz
realised, therefore, was that this definition of Hurwitz integral quaternion is the better one to operate with. This was one major step in the theory of maximal orders, the other being the remark that they will not, for a non-commutative ring such as H, be unique. One therefore needs to fix a maximal order to work with, in carrying over the concept of an algebraic integer
.
The norm
of a Hurwitz quaternion, given by , is always an integer. By a theorem of Lagrange
every nonnegative integer can be written as a sum of at most four squares. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. A Hurwitz integer is a prime element
if and only if its norm is a prime number
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Hurwitz quaternion (or Hurwitz integer) is a quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
whose components are either all integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s or all half-integer
Half-integer
In mathematics, a half-integer is a number of the formn + 1/2,where n is an integer. For example,are all half-integers. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer...
s (halves of an odd integer; a mixture of integers and half-integers is not allowed). The set of all Hurwitz quaternions is
It can be confirmed that H is closed under quaternion multiplication and addition, which makes it a subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of all quaternions H.
A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s. The set of all Lipschitz quaternions
forms a subring of the Hurwitz quaternions H.
As a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, H is free abelian
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
with generators {½(1+i+j+k), i, j, k}. It therefore forms a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
in R4. This lattice is known as the F4 lattice since it is the root lattice of the semisimple Lie algebra F4. The Lipschitz quaternions L form an index 2 sublattice of H.
The group of units in L is the order 8 quaternion group
Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
Q = {±1, ±i, ±j, ±k}. The group of units in H is a nonabelian group of order 24 known as the binary tetrahedral group. The elements of this group include the 8 elements of Q along with the 16 quaternions {½(±1±i±j±k)} where signs may be taken in any combination. The quaternion group is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of the binary tetrahedral group U(H). The elements of U(H), which all have norm 1, form the vertices of the 24-cell inscribed in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
.
The Hurwitz quaternions form an order (in the sense of ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
) in the division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
of quaternions with rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
components. It is in fact a maximal order; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an integral quaternion, also form an order. However, this latter order is not a maximal one, and therefore (as it turns out) less suitable for developing a theory of left ideals comparable to that of algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
. What Adolf Hurwitz
Adolf Hurwitz
Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...
realised, therefore, was that this definition of Hurwitz integral quaternion is the better one to operate with. This was one major step in the theory of maximal orders, the other being the remark that they will not, for a non-commutative ring such as H, be unique. One therefore needs to fix a maximal order to work with, in carrying over the concept of an algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
.
The norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
of a Hurwitz quaternion, given by , is always an integer. By a theorem of Lagrange
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that any natural number can be represented as the sum of four integer squaresp = a_0^2 + a_1^2 + a_2^2 + a_3^2\ where the four numbers are integers...
every nonnegative integer can be written as a sum of at most four squares. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. A Hurwitz integer is a prime element
Prime element
In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...
if and only if its norm is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
.
See also
- Gaussian integerGaussian integerIn number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
- Eisenstein integerEisenstein integerIn mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
- The Lie group F4
- The E8 latticeE8 latticeIn mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...