Abstract Algebra/Quaternions

The algebra of Quaternions is a structure first studied by the Irish mathematician William Rowan Hamilton which extends the two-dimensional complex numbers to four dimensions. Multiplication is non-commutative in quaternions, a feature which enables its representation of three-dimensional rotation. Hamilton's provocative discovery of quaternions founded the field of hypercomplex numbers. Suggestive methods like dot products and cross products implicit in quaternion products enabled algebraic description of geometry now widely applied in science and engineering. Definitions right|thumb|Quaternion plaque on Broom Bridge, Dublin, which says: Here as he walked byon the 16th of October 1843Sir William Rowan Hamiltonin a flash of genius discoveredthe fundamental formula forquaternion multiplication i2 = j2 = k2 = ijk = −1 & cut it on a stone of this bridge A Quaternion corresponds to an ordered 4-tuple , where . A quaternion is denoted . The sum is called the vector part of q, and a is the real part. Hamilton coined the term vector in this context. Subsequent developments have extended the usage of the term vector to any element of a linear space. The vectors in H form a 3-dimensional subspace V. The set of all quaternions is denoted by . It is straightforward to define component-wise addition and scalar multiplication on , making it a real vector space. Multiplication follows the rules of the "quaternion group" Q8 = {1, -1, i, -i, j, -j, k, -k} that Hamilton carved into a stone of Broom Bridge, Dublin: The rules for the pairwise multiplication of , , and are: (positive cyclic products) (negative cyclic products). Using these, one can define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative, is a not a field. However, every nonzero quaternion has a multiplicative inverse (see below), so the quaternions are an example of a division ring. It is important to note that the non-commutative nature of quaternion multiplication makes it impossible to define the quotient of two quaternions p and q unambiguously, as the quantities and are generally different. Like the more familiar complex numbers, the quaternions have a conjugation, often denoted by a superscript star: . The conjugate of the quaternion is . As is the case for the complex numbers, the product is always a positive real number equal to the sum of the squares of the quaternion's components. The norm of a quaternion is the square root of . If pq is the product of two quaternions, then implying that forms a composition algebra. The multiplicative inverse of a non-zero quaternion is given by where division is defined since Unlike in the complex case, the conjugate of a quaternion can be computed algebraically: . Versors and elliptic space William Kingdon Clifford used Hamilton’s quaternions to explicate rotation geometry as an elliptic space with its own variety of lines, parallels, and surfaces. The ideas were reviewed in 1948 by Lemaitre and Coxeter and that sketch has these definitions: A versor is a quaternion of norm one, thus it lies on a 3-dimensional sphere found in the 4-space of quaternions. The versors are given by Euler's formula for complex numbers where the imaginary unit is taken from the unit sphere in the 3-space of vector quaternions: The distance between two versors u and v is A right parataxy on elliptic space is effected by multiplying on the right by a versor Similarly a left parataxy arises from left multiplication. In recognition of his contribution to elliptic geometry, a parataxy is called a Clifford translation. The general displacement of elliptic space is a combination of two parataxies, one left, one right: Note that if then the real line in the quaternions is fixed and the displacement is a rotation of the 3-space of quaternion vectors. The term line is appropriated for elliptic geometry. These lines are not straight, but they are parametrized by real numbers. Each line is associated with a right versor like s when c = π/2 in v. Then is a typical elliptic line. It corresponds to the axis of the rotation Now for u not on L, there are two Clifford parallels to L through u: For fixed right versors r and s, a Clifford surface can be formed as a union of Clifford parallels or as To form elliptic space from versors, two versors u and v are equivalent if u + v = 0. Modulo this equivalence, the versors, their algebra and geometry, represent elliptic space. Linear viewpoint Quaternions may be represented by 2×2 matrices with complex number entries: the place of is taken by these arrays: One uses matrix multiplication to verify that these expressions obey the rules of presentation of Q8. M(2,C) denotes the full algebra of 2×2 complex matrices, which has eight real dimensions, and sustains a representation of as a four-dimensional subalgebra. The linear properties of and M(2,C) assure the fidelity of the representation once the copy of Q8 has been identified. Quaternions, like other associative hypercomplex systems of the 19th century, eventually were viewed as matrix algebras in the 20th century. However, in 1853 Hamilton included biquaternions in his book of Lectures on Quaternions. Biquaternions are quaternions with complex number coefficients, sometimes called complex quaternions. Biquaternions form an algebra isomorphic to M(2,C). If the rows or columns of a matrix are proportional, then the determinant is zero, and there is no inverse. Nevertheless, such matrices have been used in physical science to represent events on a light-path from the origin. Authors Silberstein and Lanczos refer to this algebra as the biquaternions, but other writers have abandoned the label: Elie Cartan used M(2,C) extensively in The Theory of Spinors (1938), and Wolfgang Pauli, in his matrix mechanics of the atom, caused himself to be associated with M(2,C). Pauli Spin Matrices Quaternions are closely related to the Pauli spin matrices of Quantum Mechanics. The Pauli matrices are often denoted as , , (Where is the well known quantity of complex numbers) The 2×2 identity matrix is sometimes taken as . Thus , the real linear span of the matrices , , and , is isomorphic to . For example, take this matrix product: Or, equivalently,   All three of these matrices square to the negative of the identity matrix. If we take , , , and , it is easy to see that the span of the these four matrices is "the same as" (that is, isomorphic to) the set of quaternions . Exercises Using the presentation equations of Q8, write out the full product of two quaternions. In other words, given and , find the components of their product Show the composition algebra property Hint: use w: Euler's four-square identity. Axial pencils thumb|right|250px|The axis of a pencil is the real axis in the 4-algebra Hamilton's quaternions provide a picture of a pencil of complex number planes that fill out his hyperspace. Another pair of pencils provide alternative descriptions of 4-space as made up of planar algebras: The hyperspace is with the first coordinate taken as the real line, and as the axis of the various pencils. The Hamilton case uses the sphere of imaginary units Any pair of antipodal points on this sphere generates a plane isomorphic to the ordinary complex plane The second and third pencils derive from the findings of James Cockle and Arthur Cayley. Cayley set up an arithmetic of matrix multiplication which has expedited modern science. For instance, the so-called imaginary units are represented by which has multiplicative square equal to the negative of the identity matrix. But then there is also by which generates an algebraic plane distinct from ordinary complex numbers. This algebra split-binarions, has inverse proportion included as a structural feature, such as found in economics or spacetime. In fact, the Lorentz boost is exhibited by a split-binarion multiplication. The inherent relation was recognized in the 19th century by J. Cockle, W.K. Clifford, and A. Macfarlane in the English world and by some Serbians. The second pencil is an imaginary one without linear representation. As Hamilton had a sphere of imaginary units, Macfarlane would have a sphere of hyperbolic units u with u2 = +1. The full algebra of split-binarions is Any pair of elements that are polar opposite on this sphere generate a plane isomorphic to the split-binarions A. The 4-algebra containing this pencil is the hyperbolic quaternion algebra. As Oliver Heaviside and Willard Gibbs advocated a positive dot product for vectors, they have been associated with hyperbolic quaternions. When this algebra drew attention in the 1890s a "great vector debate" ensued in various publications including Nature. When the failure of the algebra to satisfy the associative law of multiplication was noted, it was realized that no matrix representation would be found. Each plane of the pencil can represent a Lorentz boost. However, rotations of the vector subspace, an operation within the reach of Hamilton's structure, is beyond the means of hyperbolic quaternions, hence the Lorentz group cannot be represented with Macfarlane's algebra. The third pencil arises from both imaginary units and hyperbolic units as found in the ring of 2x2 real matrices. In this figure there must be noted nilpotent matrices such as by which correspond to dual numbers in the matrix algebra. Such planes separate the complex and split-binarion planes, and are included in the pencil. The axis of the pencil is the line of matrices that are real multiples of the identity matrix. Over an alternate basis this ring is known as split-quaternions, and the pencil has three types of planar subrings. Quaternion algebras The real numbers are part of Hamilton's quaternions. An objective of abstract algebra is to show algebraic structures in their most general form. The real numbers satisfy the axioms of a field and they have other properties such as order and completeness. Given a field F of characteristic not equal to 2, it can be associated with an abstract algebra called a quaternion algebra as follows: The elements of a quaternion algebra are in the four-dimensional space over F :  V = F 4. As in F, there is a multiplicative identity 1 in V, and hence a one-dimensional subspace The multiplication on V has the feature that basis vectors of the orthogonal complement of in V are anticommutative. More particularly, with i and j perpendicular in this orthogonal complement of , i j = − j i. With k = i j, and perpendicular to the plane of i and j, associativity of multiplication implies jk = j i j = − i j j = −k j, and k i = i j i = −i k, so k also anticommutes with i and j. The multiplication on V is determined by multiplication on its basis vectors. That multiplication is not commutative (due to anticommutivity) and includes the negative for each basis vector: There are only two non-commutative groups of order eight: the quaternion group Q8 and the dihedral group of symmetries of a square figure D4. Thus, for any field F, there are two quaternion algebras over F corresponding to the selection of multiplication on B according to Q8 or D4. The smallest ones The finite fields GF(q ) lead to finite quadratic algebras. Here q is a power of a prime number. Since i ≠ − i in the groups, and since 1 ≡ −1 (mod 2), GF(2) is too small. The field Z3 is large enough and its quaternion algebras have 81 = 34 elements. The two basis-generated groups define multiplication in the algebras. The norms are isotropic quadratic forms since both algebras have null vectors, in the Q8 case because of the finite characteristic of the field. Historic incidents The extension of quaternion theory to F = C, the complex number plane, was given by Hamilton in his Lectures on Quaternions (1853).W. Hamilton (1853) Lectures on Quaternions, Page 639, quaternions with complex number scalars There was a glitch in an early presentation, with F being the field of real numbers, and the basis group confused. In 1878 w:W. K. Clifford nearly described representation of split-quaternions with matrices:w:W. K. Clifford (1878) Elements of Dynamic, page 170 via Internet Archive He used K to express the imaginary unit . The only flaw was a missing minus sign in the equation The inexplicable mistake in a case of a finite subgroup of the general linear group hindered understanding. The paragraphs in Elements of Dynamic included expressions equivalent to the equation These equations imply that I and J are hyperbolic units since the exponential series applied to an I or J axis splits into even and odd terms, which are series for hyperbolic cosine and hyperbolic sine respectively. Then forms a hyperbola. In the plane, the hyperbola satisfies so it is a unit hyperbola. The expression is a quadratic form. Among those who study quadratic forms, this form is associated with a hyperbolic plane with its unit hyperbola. Some clarity followed from a letter to the editor from w:C. G. KnottC.G. Knott Recent Innovations in Vector Theory Nature 47 1225 590–3 1893 10.1038/047590b0 1893Natur..47R.590. free read before the Royal Society of Edinburgh 19 December 1892 and published in Proceedings 1892) He reaffirmed Hamilton's products in this proposition: If a 4-algebra on basis is associative and off-diagonal products are given by Hamilton's rules, then . Proof: , so . Cycle the letters , , to obtain . Another hitch is the notion of isomorphism of algebras, which should include preserving the field F. Here w:Leonard Dickson ventured to nullify the difference between the complex algebras ( F = C ) over the two basis groups Q8 and D4. He wrote in Linear AlgebrasLeonard Dickson (1915) Equivalence of complex quaternion and matric algebras, Linear Algebras, page 15 that the complex quaternion and matric algebras were equivalent. There is a real isomorphism between them, but that is not a complex isomorphism preserving the field. Nevertheless, Dickson was one of the first to acknowledge the existence of a four-dimensional, real subalgebra of M(2,C) other than quaternions. Auxiliaries So far algebras over a field have been described. When algebras over a ring are considered, there are other "quaternion algebras". Clifford used split-binarions for scalars of his biquaternions, an algebra designated split-biquaternions. Several authors use dual numbers as scalars, giving dual quaternions. Another case abandons the Q8 and D4 convention for basis-element products. A quasigroup of order 8 and the real field give hyperbolic quaternions. These three auxiliary algebras, using quaternion in their names, fall outside consideration of standard works using the presumption of an algebra over a field. References Abstract Algebra