Split-quaternion
In abstract algebra, split-quaternions, also known as coquaternions, are a four-dimensional associative algebra over the real numbers, introduced by James Cockle in 1849 as an extension of Hamilton's quaternions from 1843.[1] Elements of the algebra are expressed as q = a_0 + a_1 i + a_2 j + a_3 k, where a_0, a_1, a_2, a_3 \in \mathbb{R}, the basis elements satisfy the multiplication rules i^2 = -1, j^2 = k^2 = +1, and ij = k = -ji, leading to a non-commutative structure with zero divisors and an indefinite quadratic form I_q = a_0^2 + a_1^2 - a_2^2 - a_3^2.[2] Unlike the division algebra of ordinary quaternions, split-quaternions contain nilpotent elements and nontrivial idempotents, classifying them as a "split" algebra with signature (2,2).[1] Split-quaternions admit various representations, including as 2×2 matrices over the complex numbers or 4×4 real matrices, facilitating computations in linear algebra and geometry.[2] The norm is defined as N_q = \sqrt{|I_q|}, with invertible elements satisfying N_q \neq 0, and the conjugate q^* = a_0 - a_1 i - a_2 j - a_3 k enabling the inverse q^{-1} = q^* / (q q^*).[1] Unit split-quaternions, where N_q = 1, are categorized as timelike (I_q > 0), spacelike (I_q < 0), or lightlike (I_q = 0), each corresponding to distinct geometric behaviors in Minkowski space.[3] Notable applications include modeling rotations and Lorentz transformations in Minkowski 3-space, where unit timelike split-quaternions represent hyperbolic rotations, and spacelike ones handle elliptic rotations in semi-Euclidean spaces.[1] In physics, they provide an equivalent formulation of Dirac 4-spinors as 2-spinors, deriving the Dirac equation with SO(3,2;ℝ) symmetry and representing Lorentz transformations via 2×2 unitary matrices.[4] Further geometric interpretations link split-quaternions to lines and planes in Lorentzian geometry, classifying planes by the causality of their normals—timelike to the complex plane, spacelike to the hyperbolic plane, and lightlike to the dual number plane—using mutual pseudo-orthogonal bases.[3] These properties extend their use to matrix theory, differential geometry, quantum mechanics, and superstring theory.[1]Fundamentals
Definition
A split-quaternion is a four-dimensional algebra over the real numbers, consisting of elements of the form q = a + b \mathbf{i} + c \mathbf{j} + d \mathbf{k}, where a, b, c, d \in \mathbb{R}, the basis elements satisfy \mathbf{i}^2 = -1, \mathbf{j}^2 = \mathbf{k}^2 = +1, and \mathbf{i} \mathbf{j} = \mathbf{k} = - \mathbf{j} \mathbf{i}.[5] This structure was introduced by James Cockle in 1849 as an extension of Hamilton's quaternions, forming an associative but non-commutative algebra with zero divisors.[6] The basis \{1, \mathbf{i}, \mathbf{j}, \mathbf{k}\} spans the algebra as a vector space over \mathbb{R}, with multiplication defined by the relations above and extended by linearity and associativity. The full multiplication table for the basis elements is as follows:| × | 1 | \mathbf{i} | \mathbf{j} | \mathbf{k} |
|---|---|---|---|---|
| 1 | 1 | \mathbf{i} | \mathbf{j} | \mathbf{k} |
| \mathbf{i} | \mathbf{i} | -1 | \mathbf{k} | -\mathbf{j} |
| \mathbf{j} | \mathbf{j} | -\mathbf{k} | 1 | \mathbf{i} |
| \mathbf{k} | \mathbf{k} | \mathbf{j} | -\mathbf{i} | 1 |
Notation and Basic Operations
Split-quaternions are commonly represented in component form as q = (a, b, c, d), where a, b, c, d \in \mathbb{R}. This tuple notation emphasizes their structure as elements of a four-dimensional real vector space. Equivalently, a split-quaternion can be written using a basis expansion q = a + b \mathbf{e}_1 + c \mathbf{e}_2 + d \mathbf{e}_3, where the basis elements \{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\} satisfy the relations \mathbf{e}_1^2 = -1, \mathbf{e}_2^2 = \mathbf{e}_3^2 = 1, and \mathbf{e}_1 \mathbf{e}_2 = \mathbf{e}_3 = -\mathbf{e}_2 \mathbf{e}_1.[9] Addition of split-quaternions is defined component-wise. For q = (a, b, c, d) and p = (a', b', c', d'), the sum is q + p = (a + a', b + b', c + c', d + d'). In basis form, this corresponds to distributing the addition over the coefficients while preserving the basis elements. Scalar multiplication by a real number r \in \mathbb{R} scales each component: r q = (r a, r b, r c, r d), or equivalently r (a + b \mathbf{e}_1 + c \mathbf{e}_2 + d \mathbf{e}_3) = r a + r b \mathbf{e}_1 + r c \mathbf{e}_2 + r d \mathbf{e}_3. These operations make the set of split-quaternions into a vector space over \mathbb{R}.[9] The zero element of the algebra is the split-quaternion $0 = (0, 0, 0, 0), which acts as the additive identity: q + 0 = q for any q. The additive inverse of q = (a, b, c, d) is -q = (-a, -b, -c, -d), satisfying q + (-q) = 0. These properties follow directly from the component-wise nature of the operations.[9]Algebraic Properties
Multiplication and Composition
Split-quaternions form a non-commutative algebra under multiplication, where the product of two split-quaternions q = a + b i + c j + d k and p = a' + b' i + c' j + d' k, with a, b, c, d, a', b', c', d' \in \mathbb{R}, is given by \begin{align*} q p &= (a a' - b b' + c c' + d d') \\ &\quad + (a b' + b a' - c d' + d c') i \\ &\quad + (a c' - b d' + c a' + d b') j \\ &\quad + (a d' + b c' - c b' + d a') k. \end{align*} [10] This multiplication arises from the defining relations of the basis elements: i^2 = -1, j^2 = k^2 = +1, and ij = k = -ji, leading to jk = -i = -kj, ki = j = -ik.[10] To verify these basis rules, consider the product i j. Substituting q = i (so a = 0, b = 1, c = 0, d = 0) and p = j (so a' = 0, b' = 0, c' = 1, d' = 0) into the multiplication formula yields a scalar part of 0, an i-coefficient of 0, a j-coefficient of 0, and a k-coefficient of 1, confirming i j = k. Similarly, for j i, the computation gives a k-coefficient of -1, so j i = -k. The remaining rules follow analogously from the formula.[10] The algebra is associative, meaning (q r) p = q (r p) for all split-quaternions q, r, p, as the multiplication is bilinear over the reals and the basis relations preserve associativity in the generated algebra.[10] For example, the product (1 + i)(1 + j) is computed as follows: with q = 1 + i (so a = 1, b = 1, c = 0, d = 0) and p = 1 + j (so a' = 1, b' = 0, c' = 1, d' = 0), the formula gives a scalar of 1, an i-coefficient of 1, a j-coefficient of 1, and a k-coefficient of 1, resulting in $1 + i + j + k.[10]Norm, Conjugation, and Inverses
The conjugation of a split-quaternion q = a + b i + c j + d k, where a, b, c, d \in \mathbb{R}, is defined by reversing the signs of the coefficients of the imaginary units, yielding q^* = a - b i - c j - d k.[10] This operation is an anti-automorphism, satisfying (p q)^* = q^* p^* for any split-quaternions p and q.[10] The norm of q is the real number given by N(q) = q q^* = a^2 + b^2 - c^2 - d^2, which adopts a hyperbolic signature due to the indefinite nature of the underlying metric.[10] This norm is multiplicative, meaning N(p q) = N(p) N(q) for all split-quaternions p and q, a property inherited from the associative algebra structure.[10] Unlike the positive-definite norm of classical quaternions, the split-quaternion norm can take positive, negative, or zero values, reflecting the non-Euclidean geometry associated with the algebra.[1] A split-quaternion q admits a multiplicative inverse if and only if N(q) \neq 0, in which case q^{-1} = q^* / N(q).[10] This follows directly from the relation q \cdot (q^* / N(q)) = (q q^*) / N(q) = N(q) / N(q) = 1, and similarly for right multiplication due to associativity.[10] For example, consider q = 1 + i; then q^* = 1 - i and N(q) = 1^2 + 1^2 - 0^2 - 0^2 = 2, so q^{-1} = (1 - i)/2. Verification yields q q^{-1} = (1 + i)(1 - i)/2 = (1 - i^2)/2 = (1 - (-1))/2 = 2/2 = 1.[10] The units (invertible elements) in the split-quaternion algebra are precisely the elements q with N(q) \neq 0. The subgroup consisting of those with |N(q)| = 1 (i.e., N(q) = \pm 1) plays a key role in preserving the algebraic structure, analogous to the unit quaternions in the classical case but adapted to the split signature.[1]Representations
Matrix Form
Split-quaternions can be represented as 2×2 matrices with complex entries, providing an isomorphism between the split-quaternion algebra and a subalgebra of the matrix algebra M_2(\mathbb{C}). Specifically, a split-quaternion q = a + b i + c j + d k, where a, b, c, d \in \mathbb{R}, i^2 = -1, j^2 = 1, k = i j = - j i, and k^2 = 1, is mapped to the matrix \begin{pmatrix} a + b i & c + d i \\ c - d i & a - b i \end{pmatrix}. This representation preserves the algebraic structure, as matrix multiplication corresponds exactly to the non-commutative multiplication of split-quaternions. For instance, the images of the basis elements under this map satisfy the defining relations: the matrix for i is \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, which squares to -I; the matrix for j is \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, which squares to I; and the matrix for k is \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}, whose square is I. The product of the matrices for i and j yields the matrix for k, confirming the relation i j = k. One key advantage of this matrix form is that it allows the application of standard linear algebra techniques to split-quaternions, such as computing eigenvalues or singular value decompositions for split-quaternionic matrices. Notably, the determinant of the representing matrix equals the squared norm of the split-quaternion, given by \det \begin{pmatrix} a + b i & c + d i \\ c - d i & a - b i \end{pmatrix} = a^2 + b^2 - c^2 - d^2, which reflects the indefinite quadratic form characteristic of the split signature.Construction from Split-Complex Numbers
Split-complex numbers form a two-dimensional commutative algebra over the real numbers, with elements expressed as z = x + y j, where x, y \in \mathbb{R} and the basis element j satisfies j^2 = 1. The norm of such an element is given by N(z) = x^2 - y^2, which can be indefinite, reflecting the hyperbolic nature of the algebra.[11] Split-quaternions extend this structure in a manner analogous to the construction of ordinary quaternions from complex numbers, by treating split-complex numbers as the base field and adjoining a new imaginary unit i perpendicular to the existing basis, with i^2 = -1 and anticommuting with j (i.e., i j = - j i). This yields elements of the form q = z + w i, where z, w are split-complex numbers, expanding the algebra to four dimensions over \mathbb{R} with basis \{1, j, i, k\}, where k = i j.[11] Multiplication in this construction inherits the rules from the split-complex algebra for the z and w components, while cross terms incorporate the anticommutation: specifically, the product (z + w i)(z' + w' i) = z z' - \bar{w}' w i^2 + (z w' + w \bar{z}') i, adjusted for the split-complex conjugation \bar{z} = x - y j, leading to the full basis relations i^2 = -1, j^2 = 1, k^2 = 1, i j = k = -j i, j k = -i = -k j, and k i = j = -i k.[12] This approach parallels William Rowan Hamilton's 1843 construction of quaternions but adapts it to the hyperbolic geometry underlying split-complex numbers, as introduced by James Cockle in 1849 under the name coquaternions to explore algebraic extensions with mixed signature metrics.[13]Classification and Stratification
Norm-Based Stratification
Split-quaternions are stratified according to the sign of their quadratic form I_q, which serves as a fundamental invariant distinguishing distinct algebraic and geometric behaviors within the algebra. Those with I_q > 0 are termed timelike, drawing an analogy to timelike vectors in Minkowski spacetime where the metric is positive. Split-quaternions satisfying I_q < 0 are classified as spacelike, akin to spacelike vectors with negative metric values, while the case I_q = 0 defines the lightlike stratum.[1] This norm-based stratification can be visualized along the real line representing possible values of I_q, partitioned into the positive ray for timelike elements, the negative ray for spacelike elements, and the origin for lightlike elements. For instance, the real scalar q = 1 has I_1 = 1 > 0, exemplifying a timelike unit that acts as the multiplicative identity across strata.[1] Geometrically, the level sets of the quadratic form form surfaces governed by an indefinite quadratic form of signature (2,2), dividing the space into distinct sheets: timelike values yield two-sheeted hyperboloids, spacelike values produce one-sheeted hyperboloids, and lightlike values trace a double cone analogous to a light cone. This structure underscores the non-Euclidean nature of split-quaternions, contrasting with the definite norm in classical quaternions.[1] The classification bears a close resemblance to the (1,3) signature of Minkowski space in special relativity, where timelike, spacelike, and lightlike distinctions guide causal structures and Lorentz transformations, enabling split-quaternions to model rotations and boosts in Lorentzian geometries despite their (2,2) signature.[1]Nilpotent Elements
In split-quaternions, a nonzero element q is nilpotent if q^2 = 0. These elements form a subset of the lightlike vectors, characterized by having zero scalar part and zero quadratic form under the associated quadratic form.[14] Such nilpotents are precisely the pure split-quaternions q = a i + b j + c k (with a, b, c \in \mathbb{R}, not all zero) satisfying a^2 = b^2 + c^2. This condition arises from the multiplication rules i^2 = -1, j^2 = k^2 = 1, ij = k = -ji, jk = -i = -kj, ki = j = -ik, ensuring the scalar part of q^2 vanishes while cross terms cancel due to anticommutativity. For instance, q = i + j yields (i + j)^2 = i^2 + ij + ji + j^2 = -1 + k - k + 1 = 0, demonstrating nilpotency of index 2. Similarly, i - j satisfies (i - j)^2 = 0.[14][15] The set of nilpotent elements is not closed under addition; for example, (i + j) + (i - j) = 2i, and (2i)^2 = -4 \neq 0. These elements are non-invertible, as zero quadratic form precludes inverses in the algebra. They generate ideals within the split-quaternion ring and contribute to its radical, underscoring the presence of zero divisors and the algebra's departure from being a division ring.[1][14]Imaginary and Hyperbolic Units
Split-quaternions possess a distinguished basis consisting of the real unit 1 and three additional units: the imaginary unit i and the hyperbolic units j and k. These satisfy the relations i^2 = -1, j^2 = 1, k^2 = 1, ij = k = -ji, jk = -i = -kj, and ki = j = -ik, with ijk = 1.[1][16] The imaginary unit i lies in the timelike sector, where its square is negative, while the hyperbolic units j and k reside in the spacelike sector, characterized by positive squares. These basis units generate the algebra and underpin its geometric interpretations, with the quadratic form Q(q) = a^2 + b^2 - c^2 - d^2 for q = a + bi + cj + dk distinguishing their strata: Q(i) = 1 > 0, Q(j) = Q(k) = -1 < 0.[1] The multiplication among the imaginary and hyperbolic units follows a specific table that highlights their non-commutativity and interplay between timelike and spacelike behaviors:| i | j | k | |
|---|---|---|---|
| i | -1 | k | -j |
| j | -k | $1 | -i |
| k | j | i | $1 |