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Complexification

In mathematics, complexification is a fundamental construction that extends algebraic or geometric structures defined over the real numbers \mathbb{R} to analogous structures over the complex numbers \mathbb{C}, most commonly by forming the tensor product with \mathbb{C}.^[1] For a real vector space V, the complexification V_{\mathbb{C}} is defined as V \times V with complex scalar multiplication (a + bi)(u + iv) = (au - bv) + i(av + bu), equivalently V \otimes_{\mathbb{R}} \mathbb{C}, where elements are formal sums u + i v with u, v \in V.^[2] This yields a complex vector space of dimension equal to \dim_{\mathbb{R}} V, embedding V naturally via v \mapsto v + i \cdot 0.^[1] The complexification preserves key linear algebraic properties, such as extending real-linear maps \phi: V \to W uniquely to complex-linear maps \phi_{\mathbb{C}}: V_{\mathbb{C}} \to W_{\mathbb{C}} by \phi_{\mathbb{C}}(u + i v) = \phi(u) + i \phi(v), with kernels and images satisfying \ker(\phi_{\mathbb{C}}) = (\ker \phi)_{\mathbb{C}} and \im(\phi_{\mathbb{C}}) = (\im \phi)_{\mathbb{C}}.^[1] For linear operators T on V, the complexified operator T_{\mathbb{C}} has the same minimal polynomial as T but a characteristic polynomial over \mathbb{C} that factors completely, revealing complex eigenvalues even when T has none over \mathbb{R}.^[2] This enables the spectral theorem and Jordan form analysis for real operators, as every finite-dimensional real operator has a complexification with invariant subspaces of dimension 1 or 2.^[2] Beyond vector spaces, complexification extends to real Lie algebras \mathfrak{g}, yielding \mathfrak{g}_{\mathbb{C}} = \mathfrak{g} \otimes_{\mathbb{R}} \mathbb{C}, which decomposes semisimple representations into irreducible complex ones and facilitates study of real Lie groups via their complexifications, such as SL(n, \mathbb{R})_{\mathbb{C}} \cong SL(n, \mathbb{C}).^[3] In differential geometry, the complexification of a real vector bundle E \to M is E_{\mathbb{C}} = E \otimes_{\mathbb{R}} \mathbb{C}, supporting Chern classes and holomorphic structures.^[4] Applications span representation theory, where it classifies real representations through complex ones, and algebraic geometry, where real varieties embed into their complexifications for analytic continuation.^[3]^[5]

Fundamentals

Historical Background

The concept of complexification traces its early roots to 19th-century efforts in algebra to extend real number structures to solve polynomial equations, particularly through the development of complex numbers. Complex numbers emerged in the 16th century with Rafael Bombelli's work on cubic equations using Cardano's formula, which introduced imaginary units to handle roots like \sqrt{-1}, though they were initially viewed with skepticism. By the early 19th century, Carl Friedrich Gauss provided a geometric interpretation of complex numbers as points in the plane in his 1831 paper, solidifying their acceptance and motivating extensions beyond reals for algebraic solvability. This need intensified with William Rowan Hamilton's discovery of quaternions in 1843, a four-dimensional extension of complex numbers that preserved division but sacrificed commutativity, aimed at generalizing vector operations in three dimensions for physical applications like rotations.^[6]^[7]^[8] The related Cayley-Dickson construction, which includes the formation of complex numbers as an initial doubling step from the reals, was later generalized by Leonard E. Dickson in his 1919 paper on quaternions and their extensions, providing a framework for hypercomplex algebras like octonions. Detailed discussion of this doubling process appears in the Constructions section.^[9] By the mid-20th century, complexification had become integral to modern linear algebra and functional analysis, particularly through the tensor product with \mathbb{C}, facilitating spectral decompositions of real operators and unitary representations in quantum mechanics and harmonic analysis. This evolution enabled the study of real structures via their complex extensions, with key contributions in representation theory (e.g., decomposing real representations into complex irreducibles) and Lie theory (e.g., Claude Chevalley's work on complexifications of real Lie groups in the 1940s).^[10]^[11]^[12]

Formal Definition

The complexification of a real vector space V, denoted V^\mathbb{C}, is the complex vector space V \oplus V equipped with the scalar multiplication (a + bi)(w_1, w_2) = (a w_1 - b w_2, b w_1 + a w_2) for a, b \in \mathbb{R} and w_1, w_2 \in V, where i denotes the imaginary unit.^[1] This construction extends the real scalar multiplication on V to complex scalars while preserving the underlying real vector space structure.^[1] Alternatively, V^\mathbb{C} can be viewed as the set of formal sums v + i w with v, w \in V, forming a complex vector space under the operations (v + i w) + (v' + i w') = (v + v') + i (w + w') and (a + b i)(v + i w) = (a v - b w) + i (a w + b v).^[13] The embedding \iota: V \to V^\mathbb{C} is defined by \iota(v) = (v, 0) (or equivalently v + i \cdot 0), which identifies V as a real subspace of V^\mathbb{C}.^[1] This embedding ensures that V^\mathbb{C} is the smallest complex vector space containing an isomorphic copy of V as a real subspace, characterized by the universal property that any \mathbb{R}-linear map from V to a complex vector space W extends uniquely to a \mathbb{C}-linear map from V^\mathbb{C} to W.^[1] Assuming familiarity with real vector spaces and the field of complex numbers, the dimension satisfies \dim_\mathbb{C} V^\mathbb{C} = \dim_\mathbb{R} V.^[1]^[13]

Properties

Basic Properties

The complexification V^\mathbb{C} of a real vector space V has complex dimension equal to the real dimension of V. Specifically, if \dim_\mathbb{R} V = n, then \dim_\mathbb{C} V^\mathbb{C} = n. This follows from extending a real basis \{e_1, \dots, e_n\} of V to a complex basis \{e_1 \otimes 1, \dots, e_n \otimes 1\} (or equivalently \{(e_1, 0), \dots, (e_n, 0)\} in the direct sum construction) for V^\mathbb{C} over \mathbb{C}.^[1]^[14] As real vector spaces, V^\mathbb{C} is isomorphic to V \oplus iV, which has real dimension $2n. The isomorphism identifies elements of V^\mathbb{C} with pairs (v, w) for v, w \in V, where the complex structure is given by the linear operator J: V^\mathbb{C} \to V^\mathbb{C} satisfying J(v, w) = (-w, v) and J^2 = -\mathrm{Id}. This endows the twice-dimensional real space with a compatible complex multiplication, distinguishing it from the original real structure.^[1]^[14] The embedding \iota: V \to V^\mathbb{C} maps v \mapsto v \otimes 1 (or (v, 0)), and its image \iota(V) is a real subspace of V^\mathbb{C} isomorphic to V. Moreover, V^\mathbb{C} = \iota(V) + i \iota(V) decomposes as a direct sum of real subspaces, with i \iota(V) also isomorphic to V and the sum being internal over \mathbb{R}. This decomposition highlights the algebraic layering introduced by complexification.^[1]^[14] The complexification is unique up to canonical isomorphism: any two complexifications of V admit a unique \mathbb{C}-linear isomorphism commuting with the respective embeddings from V. This uniqueness stems from the universal mapping property, ensuring that linear maps from V to complex spaces extend uniquely to the complexification.^[1]^[14]

Complex Conjugation

In the context of the complexification V^\mathbb{C} of a real vector space V, complex conjugation is the unique \mathbb{R}-linear map \chi: V^\mathbb{C} \to V^\mathbb{C} such that \chi(v) = v for all v \in \iota(V) and \chi(i v) = -i v for all v \in \iota(V), where \iota: V \hookrightarrow V^\mathbb{C} denotes the canonical inclusion.^[1] This map is antilinear over \mathbb{C}, meaning \chi(z w) = \bar{z} \chi(w) for z \in \mathbb{C} and w \in V^\mathbb{C}, and it serves as the canonical involution that encodes the underlying real structure.^[15] Using the tensor product representation V^\mathbb{C} = V \otimes_\mathbb{R} \mathbb{C}, the conjugation takes the explicit form \chi(v \otimes z) = v \otimes \bar{z}, where \bar{z} is the standard complex conjugate of z.^[1] Key properties include \chi^2 = \mathrm{id}, confirming it is an involution, and its anti-holomorphic nature, which distinguishes it from \mathbb{C}-linear maps.^[16] The fixed points of \chi are precisely the kernel of \chi - \mathrm{id}, which coincides with the image \iota(V) of the real subspace V.^[1] This conjugation enables the decomposition of V^\mathbb{C} into eigenspaces: the +1-eigenspace \ker(\chi - \mathrm{id}) consists of the real part, isomorphic to V, while the -1-eigenspace \ker(\chi + \mathrm{id}) forms the imaginary part, isomorphic to iV.^[15] Thus, V^\mathbb{C} \cong V \oplus iV as real vector spaces, with \chi acting as the identity on the first summand and negation on the second, facilitating the recovery of the original real structure from the complexified space.^[16]

Constructions

Tensor Product Construction

The tensor product construction provides a concrete realization of the complexification of a real vector space V by extending the scalars from \mathbb{R} to \mathbb{C}. Specifically, the complexification V_{\mathbb{C}} is defined as the tensor product V \otimes_\mathbb{R} \mathbb{C}, initially viewed as an abelian group under the standard tensor product structure over \mathbb{R}.^[1]^[17] To equip this with the structure of a complex vector space, complex scalar multiplication is introduced by the rule \alpha \cdot (v \otimes \beta) = v \otimes (\alpha \beta) for all \alpha, \beta \in \mathbb{C} and v \in V, leveraging the bilinearity of the tensor product.^[1] This operation is well-defined because \mathbb{C} acts on itself by multiplication, and it satisfies the axioms of a complex vector space, with the natural embedding V \hookrightarrow V_{\mathbb{C}} given by v \mapsto v \otimes 1.^[17] A key feature of this construction is its universal property, which characterizes V_{\mathbb{C}} up to unique isomorphism. For any complex vector space W and any \mathbb{R}-linear map f: V \to W, there exists a unique \mathbb{C}-linear map \tilde{f}: V_{\mathbb{C}} \to W such that the diagram

\begin{CD} V @>f>> W \\ @VV{\iota}V @| \\ V_{\mathbb{C}} @>{\tilde{f}}>> W \end{CD}

commutes, where \iota: V \to V_{\mathbb{C}} is the embedding.^[1] This property ensures that the tensor product construction is canonical and facilitates the extension of \mathbb{R}-linear operators to \mathbb{C}-linear ones on the complexification.^[17] In terms of module structure, V_{\mathbb{C}} can be understood as the free \mathbb{C}-module generated by V over \mathbb{R}. If \{e_j\} is an \mathbb{R}-basis for V, then \{e_j \otimes 1\} forms a \mathbb{C}-basis for V_{\mathbb{C}}, confirming that V_{\mathbb{C}} is freely generated by the image of V with no relations beyond those imposed by the tensor product.^[1] This free module perspective highlights the minimal extension needed to incorporate complex scalars while preserving the linear independence and spanning properties from the real case. The construction also commutes with direct sums, meaning that for a family of real vector spaces \{V_i\}_{i \in I}, the complexification satisfies ( \bigoplus_{i \in I} V_i )_{\mathbb{C}} \cong \bigoplus_{i \in I} V_{i,\mathbb{C}} as complex vector spaces.^[17] This isomorphism follows from the general fact that tensor products over a field commute with arbitrary direct sums, allowing complexification to be applied componentwise without altering the overall structure.^[1]

Dickson Doubling

The Cayley-Dickson construction, also known as Dickson doubling, provides an iterative method to construct a sequence of real algebras by doubling the dimension, starting from the real numbers and yielding the complex numbers as the first extension. Given a real algebra A equipped with an involution * (a conjugate-linear anti-automorphism satisfying a^{**} = a) and a norm N: A \to \mathbb{R}, the doubled algebra A' is formed as the direct sum A \oplus A j, where j is a new imaginary unit with j^2 = -1. The multiplication in A' is defined by the rule

(a + b j)(c + d j) = (a c - \bar{d} b) + (\bar{a} d + b \bar{c}) j,

where \bar{\cdot} denotes the involution (often the same as *).^[18] This construction extends the algebra while inheriting the involution via (a + b j)^* = \bar{a} - b j and a compatible norm N(a + b j) = N(a) + N(b).^[18] Leonard Eugene Dickson formulated this doubling process in 1919 as part of his work on the arithmetics of algebras, demonstrating how it generates higher-dimensional structures from lower ones, such as constructing the octonions from quaternions.^[9] In the context of complexification, applying the construction to A = \mathbb{R} (with trivial involution and N(r) = r^2) yields the complex numbers \mathbb{C} with the familiar multiplication (a + b j)(c + d j) = (a c - b d) + (a d + b c) j. Iterating further, setting A = \mathbb{C} produces the quaternions \mathbb{H}, and continuing to A = \mathbb{H} yields the octonions \mathbb{O}, forming the sequence of Cayley-Dickson algebras.^[9]^[19] A key property of this doubling is its preservation of the division algebra structure—meaning every nonzero element has a multiplicative inverse—for dimensions up to 8 (the octonions), aligning with the Hurwitz theorem on normed division algebras over the reals.^[19] Beyond this, subsequent doublings (e.g., to the 16-dimensional sedenions) introduce zero divisors, and associativity is lost starting from the octonions themselves, though alternative associativity holds in subalgebras. This process fundamentally trades commutativity and associativity for increased dimensionality.^[19] This makes the Cayley-Dickson construction particularly useful for studying non-associative structures in algebra and geometry, with the initial step providing a concrete realization of the complexification of \mathbb{R}, distinct from the general tensor product method.^[9]

Examples and Extensions

Examples

The complexification of the finite-dimensional real vector space \mathbb{R}^n is the complex vector space \mathbb{C}^n, constructed via the extension of scalars where each component is allowed to be complex, preserving the standard Euclidean structure.^[1] If \{e_1, \dots, e_n\} denotes the standard basis of \mathbb{R}^n, then the corresponding basis for \mathbb{C}^n as the complexification is \{e_k \otimes 1 \mid k = 1, \dots, n\}, reflecting the tensor product construction with \mathbb{C} \cong \mathbb{R} \oplus i\mathbb{R}.^[2] For matrix algebras, the complexification of M_m(\mathbb{R}), the algebra of m \times m real matrices under matrix multiplication, yields M_m(\mathbb{C}), the algebra of m \times m complex matrices, obtained by applying the complexification entrywise to each matrix element.^[1] This entrywise process ensures that the resulting structure remains an associative algebra, with multiplication and addition defined componentwise in the complex entries, maintaining the original dimension m^2 over \mathbb{C}.^[1] In the realm of function spaces, the complexification of C^\infty(\mathbb{R}), the real vector space of smooth real-valued functions on \mathbb{R} equipped with pointwise addition and scalar multiplication, is the space of complex-valued smooth functions on \mathbb{R}, denoted C^\infty(\mathbb{R}, \mathbb{C}).^[20] Operations in this complexified space are likewise pointwise, allowing functions of the form f + ig where f, g \in C^\infty(\mathbb{R}) and i is the imaginary unit, thus extending differentiability and other properties to the complex domain.^[20] For infinite-dimensional cases, consider the real Hilbert space L^2(\mathbb{R}) consisting of real-valued square-integrable functions on \mathbb{R} with respect to Lebesgue measure; its complexification is the complex Hilbert space L^2(\mathbb{R}, \mathbb{C}) of complex-valued square-integrable functions. This extension preserves the inner product structure up to the standard complex conjugation and enables applications such as the complexification of Fourier transforms for analyzing signals in quantum mechanics and harmonic analysis.

Linear Transformations

Given a real linear map f: V \to W between real vector spaces, its complexification f^\mathbb{C}: V^\mathbb{C} \to W^\mathbb{C} is defined using the tensor product construction by f^\mathbb{C}(v \otimes z) = f(v) \otimes z for v \in V and z \in \mathbb{C}. This extension is \mathbb{C}-linear, as it respects scalar multiplication: for c \in \mathbb{C}, f^\mathbb{C}((v \otimes z) \cdot c) = f^\mathbb{C}(v \otimes (z c)) = f(v) \otimes (z c) = (f(v) \otimes z) \cdot c = f^\mathbb{C}(v \otimes z) \cdot c.^[1]^[21] The complexification preserves the inclusion maps \iota_V: V \to V^\mathbb{C} and \iota_W: W \to W^\mathbb{C} defined by \iota_V(v) = v \otimes 1, satisfying f^\mathbb{C} \circ \iota_V = \iota_W \circ f. Specifically, f^\mathbb{C}(\iota_V(v)) = f^\mathbb{C}(v \otimes 1) = f(v) \otimes 1 = \iota_W(f(v)). For inner product spaces with compatible real inner products, the complexification also respects adjoints: if f^* is the adjoint of f with respect to the real inner products, then (f^*)^\mathbb{C} = (f^\mathbb{C})^* with respect to the induced Hermitian inner products on V^\mathbb{C} and W^\mathbb{C}. This follows because the complex inner product on the complexified spaces extends the real one, ensuring \langle f^\mathbb{C}(u), w \rangle_{\mathbb{C}} = \langle u, (f^\mathbb{C})^*(w) \rangle_{\mathbb{C}} aligns with the extension of the real adjoint property.^[1]^[22] In terms of matrix representations, if f has a real matrix A with respect to bases of V and W, then f^\mathbb{C} has the same matrix A (viewed over \mathbb{C}) with respect to the induced bases \{e_i \otimes 1\} of V^\mathbb{C} and \{f_j \otimes 1\} of W^\mathbb{C}. The entries remain unchanged, as the action on basis elements e_i \otimes 1 maps to linear combinations with real coefficients matching A.^[21]^[1] This extension is canonical and unique: any \mathbb{C}-linear map g: V^\mathbb{C} \to W^\mathbb{C} that commutes with the inclusions (i.e., g \circ \iota_V = \iota_W \circ f) must coincide with f^\mathbb{C}. This uniqueness arises from the universal mapping property of the complexification functor, which ensures f^\mathbb{C} is the only such extension preserving the real structure.^[1]

Dual Spaces and Tensor Products

In the context of real vector spaces, the dual space of the complexification V^\mathbb{C} is canonically isomorphic to the complexification of the dual space V^*, where V^* = \Hom_\mathbb{R}(V, \mathbb{R}). This isomorphism identifies \Hom_\mathbb{C}(V^\mathbb{C}, \mathbb{C}) with \mathbb{C} \otimes_\mathbb{R} V^* and with \Hom_\mathbb{R}(V, \mathbb{C}) equipped with the complex structure (\lambda \cdot \phi)(v) = \lambda \phi(v) for \lambda \in \mathbb{C}, \phi \in V^*, and v \in V.^[23] Explicitly, for \phi \in V^*, the extension \phi^\mathbb{C} \in (V^\mathbb{C})^* is defined by linearity on pure tensors as

\phi^\mathbb{C}(v \otimes z) = z \cdot \phi(v)

for v \in V and z \in \mathbb{C}, ensuring \mathbb{C}-linearity.^[23] This construction holds algebraically for finite- or infinite-dimensional spaces without additional topology.^[23] However, in the setting of topological vector spaces, such as Banach spaces, the continuous dual of V^\mathbb{C} corresponds to the completion of the complexification of the continuous dual V^* under appropriate norms, like the Taylor norm \|x + iy\| = \sup_{0 \leq \theta \leq 2\pi} \|x \cos \theta + y \sin \theta\|_V.^[24] This ensures compatibility with bounded linear functionals, as the algebraic isomorphism may not preserve continuity without completion. Complexification extends naturally to tensor products of real vector spaces, commuting with the operation to yield (V \otimes_\mathbb{R} W)^\mathbb{C} \cong V^\mathbb{C} \otimes_\mathbb{C} W^\mathbb{C} as complex vector spaces.^[24] This isomorphism preserves multilinearity: a \mathbb{R}-bilinear map V \times W \to U extends to a \mathbb{C}-bilinear map V^\mathbb{C} \times W^\mathbb{C} \to U^\mathbb{C} via the universal property of tensor products. For Banach spaces equipped with projective or injective tensor norms, the isomorphism is isometric, maintaining boundedness and operator ideal properties.^[24] The functoriality of complexification further ensures it commutes with higher tensor constructions, including exterior and symmetric powers. Specifically, for the k-th exterior power, \bigwedge^k V^\mathbb{C} \cong (\bigwedge^k V)^\mathbb{C}, and analogously \Sym^k V^\mathbb{C} \cong (\Sym^k V)^\mathbb{C}, preserving alternation and symmetry in the multilinear structure.^[1] These isomorphisms arise from the universal properties of the respective algebras and facilitate applications in representation theory and differential geometry.

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Below is a merged summary of the sections related to Leonard Dickson, Cayley-Dickson Construction, Doubling Algebras, and Complexification, consolidating all information from the provided segments into a dense and organized format. To maximize detail retention, I will use a table in CSV format for key data points, followed by a narrative summary that integrates additional context and historical developments. Since the system has a "no thinking token allowed" constraint, I will focus on directly compiling and structuring the provided information without additional inference or synthesis beyond what is explicitly stated.
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Below is a merged response summarizing the "Complexification of the Dual Space" based on all provided segments from *Algebra I* by Alexey L. Gorodentsev. Since the document does not contain a section explicitly titled "Complexification of the Dual Space," the summary integrates relevant information from related sections (e.g., Chapters 7, 18.2.5, 20.3.2) and notes where content is missing or inferred. To retain all details efficiently, I will use a table in CSV format for key information, followed by a narrative summary and notes.
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In this chapter we show that there is a complexification procedure for the projective tensor norm, so that the complexification of the projective tensor product ...