Fact-checked by Grok 2 weeks ago

Algebra over a field

Algebra over a field, often denoted as a k-algebra where k is the base field, is a fundamental algebraic structure in mathematics consisting of a vector space A over k equipped with a bilinear multiplication operation A × A → A that is typically associative and unital, allowing elements to be combined in a way compatible with scalar multiplication from k.^[1]^[2] This structure generalizes both rings and vector spaces, embedding the field k into the center of the algebra via scalar multiplication, ensuring that multiplication distributes over addition and interacts linearly with field elements.^[3] Key examples include field extensions K/k, such as the complex numbers ℂ over the reals ℝ, which form a 2-dimensional algebra; matrix rings M_n(k), which are n²-dimensional and non-commutative for n > 1; and group algebras k[G] for a finite group G, combining group theory with linear algebra.^[1] Algebras may be commutative (multiplication satisfies ab = ba) or non-commutative, finite-dimensional or infinite-dimensional, with the latter including spaces of continuous functions C([0,1], ℝ) over ℝ.^[1]^[3] Basic concepts encompass subalgebras (closed under multiplication and addition, as k-subspaces), homomorphisms (linear ring maps preserving multiplication and the unit), and ideals (subspaces closed under multiplication by algebra elements).^[1] Important properties include the Cayley-Hamilton theorem, which states that every element a satisfies its own characteristic polynomial derived from the left multiplication map; traces and norms from the characteristic polynomial; and the classification of simple finite-dimensional algebras over algebraically closed fields as matrix rings over the field.^[1]^[2] These structures underpin diverse areas such as representation theory, where algebras act on vector spaces via endomorphisms; Lie theory, via associative enveloping algebras; and non-commutative geometry, with applications in physics like quantum mechanics through division algebras such as the quaternions over ℝ.^[3]^[2] Wedderburn's little theorem states that every finite division ring is a field. The Artin–Wedderburn theorem implies that central simple algebras over a field are isomorphic to matrix rings over central division algebras over that field.^[1]

Introduction and Definition

Motivating Examples

One of the simplest and most intuitive examples of an algebra over a field K is the polynomial ring K in one indeterminate. This structure is a vector space over K with basis \{1, x, x^2, \dots \}, making it infinite-dimensional, and the multiplication is defined by the usual extension of the product of monomials, which is bilinear with respect to scalar multiplication by elements of K.^[4] Such polynomial algebras arise naturally in algebraic geometry and commutative algebra, where they model functions on affine spaces and facilitate the study of ideals and varieties through their infinite basis and commutative multiplication.^[4] For a finite-dimensional and non-commutative instance, consider the algebra M_n(K) of n \times n matrices with entries in K. This forms a vector space of dimension n^2 over K, with the standard basis consisting of the matrix units E_{ij} (matrices with a 1 in the (i,j)-entry and zeros elsewhere), and multiplication given by matrix multiplication, which is bilinear over K but does not commute in general.^[5] Matrix algebras like M_n(K) are central in linear algebra and representation theory, capturing linear transformations on K^n and enabling the analysis of symmetries through their associative, non-commutative structure.^[5] Another motivating construction is the group algebra K[G] associated to a finite group G, where the underlying vector space has basis the elements of G and multiplication is the K-bilinear extension of the group operation.^[6] This algebra, which has dimension |G| over K, bridges group theory and linear algebra by identifying representations of G with modules over K[G], thus providing a linear algebraic framework for studying group actions and characters.^[6] Many such examples, including polynomial and matrix algebras, are unital with the identity serving as the multiplicative unit. The exterior algebra \Lambda(V) on a finite-dimensional vector space V over K offers an example with additional grading structure. It is generated by V placed in degree 1, forming a graded vector space where the multiplication (the wedge product) is bilinear, associative, and graded-commutative, meaning elements of odd degree anticommute while even-degree elements commute.^[7] With dimension $2^{\dim V} and basis the wedge products of basis elements of V, the exterior algebra models antisymmetric multilinear forms and underpins differential geometry, such as in the construction of differential forms on manifolds.^[7] This structure highlights how algebras over a field can incorporate grading to capture geometric and topological invariants.

Formal Definition

An algebra A over a field K is a vector space over K equipped with a bilinear map m: A \times A \to A, called the multiplication.^[8]^[9] The bilinearity of the multiplication means that it is linear in each argument separately. Specifically, for all \lambda, \mu \in K and a, b, c \in A,

\begin{align*} m(\lambda a + \mu b, c) &= \lambda m(a, c) + \mu m(b, c), \\ m(a, \lambda b + \mu c) &= \lambda m(a, b) + \mu m(a, c). \end{align*}

^[8]^[9] In general, there is no requirement that the multiplication be associative, commutative, or admit a unit element unless specified otherwise in a particular context.^[8]^[10] Such algebras are often denoted by (A, m) to emphasize the multiplication, or simply by A with the operation indicated by juxtaposition ab or the dot product a \cdot b.^[9]^[10]

Basic Structures and Operations

Algebra Homomorphisms

In the category of algebras over a field K, a homomorphism \phi: A \to B between two K-algebras A and B is a map that preserves both the vector space structure and the multiplication. Specifically, \phi is a K-linear map, meaning \phi(\alpha a + \beta b) = \alpha \phi(a) + \beta \phi(b) for all \alpha, \beta \in K and a, b \in A, and it satisfies the multiplicative property \phi(ab) = \phi(a)\phi(b) for all a, b \in A, and preserves the unit \phi(1_A) = 1_B.^[1]^[11] This ensures that \phi respects the ring structure while being compatible with the scalar multiplication from K, distinguishing algebra homomorphisms from mere ring homomorphisms.^[12] The kernel of an algebra homomorphism \phi: A \to B, denoted \ker(\phi) = \{a \in A \mid \phi(a) = 0\}, forms a two-sided ideal in A. This follows from the fact that algebra homomorphisms are ring homomorphisms, and the kernel of any ring homomorphism is an ideal, with the K-linearity ensuring the ideal is also a K-subspace.^[13]^[1] Conversely, the image \operatorname{im}(\phi) = \{\phi(a) \mid a \in A\} is a subalgebra of B, as it is closed under addition, scalar multiplication, and the multiplication in B, inheriting the algebraic structure from A via \phi.^[1] These properties enable the first isomorphism theorem for algebras: if \phi is surjective, then A / \ker(\phi) \cong B as K-algebras.^[13] An algebra isomorphism is a bijective algebra homomorphism whose inverse is also an algebra homomorphism. Such a map preserves all algebraic structures, including the field scalar multiplication, addition, and multiplication, establishing an equivalence between the algebras.^[1] Isomorphisms are central to classifying algebras up to structural similarity, as they identify algebras that are essentially the same despite different presentations. The free algebra on a K-vector space V is the universal object for algebra homomorphisms from V, satisfying a universal mapping property: for any K-algebra B and any K-linear map f: V \to B, there exists a unique algebra homomorphism \tilde{f} from the free algebra to B extending f. This free algebra can be constructed as the tensor algebra T(V), the quotient of the free unital associative algebra on V by no relations beyond those imposed by the field.^[14]^[15]

Subalgebras and Ideals

In an algebra A over a field F, a subalgebra is a subset S \subseteq A that forms a subspace over F and is closed under the multiplication of A, thereby inheriting the full algebra structure from A.^[12] More precisely, S must be an F-subspace such that S \cdot S \subseteq S, and if A is unital, S typically contains the unit element, though non-unital subalgebras are also considered in general contexts.^[16] This closure ensures that S is itself an algebra over F, allowing the study of algebraic structures within larger ones; for example, the set of scalar matrices forms a subalgebra isomorphic to F inside the matrix algebra M_n(F).^[12] Ideals in an algebra A over F are subspaces that absorb multiplication from A, generalizing the notion from ring theory while leveraging the vector space structure. A left ideal I \subseteq A satisfies A \cdot I \subseteq I, a right ideal satisfies I \cdot A \subseteq I, and a two-sided ideal satisfies both conditions simultaneously.^[16] Since F is commutative and acts centrally, any ring ideal in A is automatically an F-subspace, making the definitions align seamlessly.^[12] For instance, in the polynomial algebra F, the subspace generated by x^n is a two-sided ideal, as multiplication by any polynomial shifts degrees without escaping the span.^[16] Given a two-sided ideal I in A, the quotient A/I is defined as the quotient vector space equipped with the induced multiplication (a + I)(b + I) = ab + I, forming an algebra over F via the natural projection map, which is an algebra homomorphism.^[12] This construction satisfies a universal property: any algebra homomorphism from A to another algebra B with kernel containing I factors uniquely through A/I.^[16] An example is the quotient F/(x^n), which yields a finite-dimensional algebra of truncated polynomials, useful for studying nilpotent elements.^[12] A maximal ideal in A is a proper two-sided ideal not contained in any larger proper two-sided ideal; if A is finite-dimensional over F, the quotient A/M by a maximal ideal M is a simple algebra.^[16] In the commutative case, A/M is a field extension of F. A simple algebra over F is a finite-dimensional algebra with no nontrivial two-sided ideals other than \{0\} and itself.^[17] Matrix algebras M_n(F) exemplify simple algebras, as their only two-sided ideals are trivial due to the density of invertible matrices.^[17]

Constructions and Extensions

Extension of Scalars

In the context of algebras over fields, the extension of scalars provides a method to "change the base field" from a field K to a larger field L containing K, while preserving the algebraic structure. Given a field extension L/K and a K-algebra A, the extension of scalars is defined as the tensor product A_L = A \otimes_K L, where L is viewed as a K-vector space. This construction equips A_L with an L-algebra structure by extending the multiplication bilinearly: for a, b \in A and \lambda, \mu \in L, the product is given by

(a \otimes \lambda)(b \otimes \mu) = (ab) \otimes (\lambda \mu).

The scalar multiplication by elements of L is defined via \nu \cdot (a \otimes \lambda) = a \otimes (\nu \lambda) for \nu \in L.^[1]^[18] As an L-algebra, A_L inherits key properties from A. In particular, if A is finite-dimensional as a K-vector space with dimension n = [A : K], then A_L is finite-dimensional as an L-vector space with the same dimension n = [A_L : L], since the tensor product preserves the dimension in this setting. Additionally, the extension operation is functorial: for a K-algebra homomorphism f: A \to B, there is an induced L-algebra homomorphism \mathrm{id}_A \otimes f: A_L \to B_L, and for a field homomorphism \phi: K \to L, the base change is compatible via \mathrm{id}_A \otimes \phi. This functoriality ensures that the construction respects morphisms and makes extension of scalars a covariant functor from K-algebras to L-algebras.^[1]^[19]^[18] Concrete examples illustrate the utility of this construction. For the polynomial algebra A = K, the extension yields K \otimes_K L \cong L, the polynomial ring over L, which allows studying polynomials over larger fields without altering their structure. Similarly, for the matrix algebra A = M_n(K), the full matrix ring over K, the extension is isomorphic to M_n(L), the full matrix ring over L; this isomorphism arises because matrix multiplication extends naturally via the tensor product, preserving the non-commutative structure. These examples demonstrate how extension of scalars facilitates the transfer of algebraic properties, such as representations or ideals, to a broader scalar field.^[1]^[18]

Tensor Products of Algebras

Let A and B be algebras over a field K. The tensor product A \otimes_K B is the K-vector space generated by symbols a \otimes b for a \in A, b \in B, subject to the relations of bilinearity over K: (a_1 + a_2) \otimes b = a_1 \otimes b + a_2 \otimes b, a \otimes (b_1 + b_2) = a \otimes b_1 + a \otimes b_2, and (c a) \otimes b = a \otimes (c b) = c (a \otimes b) for c \in K. This space is equipped with an algebra multiplication defined by

(a_1 \otimes b_1)(a_2 \otimes b_2) = (a_1 a_2) \otimes (b_1 b_2)

for all a_i \in A, b_i \in B, extended linearly; this operation is bilinear over K and associative, making A \otimes_K B into a K-algebra.^[20] The tensor product satisfies a universal property with respect to algebra homomorphisms: given any K-algebra C and K-algebra homomorphisms \phi: A \to C, \psi: B \to C, there exists a unique K-algebra homomorphism \theta: A \otimes_K B \to C such that \theta(a \otimes b) = \phi(a) \psi(b) for all a \in A, b \in B. This property characterizes A \otimes_K B up to unique isomorphism as the coproduct in the category of K-algebras.^[20] If A and B are unital K-algebras, then A \otimes_K B is unital with multiplicative identity $1_A \otimes 1_B. Moreover, if A and B are finite-dimensional as K-vector spaces with \dim_K A = m and \dim_K B = n, then \dim_K (A \otimes_K B) = m n.^[20] Representative examples illustrate the construction. The tensor product of polynomial algebras is K \otimes_K K \cong K[x, y], the polynomial ring in two commuting variables over K, via the map sending x \otimes 1 to x and $1 \otimes y to y. For matrix algebras, M_m(K) \otimes_K M_n(K) \cong M_{m n}(K), where the isomorphism arises from the identification of simple tensors of matrix units with larger matrix units.^[20]

Types and Examples

Unital Algebras

A unital algebra, also known as a unitary algebra, over a field K is a K-algebra A equipped with a distinguished element $1_A \in A, called the multiplicative identity or unit, satisfying $1_A \cdot a = a \cdot 1_A = a for all a \in A. This unit element ensures that the multiplication in A behaves compatibly with the scalar multiplication from K, preserving the algebraic structure while allowing for identity-preserving operations. The presence of the unit distinguishes unital algebras from more general algebras, enabling concepts like invertibility and direct sums in a straightforward manner.^[21] For any non-unital algebra A over a field K, the unitization (or minimal unitization) A^+ provides a canonical way to adjoin a unit. As a vector space, A^+ = A \oplus K, with multiplication defined by

(a, \lambda) \cdot (b, \mu) = (ab + \lambda b + \mu a, \lambda \mu)

for a, b \in A and \lambda, \mu \in K. The element (0, 1) serves as the unit in A^+, and the original algebra A embeds as the ideal \{(a, 0) \mid a \in A\}. This construction is universal: any homomorphism from A to a unital algebra extends uniquely to a unital homomorphism from A^+ to that algebra.^[22]^[21] Unital algebras exhibit key properties regarding their morphisms and substructures. A homomorphism \phi: A \to B between unital algebras over the same field K must preserve the unit, meaning \phi(1_A) = 1_B, ensuring that the map respects the identity operation. This requirement aligns with the standard definition of ring homomorphisms for unital rings, extended to the vector space setting of algebras. Subalgebras of a unital algebra A need not contain $1_A; those that do are themselves unital, inheriting the unit, while others form non-unital subalgebras.^[23]^[24] Prominent examples of unital algebras include the polynomial algebra K, where the constant polynomial $1 acts as the unit, and the matrix algebra M_n(K) of n \times n matrices over K, with the identity matrix I_n as the unit. These structures are fundamental in linear algebra and representation theory, illustrating how the unit facilitates computations like solving linear systems or defining group representations.^[22]

Associative Algebras

An associative algebra over a field K is a vector space A over K equipped with a bilinear multiplication operation A \times A \to A that satisfies the associative law: (ab)c = a(bc) for all a, b, c \in A.^[1] This structure generalizes both rings and vector spaces, allowing scalar multiplication from K to distribute over the algebra's product. Many associative algebras are unital, possessing a multiplicative identity element $1_A \in A such that $1_A a = a 1_A = a for all a \in A.^[18] In an associative algebra, the associator [a, b, c] = (ab)c - a(bc) vanishes identically, implying that the nucleus N(A) = \{ n \in A \mid [n, A, A] = [A, n, A] = [A, A, n] = 0 \} coincides with the entire algebra A. Concepts from ring theory extend naturally to associative algebras: an associative algebra A is Artinian if it satisfies the descending chain condition on left (or right) ideals, and Noetherian if it satisfies the ascending chain condition on left (or right) ideals, mirroring the definitions for rings but leveraging the underlying vector space structure.^[25] Central simple algebras form a key class of finite-dimensional associative algebras over a field K. A central simple algebra over K is a simple associative K-algebra (i.e., with no nontrivial two-sided ideals) whose center is precisely K.^[26] By the Artin-Wedderburn theorem, every finite-dimensional central simple algebra over K is isomorphic to a matrix algebra M_n(D), where D is a central division algebra over K and n \geq 1.^[18] Classic examples include the full matrix algebras M_n(K) over K, which are central simple with dimension n^2.^[3] Prominent examples of associative algebras include group algebras and Weyl algebras. The group algebra K[G] of a group G over K consists of formal finite linear combinations \sum_{g \in G} a_g g with a_g \in K, under componentwise addition and multiplication extended from the group operation, yielding an associative algebra of dimension |G| if G is finite.^[27] The Weyl algebra A_1(K), over a field K of characteristic zero, is the associative K-algebra generated by elements x and \partial satisfying the relation \partial x - x \partial = 1, modeling the algebra of differential operators on the affine line and serving as a simple, infinite-dimensional example.^[28]

Non-Associative Algebras

Non-associative algebras over a field K are vector spaces equipped with a bilinear multiplication that does not satisfy the associative law (ab)c = a(bc) for all elements a, b, c. These structures generalize associative algebras by relaxing the full associativity condition, allowing for specialized identities that ensure useful properties in subexpressions or powers. Notable subclasses include power-associative algebras, where the subalgebra generated by any single element is associative, meaning powers of an element x satisfy x^{m+n} = x^m x^n unambiguously for positive integers m, n, and alternative algebras, which obey the left and right alternative laws (xx)y = x(xy) and (yx)x = y(xx) for all x, y, ensuring that products involving repeated factors associate in certain ways.^[29]^[30]^[31] Lie algebras represent a fundamental class of non-associative algebras, defined as vector spaces over K (typically of characteristic not 2 or 3) with a bilinear bracket operation [ \cdot, \cdot ]: g \times g \to g that is alternating, so [a, a] = 0 for all a \in g, and satisfies the Jacobi identity [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a, b, c \in g. The bracket often arises as the commutator [a, b] = ab - ba in an underlying associative algebra, capturing infinitesimal symmetries in Lie groups and applications in physics, such as symmetry groups in particle physics. A classic example is the Heisenberg algebra, a 3-dimensional nilpotent Lie algebra over K with basis \{p, q, z\} and relations [p, q] = z, [p, z] = [q, z] = 0, where nilpotency follows from the lower central series terminating at the center spanned by z. This structure models the canonical commutation relations in quantum mechanics and exemplifies solvable Lie algebras of low dimension.^[32]^[33] Jordan algebras, another key non-associative type, are commutative unital algebras over K (often real or complex) satisfying the Jordan identity (a^2 b) a = a^2 (b a) for all a, b, which ensures a quadratic form compatible with the multiplication and supports spectral decomposition analogous to associative cases. Introduced to formalize observables in quantum mechanics, where self-adjoint operators form a Jordan algebra under the symmetrized product a \circ b = \frac{1}{2}(ab + ba), they provide an algebraic framework for non-commutative measurements without full associativity. The seminal work establishing their connection to quantum formalism showed that finite-dimensional formally real Jordan algebras decompose into sums of simple ones, including matrix algebras over reals, complexes, quaternions, and the exceptional 27-dimensional Albert algebra.^[34] Examples of non-associative algebras include the octonions \mathbb{O}, an 8-dimensional alternative division algebra over the reals, constructed via the Cayley-Dickson process from quaternions, with basis \{1, e_1, \dots, e_7\} and multiplication rules ensuring no zero divisors and a norm making it a composition algebra, though non-associativity appears in triples like (e_1 e_2) e_4 \neq e_1 (e_2 e_4). Unlike the associative complexes and quaternions, octonions lose full associativity but retain alternativity, enabling applications in exceptional Lie groups and string theory.^[35]

Relation to Rings

Algebras as Ring Extensions

In the context of algebra over a field K, an algebra A is fundamentally a ring (typically associative and unital) that is also equipped with a compatible K-module structure, meaning there is a scalar multiplication operation K \times A \to A, denoted (\alpha, a) \mapsto \alpha a, such that \alpha(ab) = (\alpha a)b = a(\alpha b) for all \alpha \in K and a, b \in A. This compatibility ensures that the ring multiplication in A is bilinear over K, i.e., it is linear in each argument when the other is fixed: a(\alpha b + \beta c) = \alpha (a b) + \beta (a c) and (\alpha b + \beta c)d = \alpha (b d) + \beta (c d) for \alpha, \beta \in K and a, b, c, d \in A. Consequently, every element of K commutes with every element of A, embedding K into the center of A.^[1] This structure implies that A is a vector space over K, with the ring addition serving as the vector addition and the scalar multiplication as defined. The dimension of A as a K-vector space, denoted [A : K], plays a central role in many properties; for instance, if A is finite-dimensional, each element a \in A satisfies a monic characteristic polynomial \chi_a(X) \in K[X] of degree [A : K], defined via the action of left multiplication by a on A. Moreover, if K has characteristic zero, then A also has characteristic zero, as the multiplicative identity $1_A satisfies n \cdot 1_A = n \cdot 1_K \neq 0 for any nonzero integer n, ensuring no torsion in the additive group.^[1]^[1] A canonical example of a free K-algebra is the tensor algebra T(V) generated by a K-vector space V, constructed as the direct sum

T(V) = \bigoplus_{n=0}^\infty T^n(V),

where T^0(V) = K, T^1(V) = V, and T^n(V) = V \otimes_K V \otimes_K \cdots \otimes_K V (n factors) for n \geq 2, with multiplication given by concatenation of pure tensors extended linearly: (x_1 \otimes \cdots \otimes x_n) \cdot (y_1 \otimes \cdots \otimes y_m) = x_1 \otimes \cdots \otimes x_n \otimes y_1 \otimes \cdots \otimes y_m. This algebra is freely generated by V in the sense that it imposes no relations on elements of V beyond those required by bilinearity and associativity of the tensor product, making it the universal object among K-algebras containing V as a subspace.^[36]^[36] Viewing A as a module over itself (with the natural left A-module structure via ring multiplication), the K-linear endomorphisms of A form the endomorphism ring \operatorname{End}_K(A), which consists of all K-linear maps \phi: A \to A. The algebra A embeds into \operatorname{End}_K(A) via the maps m_a: A \to A defined by left multiplication m_a(b) = a b, turning \operatorname{End}_K(A) into a ring that contains A as a subring and respects the K-vector space structure. This perspective highlights how the algebraic structure enriches the underlying ring with linear algebra tools.^[1]

Ideals in Algebras versus Rings

In algebras over a field K, denoted K-algebras, ideals possess additional structure compared to those in general rings. Specifically, every (two-sided) ideal I of a K-algebra A is a K-subspace of A, meaning it is closed under addition and scalar multiplication by elements of K, in addition to the usual absorption property A I A \subseteq I. This vector space structure arises because the ring multiplication in A is K-bilinear, ensuring that for any a \in A, \lambda \in K, and i \in I, \lambda i \in I and a (\lambda i) = \lambda (a i) \in I. In contrast, ideals in arbitrary rings lack this inherent vector space compatibility, as rings may not admit a compatible scalar action from a field.^[3] For prime and maximal ideals, the field base introduces significant simplifications, particularly in finite-dimensional cases. In a finite-dimensional K-algebra A, the maximal two-sided ideals correspond precisely to the annihilators of the irreducible representations of A, which are the simple left (or right) A-modules. This correspondence follows from the fact that finite-dimensional algebras are artinian, so their simple modules determine the primitive ideals, and maximal ideals among these are the kernels of surjective homomorphisms onto simple matrix algebras over division rings. Prime ideals, being those where the quotient is a prime ring, also align with indecomposable representations in this setting, unlike in general rings where such ideals may not relate directly to modular structure without additional finiteness assumptions.^[37] Nilpotent ideals in K-algebras exhibit enhanced tractability due to the finite-dimensionality often assumed or implied. A nilpotent ideal I satisfies I^n = 0 for some positive integer n, and in finite-dimensional K-algebras, the Jacobson radical—the intersection of all maximal ideals—is itself nilpotent, admitting a composition series where each factor is a simple module. This nilpotency index is bounded by the dimension of A over K, allowing explicit computations via descending central series or powers, which is not generally feasible in infinite-dimensional or non-vectorial ring contexts without field-induced bounds.^[38] A key distinction arises in semisimple K-algebras, where the field base enables a clean decomposition absent in general rings. By the Artin-Wedderburn theorem, a finite-dimensional semisimple associative K-algebra decomposes as a direct sum of matrix algebras over division K-algebras, and if K is algebraically closed, these are simply full matrix rings over K itself. This structure theorem relies on the vector space properties to classify minimal ideals as matrix units, contrasting with semisimple artinian rings in general, which decompose into matrix rings over arbitrary division rings without a unified field scalar action.^[39]

Representation Theory

Structure Coefficients

In a finite-dimensional algebra A over a field K of dimension n, select a basis \{e_1, \dots, e_n\}. The multiplication in A is then determined by the structure constants c_{ij}^k \in K via the formula

e_i e_j = \sum_{k=1}^n c_{ij}^k e_k

for all i, j = 1, \dots, n. These constants fully encode the bilinear multiplication map A \times A \to A, allowing the algebra to be represented concretely as a vector space with a specified product table.^[1] The bilinearity of the multiplication implies linearity in each factor: for \alpha \in K and a, b \in A,

(\alpha a) b = a (\alpha b) = \alpha (a b).

This property follows directly from the algebra axioms and ensures that the structure constants satisfy linear relations when scalars are involved. If the algebra is associative, the constants must obey additional quadratic conditions derived from (e_i e_j) e_k = e_i (e_j e_k) for all i, j, k, which impose constraints on the c_{ij}^\ell to guarantee compatibility with the associative law.^[1]^[40] Under a change of basis given by an invertible matrix P \in \mathrm{GL}_n(K), with new basis elements f_m = \sum_p P_{pm} e_p, the structure constants transform via

\tilde{c}_{rs}^t = \sum_{i,j,k} P_{ir} P_{js} c_{ij}^k (P^{-1})_{kt},

resembling a similarity transformation but depending on both P and P^{-1}; consequently, the structure constants are not invariant under basis changes. Structure constants facilitate the classification of algebras by reducing the problem to solving algebraic equations over K for the c_{ij}^k that satisfy the required identities, such as those for associativity or commutativity; an open dense subset of such constants often yields canonical forms for the algebras.^[41] This method is applied in classifying low-dimensional algebras over fields.^[41]

Classification of Low-Dimensional Algebras

In dimension 1, the only unital associative algebra over the complex numbers ℂ up to isomorphism is ℂ itself, which is a commutative division algebra.^[42] In dimension 2, there are exactly two isomorphism classes of unital associative algebras over ℂ. The semisimple example is the direct product ℂ × ℂ, whose center has dimension 2 and which decomposes into two minimal ideals. The other is the local algebra of dual numbers ℂ[ε] with the relation ε² = 0, which has a unique maximal ideal of dimension 1 generated by ε and radical of dimension 1.^[42] In dimension 3, the unital associative algebras over ℂ fall into several isomorphism classes, including direct sums and non-semisimple examples. Semisimple cases include the direct sum ℂ ⊕ ℂ ⊕ ℂ, with center dimension 3. Non-semisimple ones encompass ℂ ⊕ (ℂ[ε]/(ε² = 0)), where the radical has dimension 1, and the algebra of 2 × 2 upper triangular matrices over ℂ, which has basis consisting of the identity matrix, the nilpotent matrix with 1 in the (1,2) entry, and the difference of the diagonal projections; this algebra is non-commutative with radical dimension 1. A nilpotent representative is the 3-dimensional Heisenberg algebra, with basis {1, x, y} where x² = y² = xy = yx = 0, yielding radical dimension 2 and radical squared zero. Isomorphism is determined by invariants such as radical dimension and the action of the semisimple quotient on the radical.^[42]^[43] In dimension 4, the classification includes both semisimple and non-semisimple unital associative algebras over ℂ, with isomorphism classes distinguished by invariants like the dimension of the center and the radical structure. Semisimple examples are ℂ⁴ (center dimension 4), ℂ² × ℂ² (center dimension 2), and M₂(ℂ) (center dimension 1, simple). The quaternion algebra ℍ over ℝ tensorized with ℂ is isomorphic to M₂(ℂ). Non-semisimple cases involve extensions such as upper triangular 2 × 2 matrix algebras with additional nilpotent components or direct sums like ℂ ⊕ (3-dimensional nilpotent algebra), where the radical dimension ranges from 1 to 3, and associativity constraints limit the possible multiplication on the radical. Full isomorphism criteria rely on the dimension of the center (which equals the number of simple components in the semisimple quotient) and the representation of the semisimple part on the radical.^[42] Over ℂ, every finite-dimensional semisimple associative algebra is isomorphic to a direct sum of matrix algebras ⊕ M_{n_i}(ℂ) by the Artin-Wedderburn theorem, which decomposes such algebras into simple components; for low dimensions, this restricts the possible block sizes (e.g., no M₃(ℂ) in dimension ≤ 4). Non-semisimple algebras are then extensions of these by their nilpotent radicals.^[44]

Generalizations

Algebras over Commutative Rings

An algebra over a commutative ring R, often denoted an R-algebra, is an R-module A equipped with a bilinear multiplication m: A \times A \to A, satisfying m(ra, b) = r m(a, b) = m(a, rb) for all r \in R and a, b \in A. Equivalently, it can be defined as a (typically unital) ring A together with a ring homomorphism \phi: R \to Z(A), where Z(A) is the center of A, inducing the R-module structure via r \cdot a = \phi(r) a. These perspectives emphasize the compatibility between the ring and module structures, generalizing the vector space framework of field-based algebras.^[45] Prominent examples include the polynomial ring R, which arises as the free commutative R-algebra on one generator, with multiplication extending the usual polynomial operations over R. Another is the matrix ring M_n(R) of n \times n matrices with entries in R, equipped with matrix addition and multiplication, forming an R-algebra via entrywise scalar multiplication. For R = \mathbb{Z}, the ring of integers, M_n(\mathbb{Z}) exemplifies an integral R-algebra, relevant in number theory and representation contexts.^[1] In contrast to algebras over fields, where the base is a vector space, R-algebras are R-modules, which need not be free and may exhibit torsion if R has zero divisors or is not a principal ideal domain. Even when R is an integral domain, A can possess zero divisors; for example, in M_n(R) with n \geq 2, non-zero matrices exist whose product is the zero matrix, such as \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}.^[1] A key technique for extending R-algebras to fields involves localization: for a multiplicative subset S \subseteq R, the localized module S^{-1}A inherits a bilinear multiplication from A, making it an algebra over the localized ring S^{-1}R. If S is chosen such that S^{-1}R = K is a field (for example, when R is an integral domain and S is the set of all nonzero elements of R, yielding the field of fractions K = \mathrm{Frac}(R)), then S^{-1}A becomes a K-algebra, bridging ring-based and field-based algebraic structures.^[46]

Non-Unital and Non-Associative Generalizations

In the theory of algebras over a field K, non-unital and non-associative generalizations extend the basic structure by relaxing the requirements of a multiplicative identity and the associative law. Such an algebra A is a vector space over K equipped with a bilinear multiplication A \times A \to A, where no element serves as a two-sided identity (i.e., there is no e \in A satisfying e a = a e = a for all a \in A) and the multiplication need not satisfy (a b) c = a (b c) for all a, b, c \in A. These structures arise naturally in contexts where the full associative and unital properties are unnecessary or counterproductive, such as in the study of derivations or infinitesimal symmetries.^[47] A canonical class of non-unital non-associative algebras over a field K is that of Lie algebras, defined by a bilinear product [ \cdot, \cdot ]: A \times A \to A that is alternating ([a, a] = 0 for all a \in A) and satisfies the Jacobi identity [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a, b, c \in A. Lie algebras lack units over fields of characteristic not 2, as assuming such an e leads to [e, a] = a = -a for all a (by skew-symmetry of the bracket), implying the algebra is trivial. A concrete example is the Lie algebra \mathfrak{so}(3) over \mathbb{R}, realized as \mathbb{R}^3 with the cross product as multiplication: for standard basis vectors e_1, e_2, e_3, we have [e_1, e_2] = e_3, [e_2, e_3] = e_1, [e_3, e_1] = e_2, and cyclic permutations, capturing rotations in three dimensions without an identity.^[48] Non-unital non-associative algebras also appear as ideals or radicals within larger structures. For instance, in alternative algebras over a field F of characteristic not 2 or 3, the radical \mathfrak{N} (the maximal nil ideal) forms a non-unital subalgebra satisfying the alternative laws a^2 b = a (a b) and b a^2 = (b a) a but failing associativity in general. Semisimple alternative algebras decompose as direct sums of simple ideals, each of which may be non-unital if derived from split forms like certain Cayley algebras with zero divisors. Similarly, non-commutative Jordan algebras over fields of characteristic not 2 or 3 include non-unital examples such as nodal ones, constructed as J = F \cdot 1 \oplus N where N is a nilpotent ideal of index 3, with multiplication defined via partial derivatives to satisfy the Jordan identity (a b) a^2 = a (b a^2).^[47] These generalizations facilitate the study of broader algebraic phenomena, such as derivations and representations, without the constraints of units or associativity. For example, the derivation algebra D(A) of a non-unital non-associative algebra A over a field of characteristic not 2 or 3 forms a Lie algebra, enabling connections to symmetry groups via the Baker-Campbell-Hausdorff formula in characteristic zero. Finite-dimensional simple non-unital non-associative algebras over algebraically closed fields are classified up to isomorphism in low dimensions, often as deformations of associative ones, highlighting their role in structure theory.^[49]

References

[1]
[PDF] ALGEBRAS 1. Definitions and Examples Let k be a ... - Keith Conrad
We focus here on algebras over a field for simplicity (e.g., bases are automatically available). For a k-algebra A, α 7→ α · 1A for α ∈ k is a ring homomorphism ...
[2]
Algebra over a Field - an overview | ScienceDirect Topics
An algebra over a field is defined as a vector space equipped with a bilinear product, allowing operations that combine elements of the vector space while ...
[3]
Algebras over a field - Harvard Mathematics Department
An algebra over k, or more simply a k-algebra, is an associative ring A with unit together with a copy of k in the center of A.
[4]
[PDF] Advanced Algebra
Sections 7–10 concern Gröbner bases, which are finite generating sets of a special kind for ideals in a polynomial algebra over a field. Section 7 sets the ...
[5]
[PDF] Modules and Wedderburn Theory - Alexander Duncan
Apr 13, 2023 · 1 Warm up: Matrix Algebra over a Field. In this section, we consider the particular case of the non-commutative k- algebra Mn(k) where k is a ...
[6]
[PDF] Introduction to representation theory - MIT Mathematics
Jan 10, 2011 · (which we will explain below), Frobenius created representation theory of finite groups. ... The group algebra A = k[G] of a group G. Its basis is ...
[7]
[PDF] extra material on tensor, symmetric and exterior algebras
graded. By definition, it is a symmetric algebra of a module M,. Sym. •. (M) def ... θiθj + θjθi = 0, i, j = 1,...,n. 4.2. Exterior algebra of a vector space.
[8]
[PDF] An Introduction to Nonassociative Algebras - Project Gutenberg
We define an algebra A over a field F to be a vector space over. F with a bilinear multiplication (that is, a multiplication satisfying. (2) and (3)). We shall ...
[9]
[PDF] math 223a notes 2011 lie algebras - Brandeis
Sep 6, 2011 · By an (nonassociative) algebra over a field F we mean a vector space. A together with an F-bilinear operation A×A → A which is usually written ( ...
[10]
[PDF] Introduction to Lie Algebras - UCI Mathematics
bilinear map, the Lie bracket. L × L → L, (x, y) → [x, y], satisfying the ... An algebra over a field F is a vector space A over F together with a ...
[11]
[PDF] representation theory, chapter 0. basics - Yale Math
A homomorphism of k-algebra is a k-linear homomorphism of unital rings. Definition 1.3. Let A, B be (associative, unital) rings and M be an abelian group. • By ...
[12]
[PDF] Algebras over a field
Oct 14, 2014 · Roughly speaking, an algebra over a field F is just a ring R with F contained in the center of R. In particular R is an F-vector space, ...
[13]
[PDF] 26 Homomorphisms, Ideals and Factor Rings - UCI Mathematics
Theorem 26.7 (Fundamental Homomorphism Theorem). Every kernel of a homomorphism is an ideal and vice versa. Specifically: 1. Let φ : R → S be a ...
[14]
[PDF] The structure of free algebras - Department of Mathematics
The algebra FV(X) may be defined as an algebra in V generated by X that has the universal mapping property: For every A ∈ V and every function v : X → A there ...
[15]
[PDF] Chapter 4. Some well known algebras.
satisfies the universal property needed to be the free algebra generated by A and proves the proposition. Comments 1.15: In view of the structure of T(A) as ...
[16]
[PDF] An introduction to the algebra of rings and fields
This is an introduction to rings and fields, written for a quarter-long undergraduate course. It includes the basic properties of ideals, modules, algebras and ...
[17]
[PDF] CENTRAL SIMPLE ALGEBRA SEMINAR 1. Lecture (1/9)
Definition 1.26. An F-algebra A is called a central simple algebra over F (CSA/F) if A is simple and Z(A) = F. 2. LECTURE (1/16): TENSORS AND CENTRALIZERS.<|control11|><|separator|>
[18]
[PDF] Basics of associative algebras - OU Math
Jan 2, 2016 · Let A be an F-algebra and K/F a field extension of possibly infinite degree. Then the extension of scalars A ⌦F K is a K-algebra of dimension ...
[19]
[PDF] 1 Extension of Scalars
Hence, we have restricted the scalars from C to R. This works the same way for any field extension. Now we go the other way, which requires the tensor ...
[20]
[PDF] TENSOR PRODUCTS 1. Introduction: R-modules
The tensor product is the first concept in algebra whose properties make consistent sense only by a universal mapping property, which is: M ⊗R N is the ...
[21]
unitalization in nLab
Apr 24, 2021 · The unitalization of a non-unital algebra is a unital algebra with a “unit” (an identity element) freely adjoined. This is a free functor: Rng → \to Ring.
[22]
[PDF] Graph algebras
For any positive integer n and any field k, the set Mn(k) of all n × n matrices with coefficients in k is a unital algebra over k. The multiplication ...
[23]
Consequences of not requiring ring homomorphisms to be unital?
Aug 3, 2010 · As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: f(1R)=1S. But there has been a ...Unital *-homomorphisms between matrices - MathOverflowRealize a $K_0$-group homomorphism by a unital $\astMore results from mathoverflow.net
[24]
[PDF] arXiv:1404.0126v3 [math.RA] 10 Apr 2014
Apr 10, 2014 · Every commutative unital R-algebra A has the underlaying structure of an associative R-algebra, likewise any unit preserving homomor- phism of ...
[25]
[PDF] Artinian and Noetherian Rings
Apr 10, 2012 · Artinian and Noetherian rings have some measure of finiteness. Noetherian rings satisfy the Ascending Chain Condition (ACC), and Artinian rings ...Missing: nucleus | Show results with:nucleus
[26]
[PDF] A Crash Course in Central Simple Algebras - Evan Dummit
Oct 24, 2011 · ◦ Any field is a central simple algebra over itself. ◦ The quaternions are a real central simple algebra; in fact, they are essentially ...
[27]
Group algebra - Encyclopedia of Mathematics
Apr 29, 2012 · The group algebra of a group G over a field K is the associative algebra over K whose elements are all possible finite sums of the type ∑g ...<|separator|>
[28]
On the hom-associative Weyl algebras - ScienceDirect.com
The first Weyl algebra A 1 over a field K of characteristic zero, from now on just referred to as the Weyl algebra, is the free, associative and unital algebra ...
[29]
non-associative algebra - PlanetMath
Mar 22, 2013 · A non-associative algebra is an algebra in which the assumption of multiplicative associativity is dropped. From this definition ...
[30]
power-associative algebra - PlanetMath.org
Mar 22, 2013 · A non-associative algebra is power-associative if, [B,B,B]=0 [ B , B , B ] = 0 for any cyclic subalgebra B of A , where [−,−,−] is the ...
[31]
alternative algebra - PlanetMath
Mar 22, 2013 · ... Jordan identity is satisfied. •. Alternativity can be defined for a general ring R R : it is a non-associative ring such that for any a,b∈R ...
[32]
Jacobi Identities -- from Wolfram MathWorld
The elements of a Lie algebra satisfy this identity. Relationships between the Q-functions Q_i are also known as Jacobi identities: Q_1Q_2Q_3=1,. (2) ...
[33]
Heisenberg algebra - PlanetMath
Mar 22, 2013 · It is easy to see that the product [⋅,⋅] also fulfills the Jacobi identity , so a Heisenberg algebra is actually a Lie algebra of rank |I|+1 ( ...
[34]
Jordan algebra - PlanetMath.org
Mar 22, 2013 · 1. A A is commutative: ab=ba a ⁢ b = b ⁢ a , and · 2. A A satisfies the Jordan identity: (a2b)a=a2(ba) ( a 2 ⁢ b ) ⁢ a = a 2 ⁢ ( b ⁢ a ) ,.
[35]
[PDF] The Octonions - SISSA People Personal Home Pages
Dec 21, 2001 · The octonions are the largest of the four normed division algebras. ... R, C, H, and O are the only alternative division algebras. The first ...
[36]
Section 10.13 (00DM): Tensor algebra—The Stacks project
The tensor algebra of M over R is a noncommutative R-algebra, defined as the sum of T^n(M), where T^0(M) = R, T^1(M) = M, T^2(M) = M \otimes _ R M, and so on.
[37]
[PDF] Drozd and Kirichenko's "Finite Dimensional Algebras"
... maximal ideals of the algebra A. But. A is a semisimple algebra, and thus J I ... all irreducible representations of the algebra KG, and thus n = [KG: K] ...
[38]
[PDF] 7350: topics in finite-dimensional algebras - Cornell Mathematics
Let A be a finite-dimensional algebra. Then the Jacobson radical of A is the largest two-sided ideal I of A such that every x ∈ I is nilpotent. Proof ...
[39]
Wedderburn-Artin Theory | SpringerLink
Abstract. Modern ring theory began when J.H.M. Wedderburn proved his celebrated classification theorem for finite dimensional semisimple algebras over fields.Missing: paper | Show results with:paper
[40]
On the Deformation Theory of Structure Constants for Associative ...
Jul 14, 2010 · For the undeformed structure constants the associativity conditions (1.2) are nothing else than the compatibility conditions for the table of ...
[41]
[1504.01194] On classification of finite dimensional algebras - arXiv
Apr 6, 2015 · An invariant open, dense (in the Zariscki topology) subset of the space of structural constants is defined. The algebras with structural ...
[42]
Complete lists of low dimensional complex associative algebras
Oct 6, 2009 · In this paper we present a complete classification (isomorphism classes with some isomorphism invariants) of complex associative algebras up to dimension five.Missing: ℂ 1-4
[43]
A complete classification of three-dimensional algebras over ... - arXiv
Mar 5, 2019 · We provide a complete classification of three-dimensional associative algebras over the real and complex number fields based on a complete elementary proof.
[44]
[2405.04588] The Wedderburn-Artin Theorem - arXiv
May 7, 2024 · The celebrated Wedderburn-Artin theorem states that a simple left artinian ring is isomorphic to the ring of matrices over a division ring.
[45]
algebras - PlanetMath
Mar 22, 2013 · As a converse, given a unital ring R (associativity is necessary), the center of the ring forms a commutative unital subring over which R is an ...
[46]
Equivalence of two definitions of an algebra over a ring.
Dec 7, 2012 · The difference between the two definitions of algebra over A is the Atiyah-Macdonald wants all rings to have a unit, all ring homomorphisms ...Question about R-algebra "generated by" - Math Stack ExchangeDoubt in Atiyah-Macdonald proposition 5.13 - Math Stack ExchangeMore results from math.stackexchange.com
[47]
Section 10.9 (00CM): Localization—The Stacks project
Localization of a ring A with respect to a multiplicative subset S is the ring S^{-1}A, defined using equivalence classes.
[48]
[PDF] An Introduction to Nonassociative Algebras - Project Gutenberg
The Project Gutenberg EBook of An Introduction to Nonassociative Algebras, ... R.D. Schafer, Structure and representation of nonassociative algebras,. Bull.
[49]
LIE ALGEBRAS OF CHARACTERISTIC 2Q
Basic definitions and axioms. A Lie algebra over a field is a vector space over the field possessing a bilinear product which is antisymmetric and satisfies the.
[50]
Chapter 4 Lie Groups and Lie Algebras
1.1 Definition and Examples. We can define Lie algebras over any field. We restrict ourselves to the real or complex case. So let K = R or C. Definition 1.1.
[51]
[PDF] schafer-nonassociative-algebras.pdf - MIT Mathematics
Our definition of inner derivation for a nonassociative algebra agrees with ... A symmetric bilinear form (x, y) defined on an arbitrary algebra 91 is.