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Associative algebra

In , an associative algebra over a commutative ring R is an R- A equipped with an R- A \times A \to A, denoted (a, b) \mapsto ab, such that the multiplication is associative, meaning a(bc) = (ab)c for all a, b, c \in A. This structure generalizes both rings and by incorporating scalar multiplication from R into the center of the ring via a ring homomorphism R \to Z(A), where Z(A) is the center of A. Associative algebras are often assumed to be unital, possessing a multiplicative identity element $1 \in A satisfying $1a = a1 = a for all a \in A, which is unique if it exists. Key examples of associative algebras include the ring R itself, polynomial rings R[x_1, \dots, x_n], the algebra of endomorphisms \operatorname{End}_R(M) of an R-module M, free algebras R\langle X_1, \dots, X_n \rangle generated by non-commuting indeterminates, and group algebras R[G] formed by formal linear combinations of elements from a group G. Subalgebras are submodules closed under multiplication and containing the unit, while algebra homomorphisms are R-linear maps preserving both addition and multiplication (and the unit, if unital). These structures form a category where objects are associative algebras and morphisms are homomorphisms, enabling the study of tensor products and modules over them. Associative algebras play a foundational role in various branches of mathematics, including —where they encode linear actions of groups or other structures—and , where they facilitate the analysis of rings of integers and ideals. They also underpin and contribute to understanding symmetries via connections to Hopf algebras and Tannaka duality, which reconstruct algebras from categories of their modules. In applied contexts, such as , associative algebras model algebras on Hilbert spaces, bridging abstract algebra with physical systems.

Definition and Basics

Definition via ring homomorphisms

An associative algebra A over a K is a over K equipped with a m: A \times A \to A, called , such that the is associative, meaning (ab)c = a(bc) for all a, b, c \in A, and unital, meaning there exists an element $1 \in A such that $1a = a1 = a for all a \in A. The bilinearity of requires that m(\lambda a + \mu b, c) = \lambda m(a, c) + \mu m(b, c), \quad m(a, \lambda b + \mu c) = \lambda m(a, b) + \mu m(a, c) for all \lambda, \mu \in K and a, b, c \in A, ensuring compatibility with the vector space structure. This setup makes A into a whose addition and scalar multiplication coincide with those of the vector space, with the K acting centrally via the \iota: K \to Z(A) into Z(A) = \{ z \in A \mid za = az \ \forall a \in A \}. From the ring homomorphism perspective, the algebra structure on A corresponds to a unital \phi: A \to \operatorname{End}_K(A), where \operatorname{End}_K(A) is the ring of K-linear of A, defined by \phi(a)(x) = ax for a, x \in A. This map \phi is well-defined because left multiplication by a is K-linear due to bilinearity, and the homomorphism property \phi(ab) = \phi(a) \circ \phi(b) encodes the associativity condition, as \phi(ab)(x) = (ab)x = a(bx) = \phi(a)(\phi(b)(x)). The unit $1 satisfies \phi(1) = \operatorname{id}_A, the identity . This viewpoint highlights how the multiplication turns A into a of its , unifying the ring and structures. The formulation emphasizing the ring structure over the vector space arose in the 1920s through Emil Artin's work on hypercomplex numbers, where he extended earlier results on associative rings to broader algebraic systems. In particular, Artin's 1927 papers developed the arithmetic and theory of such systems, laying groundwork for modern associative algebra by focusing on linear representations and non-commutative extensions. His lectures at during this period further propagated this abstract ring-theoretic approach, influencing the standardization of algebra.

Definition as monoid objects in modules

In the categorical framework, an associative algebra over a commutative ring K can be defined as a monoid object in the monoidal category of K-modules, where the monoidal structure is given by the tensor product \otimes_K over K and the unit object is K itself. Specifically, such an algebra A consists of a K-module equipped with a multiplication morphism \mu: A \otimes_K A \to A and a unit morphism \eta: K \to A, satisfying the associativity and unit axioms as in the theory of monoids in monoidal categories. The associativity condition requires that the following diagram commutes, expressing the pentagon identity for the multiplication: 
          μ ⊗ id_A
A ⊗ A ⊗ A ──────────→ A ⊗ A
   ↓ id_A ⊗ μ          ↓ μ
A ⊗ A ──────→ A
      μ
This ensures that (a \cdot b) \cdot c = a \cdot (b \cdot c) for elements a, b, c \in A, where \cdot denotes the multiplication induced by \mu. The unit axioms similarly require commutativity of diagrams ensuring that \eta acts as a left and right identity for \mu. A key aspect of this definition is its universal property, exemplified by the free associative algebra on a K-module M, which is the tensor algebra T(M) = \bigoplus_{n=0}^\infty M^{\otimes n}, where M^0 = K. This structure satisfies the universal property: for any associative K-algebra A and any K-module homomorphism f: M \to A, there exists a unique algebra homomorphism \tilde{f}: T(M) \to A extending f such that the diagram 
M ──ι──→ T(M)
  f     ↓ ̃f
  A ──id──→ A
commutes, with the into the . This categorical perspective views as rings internalized to the of modules, facilitating generalizations to other such as sheaves of modules or spectra, where analogous objects capture algebraic structures adapted to the ambient .

Basic properties and axioms

An associative algebra A over a k is unital, meaning it possesses a multiplicative $1_A such that $1_A \cdot a = a \cdot 1_A = a for all a \in A. This is unique, and the structure map \iota: k \to A satisfies \iota(1_k) = 1_A, ensuring compatibility with the ring operations. In the unital case, the regular _A A_A is well-defined as an A-A-bimodule (_A A_A), which facilitates the study of bimodules and unifies certain aspects of left and right theory, though left and right modules remain distinct in the non-commutative case. A B of A is a that is closed under , , and by elements of k, forming a k- under the induced operations. in A are defined relative to the structure: a left I is an additive such that A \cdot I \subseteq I; a right satisfies I \cdot A \subseteq I; and a two-sided satisfies both. For instance, in the polynomial k over a k, the principal generated by a polynomial f(x) consists of all multiples g(x) f(x) for g(x) \in k, which is two-sided since k is commutative. The center of an algebra A, denoted Z(A), is the subalgebra consisting of elements that commute with every element of A: Z(A) = \{ z \in A \mid z a = a z \ \forall \, a \in A \}. This forms a commutative , and the image of the structure map \iota(k) lies in Z(A), making Z(A) a k- in its own right. The associativity axiom (a b) c = a (b c) for all a, b, c \in A ensures that the is a well-defined , and in unital associative algebras, this property implies alternativity, where identities like (a b) b = a (b^2) hold, though the focus here remains on the core associative identity itself. While classical treatments often restrict to fields, modern definitions generalize to commutative rings k for broader applicability in and .

Morphisms and Structures

Algebra homomorphisms

In the context of associative algebras over a K (often a ), an homomorphism \phi: A \to B between two K-algebras A and B is a K- that preserves both the and the multiplicative . Specifically, \phi(ab) = \phi(a)\phi(b) for all a, b \in A, and \phi(1_A) = 1_B. This definition ensures that \phi is also a unital ring homomorphism, compatible with the K-module structures. Since both A and B are associative, the image \phi(A) inherits associativity automatically: for any a, b, c \in A, \phi((ab)c) = \phi(a(bc)) \implies (\phi(a)\phi(b))\phi(c) = \phi(a)(\phi(b)\phi(c)), as the equality on the left follows from the associativity in A and the multiplicativity of \phi. The kernel of \phi, defined as \ker(\phi) = \{a \in A \mid \phi(a) = 0\}, is a two-sided ideal in A. The image \operatorname{im}(\phi) = \phi(A) forms a subalgebra of B. The first isomorphism theorem for rings, applicable here since associative algebras are unital rings over K, states that A / \ker(\phi) \cong \operatorname{im}(\phi) as K-algebras. An algebra isomorphism is a bijective homomorphism, and an automorphism is an isomorphism \phi: A \to A. Among automorphisms, inner automorphisms are those induced by conjugation with an invertible element g \in A^\times, given by \phi_g(a) = g a g^{-1} for all a \in A. These form a normal subgroup of the automorphism group and play a key role in the structure of central simple algebras. For non-unital associative algebras, where no multiplicative unit is required, a homomorphism \phi: A \to B is defined analogously as a K-linear map preserving multiplication, \phi(ab) = \phi(a)\phi(b), but without the unit preservation condition. The kernel remains a two-sided ideal, and the first isomorphism theorem holds, though the image is a subalgebra without necessarily containing a unit.

Representations of algebras

In the context of associative algebras, a representation provides a way to interpret the abstract algebraic structure concretely through linear transformations on vector spaces. Specifically, for an associative algebra A over a field K, a representation of A on a K-vector space V is a unital algebra homomorphism \rho: A \to \operatorname{End}_K(V), where \operatorname{End}_K(V) denotes the algebra of K-linear endomorphisms of V. This induces a left A-module structure on V via the action a \cdot v = \rho(a)(v) for a \in A and v \in V. Conversely, every left A-module M that is a K-vector space yields a representation \rho_M: A \to \operatorname{End}_K(M) defined by \rho_M(a)(m) = a \cdot m, establishing a bijective correspondence between representations of A and left A-modules that are K-vector spaces. The dimension of such a representation is defined as \dim_K(V), which corresponds to the dimension of the underlying module. Representations are classified by properties such as and irreducibility, which highlight their structural significance. A representation \rho is if its is the zero ideal, \ker(\rho) = 0, meaning that the is injective and A embeds into \operatorname{End}_K(V). It is irreducible if the only A- subspaces of V are \{0\} and V itself; equivalently, V is a left A- with no nontrivial submodules. These notions extend the ideas from linear , where representations of algebras arise naturally, and representations of algebras can be viewed as special cases of homomorphisms into endomorphism rings. For semisimple algebras, the Artin-Wedderburn theorem provides a profound insight into their , stating that such an algebra is isomorphic to a finite of algebras over division rings. This decomposition implies that the irreducible representations of a semisimple algebra correspond precisely to the simple components in this sum, allowing every representation to be analyzed as a of irreducibles. The theorem originated with Wedderburn's 1905 classification of finite-dimensional semisimple algebras over algebraically closed fields and was generalized by Artin in 1928 to arbitrary semisimple Artinian rings. The study of representations traces its roots to Ferdinand Georg Frobenius's work in the late on finite-dimensional division algebras over the reals, where he classified them as \mathbb{R}, \mathbb{C}, and the quaternions, laying groundwork for understanding irreducible representations as faithful actions on minimal modules. This was extended in the early to general associative algebras through the development of module theory and representation frameworks, influenced by the needs of group representation theory.

Examples and Applications

Classical algebras and representation theory

Classical examples of associative algebras abound in pure mathematics, particularly in the study of symmetries and linear structures. The full matrix algebra M_n(K), consisting of all n \times n matrices with entries in a K, exemplifies a finite-dimensional simple associative algebra over K. It is semisimple and, by the Artin-Wedderburn theorem, decomposes as a single over the division algebra K itself when K is algebraically closed. Representations of M_n(K) correspond to modules over it, which are equivalent to vector spaces of dimension n over K, up to similarity classes of matrices. Group algebras provide another cornerstone, linking to associative algebra. For a G and K whose does not divide |G|, the group algebra K[G] is the K- with basis \{ g \mid g \in G \} and multiplication extended from the group operation. It is semisimple, and its —where K[G] acts on itself by left multiplication—decomposes as a of all irreducible representations, each appearing with multiplicity equal to its dimension. facilitates this decomposition: the character of the is |G| at the and zero elsewhere, and inner products of characters yield multiplicities via relations. Commutative associative algebras are illustrated by polynomial algebras, which highlight connections without non-commutativity. The polynomial algebra K[x_1, \dots, x_n] over a K is a commutative unital associative algebra, generated by indeterminates with the usual and multiplication. Representations of this algebra are K-modules equipped with commuting endomorphisms corresponding to each x_i, and evaluation maps at points (a_1, \dots, a_n) \in K^n yield homomorphisms to K, associating geometric points to one-dimensional representations. Over the real numbers, the quaternion algebra \mathbb{H} = \{ a + bi + cj + dk \mid a,b,c,d \in \mathbb{R} \} with relations i^2 = j^2 = k^2 = -1, ij = k = -ji, stands out as a non-commutative . It is four-dimensional over \mathbb{R}, associative, and every non-zero element has a two-sided , making it a with \mathbb{R}. By Frobenius's theorem, \mathbb{H} is unique up to among finite-dimensional division algebras over \mathbb{R} with \mathbb{R}, distinct from the commutative fields \mathbb{R} and \mathbb{C}. In , associative algebras underpin the classification of representations for finite groups. Semisimple associative algebras like K[G] for finite G decompose via the Artin-Wedderburn theorem into direct sums of matrix algebras over division rings, where each simple component corresponds to an . This structure theorem classifies all finite-dimensional representations of G as direct sums of these irreducibles, with characters providing the necessary decomposition tools.

Algebras in analysis

In , associative algebras equipped with a complete that is submultiplicative play a central role, particularly in the study of operators on infinite-dimensional spaces. A is a normed algebra over the real or complex numbers that is complete with respect to the and satisfies \|ab\| \leq \|a\| \|b\| for all elements a, b in the algebra. This structure generalizes both Banach spaces and associative algebras, allowing the analysis of convergence and continuity in algebraic operations. A classic example is the algebra C([0,1]) of continuous functions on the closed [0,1], equipped with pointwise multiplication and the supremum \|f\| = \sup_{x \in [0,1]} |f(x)|, which satisfies the submultiplicativity condition since |f(x)g(x)| \leq \|f\| \|g\| for all x. A significant subclass consists of C*-algebras, which are involutive Banach algebras (possessing an a \mapsto a^* satisfying (a^*)^* = a, (ab)^* = b^* a^*, and \lambda a^* = \overline{\lambda} a^* for scalars \lambda) that additionally obey the C*-identity \|a^* a\| = \|a\|^2. This condition ensures the norm is compatible with the , reflecting self-adjointness in operator contexts. C*-algebras form the foundation of noncommutative topology, where commutative C*-algebras correspond to continuous functions on compact Hausdorff spaces via the Gelfand-Naimark theorem, and noncommutative examples model "noncommutative spaces" through their ideals and . The in C*-algebras extends classical eigenvalue results to infinite dimensions, particularly for elements (those commuting with their , aa^* = a^* a). For such an element a, the \sigma(a) is a nonempty compact subset of the , and the Gelfand transform \hat{a} on the relates the element's behavior to continuous functions, enabling where f(a) is defined for suitable f. In the commutative case, the Gelfand transform provides an isometric *-isomorphism to C(\Delta(A)), the continuous functions on the Gelfand \Delta(A), linking algebraic spectra to topological ones. A prototypical example is B(H), the algebra of bounded linear operators on a complex H, with as , the \|T\| = \sup_{\|x\|=1} \|Tx\|, and the T^* defined by \langle Tx, y \rangle = \langle x, T^* y \rangle. This forms a unital , as the provides the and \|T^* T\| = \|T\|^2 holds by properties of the . Representations of general often realize them as subalgebras of B(H) via *-homomorphisms, such as operators on L^2 spaces corresponding to commutative cases. Extending this framework, algebras are weak*-closed *-subalgebras of B(H) containing the identity, which are central in quantum measurement theory and due to their rich projection lattices. They are classified into types I, II, and III based on the structure of their factors (central simple subalgebras): type I factors are isomorphic to B(H) for some H (finite or infinite-dimensional); type II factors admit a finite trace but no minimal projections (subdivided into finite II_1 and infinite II_\infty); and type III factors lack a semifinite trace, characterized by modular theory and Connes' invariant S(M) measuring asymptotic behavior of weights. This classification, developed by and in the 1930s and refined by later works, decomposes any von Neumann algebra uniquely into a of type I, II, and III components, providing tools for analyzing infinite factors beyond the type I case of B(H).

Algebras in geometry and combinatorics

Associative algebras play a fundamental role in through structures like path algebras, which model directed graphs known as . For a Q consisting of vertices and directed arrows, the algebra kQ over a k is the associative algebra with basis given by all finite paths in Q, including trivial paths at each vertex, where multiplication is defined by concatenation of paths (with the product zero if the paths do not compose). This algebra is finite-dimensional if Q has finitely many vertices and no oriented cycles. Quotients of algebras by ideals generated by relations correspond to geometric objects, such as projective varieties defined by the relations, and their representations as modules over kQ capture representations in a single sentence. In , incidence algebras provide a for studying partially ordered sets (posets). The incidence algebra I(P) of a poset P over a R (typically \mathbb{Z} or a ) consists of functions f: P \times P \to R supported on pairs (x,y) with x \leq y, with product (f * g)(x,y) = \sum_{x \leq z \leq y} f(x,z) g(z,y), making I(P) an associative R-algebra. The function \zeta \in I(P) is defined by \zeta(x,y) = 1 if x \leq y and $0 otherwise, serving as the multiplicative identity for the subalgebra of interval functions. Its inverse is the \mu, enabling Möbius inversion: if g(x) = \sum_{y \leq x} f(y), then f(x) = \sum_{y \leq x} \mu(y,x) g(y), a key tool for combinatorial on posets. Clifford algebras bridge geometry and algebra by encoding quadratic forms. For a vector space V over a field k (characteristic not 2) equipped with a quadratic form q: V \to k, the Clifford algebra \mathrm{Cl}(V,q) is the associative algebra generated by V subject to relations v^2 = q(v) \cdot 1 for all v \in V, quotienting the tensor algebra T(V) by the ideal generated by these elements. Despite their origins in geometric algebras for multivectors, Clifford algebras are associative but generally non-commutative, with dimension $2^{\dim V}, and they classify quadratic forms up to isomorphism in many cases. They arise in geometric constructions, such as describing orthogonal groups and spin structures on manifolds. In , cohomology rings of varieties like function as associative algebras that facilitate counting problems. The cohomology ring H^*(Gr(k,n);\mathbb{Z}) of the Grassmannian of k-planes in \mathbb{C}^n is generated by Chern classes of the tautological bundle, with relations given by the Whitney sum formula, forming a graded commutative associative algebra whose structure encodes intersection numbers for enumerating curves or subvarieties. Associativity of this ring multiplication ensures consistency in recursive computations, resolving nontrivial counts like the number of rational curves through points. Modern applications extend to , where associative algebras over the (with min or max as "addition" and plus as "multiplication") model degenerations of classical varieties. Tropical path algebras and incidence structures on posets tropicalize classical enumerative invariants, aiding computations in mirror symmetry and moduli spaces. These algebras preserve associativity, enabling efficient algorithms for tropical on Grassmannians and other geometric objects.

Algebras in mathematical physics

Associative algebras play a central role in by providing algebraic structures that model the non-commutative nature of quantum observables and symmetries in physical systems. The Weyl algebra exemplifies this, defined as the associative algebra generated by multiplication by the coordinate x and \partial, acting on the space of polynomials, with the defining relation [\partial, x] = 1. This structure captures the canonical commutation relations of , where x represents position and \partial (up to scaling) represents momentum, thereby modeling the of a quantum particle. In physics, the Weyl algebra underlies the quantization of classical systems on , facilitating the transition from brackets to commutators via the Weyl quantization map. Quantum groups, as deformations of enveloping algebras, extend this framework to incorporate q-deformed symmetries in integrable models. Specifically, the quantum enveloping algebra U_q(\mathfrak{sl}_2) is a Hopf algebra generated by elements E, F, K, and K^{-1} satisfying relations such as KE = qE K and [E, F] = \frac{K - K^{-1}}{q - q^{-1}}, along with a compatible coproduct that preserves the Hopf structure. These algebras arise in the study of integrable quantum systems, such as those in two-dimensional statistical mechanics and Yang-Baxter integrable models, where the deformation parameter q encodes quantum corrections to classical Lie algebra symmetries. The Hopf structure enables the formulation of scattering processes and conservation laws in these systems. In (QFT), serve as underlying the description of particle interactions. The T(V) over a V of test functions or field configurations is the generated by V, with the symmetric or exterior variants quotiented for bosonic or fermionic statistics to form Fock spaces. Perturbative QFT imposes relations on this algebra derived from Feynman diagrams, which encode interaction vertices and propagators, effectively quotienting the free structure to compute scattering amplitudes and correlation functions. This algebraic approach systematizes the and unitarity constraints in interacting theories. A key example is the Dirac algebra, realized as the Clifford algebra \mathrm{Cl}(1,3) over the Minkowski space \mathbb{R}^{1,3}, generated by Dirac matrices \gamma^\mu (\mu = 0,1,2,3) obeying the anticommutation relations \{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}, where \eta^{\mu\nu} is the with (+,-,-,-). This algebra provides the spinor representation for relativistic fermions, essential for describing electrons and other spin-1/2 particles in , with the acting on 4-dimensional complex spaces. Associative algebras also feature prominently in the integrability of models like the Calogero-Moser systems, where operators take values in such algebras to guarantee the existence of conserved quantities. The pair formulation involves an operator L(t) satisfying a \frac{dL}{dt} = [L, A] for some connection A, the nonlinear dynamics into a flat connection on a , with the algebra ensuring the zero-curvature condition for integrability. This construction applies to rational, trigonometric, and elliptic variants of the Calogero-Moser model, modeling interacting particles with inverse-square potentials.

Key Constructions

Enveloping algebras

In the context of Lie algebras, the universal enveloping algebra provides a canonical way to embed a Lie algebra into an associative algebra while preserving its bracket structure. Given a Lie algebra \mathfrak{g} over a field K of characteristic zero, the tensor algebra T(\mathfrak{g}) is the free associative algebra generated by \mathfrak{g}, graded by the direct sum T(\mathfrak{g}) = \bigoplus_{n=0}^\infty \mathfrak{g}^{\otimes n} with \mathfrak{g}^{\otimes 0} = K. The universal enveloping algebra U(\mathfrak{g}) is defined as the quotient T(\mathfrak{g}) / I, where I is the two-sided ideal generated by elements of the form x \otimes y - y \otimes x - [x, y] for all x, y \in \mathfrak{g}. This construction equips U(\mathfrak{g}) with a natural Lie algebra homomorphism \iota: \mathfrak{g} \to U(\mathfrak{g}), and the relations in U(\mathfrak{g}) imply that for any x, y \in \mathfrak{g}, the commutator satisfies xy - yx = [x, y] in the associative product of U(\mathfrak{g}). The defining feature of U(\mathfrak{g}) is its universal property: for any associative algebra A over K and any Lie algebra homomorphism \phi: \mathfrak{g} \to A (where A is viewed as a Lie algebra via the commutator [a, b] = ab - ba), there exists a unique associative algebra homomorphism \tilde{\phi}: U(\mathfrak{g}) \to A such that \tilde{\phi} \circ \iota = \phi. This property ensures that U(\mathfrak{g}) is the "largest" associative algebra containing \mathfrak{g} as a Lie subalgebra, making it a universal envelope. Representations of the Lie algebra \mathfrak{g} on vector spaces extend uniquely to representations of U(\mathfrak{g}) via this homomorphism. A fundamental result concerning the structure of U(\mathfrak{g}) is the Poincaré-Birkhoff-Witt (PBW) theorem, which provides an explicit basis. If \{x_i \mid i \in I\} is a basis of \mathfrak{g} with I totally ordered, then the monomials x_{i_1}^{e_1} x_{i_2}^{e_2} \cdots x_{i_k}^{e_k}, where i_1 \leq i_2 \leq \cdots \leq i_k and each e_j \geq 1, together with the unit element, form a K-basis for U(\mathfrak{g}). This basis consists of ordered products respecting the on the indices, highlighting the non-commutative nature induced by the Lie bracket. The algebra U(\mathfrak{g}) inherits a filtration from the grading on T(\mathfrak{g}), where the n-th level F_n U(\mathfrak{g}) is the image of \bigoplus_{k=0}^n \mathfrak{g}^{\otimes k} under the quotient map. The associated graded algebra \mathrm{gr} \, U(\mathfrak{g}) = \bigoplus_{n=0}^\infty F_n U(\mathfrak{g}) / F_{n-1} U(\mathfrak{g}) is isomorphic to the symmetric algebra S(\mathfrak{g}), which is the free commutative algebra generated by \mathfrak{g}. This isomorphism underscores the PBW theorem, as the basis monomials in U(\mathfrak{g}) project to the standard monomial basis in S(\mathfrak{g}).

Dual associative algebras

In the finite-dimensional case over a k, the linear A^* of an associative algebra A acquires a natural coalgebra structure, making it coassociative and counital. The comultiplication \Delta: A^* \to A^* \otimes A^* is defined by (\Delta(\phi))(a \otimes b) = \phi(ab) for all \phi \in A^* and a, b \in A, while the counit \varepsilon: A^* \to k is given by \varepsilon(\phi) = \phi(1_A). This construction dualizes the multiplication and unit maps of A, ensuring coassociativity via the associativity of A. For infinite-dimensional associative algebras, the full algebraic dual A^* often proves unwieldy due to its size, prompting consideration of restricted duals such as the finite dual or topological variants. The finite dual A^\circ, introduced by Heyneman and Sweedler, consists of those linear functionals \phi \in A^* such that the image of \Delta(\phi) (defined analogously) lies in the tensor product of finite-dimensional subspaces; it forms a coalgebra dense in A^* in suitable topologies. In Banach algebra settings, the continuous (topological) dual can inherit a coalgebra structure under completeness assumptions, though the algebraic dual remains the primary focus for non-topological extensions. A fundamental duality theorem states that for finite-dimensional associative algebras A and B over k, the dual of their tensor product algebra A \otimes_k B is isomorphic as a coalgebra to the tensor product of the dual coalgebras A^* \otimes_k B^*, with the coalgebra operations induced componentwise from those on A^* and B^*. This isomorphism arises from the natural identification (A \otimes_k B)^* \cong A^* \otimes_k B^* of vector spaces, preserving the dualized multiplications. In finite dimensions, associative algebras are thus precisely dual to coassociative coalgebras, with commutativity of the algebra corresponding to cocommutativity of the dual coalgebra. Radford's construction addresses pairs of and coalgebras by forming a B \#_\pi H from an B and a H equipped with compatible actions and a twisting cocycle \pi, yielding a framework for recovering structures in Hopf settings; this extends classical duality by enabling explicit reconstructions of and integrals in the pair.

Separable algebras

A separable over a K is an associative K- A such that the canonical homomorphism \mu: A \otimes_K A^{\mathrm{op}} \to \mathrm{End}_K(A), defined by (a \otimes b^{\mathrm{op}})(x) = a x b for a, b, x \in A, admits a section \sigma: \mathrm{End}_K(A) \to A \otimes_K A^{\mathrm{op}} that is a of A-A^{\mathrm{op}}-bimodules; this section necessarily satisfies \mu \circ \sigma = \mathrm{id}, ensuring the inclusion of the unit map from K to A. This condition is equivalent to A being projective as a bimodule over its enveloping algebra A^e = A \otimes_K A^{\mathrm{op}}, meaning the projective dimension \mathrm{pd}_{A^e}(A) = 0. Equivalently, there exists a separability idempotent e \in A \otimes_K A^{\mathrm{op}} satisfying (a \otimes 1) e = e = e (1 \otimes a) for all a \in A and such that the multiplication map m: A \otimes_K A^{\mathrm{op}} \to A yields m(e) = 1_A. In the commutative case, where A^{\mathrm{op}} \cong A, separability reduces to A being a finite étale K-algebra, equivalent to A being a product of separable field extensions of K. Over a K (such as one of characteristic zero), a finite-dimensional A is separable if and only if it is semisimple. A key application arises in the study of Azumaya algebras, which are finite-dimensional central separable K-; these form the Brauer group \mathrm{Br}(K) under tensor product over K, classifying central simple algebras up to .

Special Classes

Finite-dimensional associative algebras

Finite-dimensional associative algebras over a k form a fundamental class in , characterized by their complete reducibility into semisimple and components. These algebras admit a precise structural decomposition that reveals their representations and idempotent structure, distinguishing them from infinite-dimensional counterparts. The Jacobson J(A) of a finite-dimensional associative algebra A over k is defined as the of all maximal left ideals of A. This ideal is nilpotent, meaning there exists a positive m such that J(A)^m = 0, and A is semisimple precisely when J(A) = 0. Semisimple finite-dimensional algebras admit a canonical decomposition given by the Artin-Wedderburn theorem: A \cong \bigoplus_i M_{n_i}(D_i), where each M_{n_i}(D_i) is the algebra of n_i \times n_i matrices over a finite-dimensional D_i with k, and the n_i and D_i are unique up to permutation. This isomorphism implies that the dimension of A satisfies \dim_k A = \sum_i n_i^2 \dim_k D_i. Over an k, each D_i must be isomorphic to k itself, simplifying the decomposition to A \cong \bigoplus_i M_{n_i}(k). For a general finite-dimensional algebra A, the quotient A / J(A) is semisimple, providing the semisimple "top" of the structure, while the nilpotent radical encodes the non-semisimple obstructions. Local algebras, where J(A) is the unique maximal two-sided ideal, are indecomposable and play a key role; here, ensures that idempotents in A / J(A) lift to idempotents in A, facilitated by the modular law for finitely generated modules over local rings. Recent computational advances, such as those implemented in the package ModIsom, enable the explicit and testing of finite-dimensional associative algebras up to low dimensions (e.g., dimension 5 over \mathbb{C}), aiding verification of theoretical predictions.

Non-unital associative algebras

A non-unital associative algebra over a K (often a ) is a K- A equipped with a bilinear multiplication A \times A \to A that is associative, meaning (ab)c = a(bc) for all a, b, c \in A, but without the existence of a multiplicative identity element. Ideals in such an algebra are defined as subsets closed under addition and multiplication by elements of A from either side, analogous to the unital case, and modules over A are defined using left or right actions via the multiplication, with bimodules requiring compatibility between the actions. A standard construction to relate non-unital algebras to unital ones is unitization. For a non-unital associative algebra A over K, the unitization A^+ = A \oplus K is defined with componentwise addition and multiplication given by (a, \lambda)(b, \mu) = (ab + \lambda b + \mu a, \lambda \mu) for a, b \in A and \lambda, \mu \in K. This makes A^+ a unital associative algebra with identity (0, 1), and A embeds injectively as the two-sided ideal \{(a, 0) \mid a \in A\}. The unitization is unique up to isomorphism in the sense that any unital algebra containing A as an ideal with A absorbing the scalars from the unit is isomorphic to A^+. Every non-unital associative algebra embeds as a two-sided in its unitization, which is unital, demonstrating that the of associative algebras without is "contained" in the unital via this free adjunction. However, properties like the Jacobson may differ: if J(A) denotes the Jacobson of A (the of all maximal modular right ideals), then J(A^+) \cap A = J(A), so the may extend beyond the embedded copy of A depending on the structure. Associativity in the non-unital setting still yields modular identities for bimodules, such as (am)r = a(mr) for m a left A-module and r \in A, but a one-sided module need not extend to a bimodule without further conditions on the action. A key example is the algebra \mathfrak{n}_n(K) of n \times n strictly upper triangular matrices over K, with the usual and . This is associative and non-unital, as all elements are with \mathfrak{n}_n(K)^n = 0, and the lies outside it. Another example arises in operator algebras, such as the algebra of nilpotent shift operators on a space with finite support, which forms a non-unital associative algebra under .

Lattices and orders in associative algebras

A lattice-ordered algebra, also known as an l-algebra, is a real vector lattice equipped with an associative multiplication operation that is compatible with the order structure, meaning that if x, y \in A^+ (the positive cone), then xy \in A^+. This compatibility ensures that the multiplication preserves the partial order, with the positive elements forming a proper cone in the underlying vector space. Such algebras generalize ordered rings by incorporating the lattice structure, where every pair of elements has a supremum and infimum with respect to the order. A special class within lattice-ordered algebras are f-algebras (or f-rings in the context), where the distributes over the operations for positive elements: specifically, for z \in A^+, by z acts as a homomorphism, satisfying (x \vee y)z = xz \vee yz and z(x \vee y) = zx \vee zy. Additionally, in f-algebras, if x \wedge y = 0, then xz \wedge y = 0 and zx \wedge y = 0 for all z \in A^+. These properties make f-algebras particularly useful in , as they allow for subdirect representations as sums of totally ordered s. A prototypical example is the algebra C(X) of continuous real-valued functions on a compact X, equipped with pointwise addition, multiplication, and the pointwise order f \leq g if f(t) \leq g(t) for all t \in X. This forms a Riesz space (Dedekind complete vector lattice under ) and an f-algebra, with the constant function 1 serving as the multiplicative unit. In the context of Banach lattices, which are complete normed lattice-ordered vector spaces, an order unit e > 0 induces the order-unit \|x\|_e = \inf\{\lambda > 0 \mid -\lambda e \leq x \leq \lambda e\}, a monotone equivalent to the given Banach when e is a strong order unit. Archimedean Banach lattices with an order unit and the Levi property can often be represented as closed sublattices of function algebras like C(K) for some K, via the order-unit . Recent developments extend these concepts to noncommutative ordered algebras, where structures are adapted to noncommutative associative algebras, finding applications in the algebraic framework of quantum groups and . For instance, noncommutative lattice-ordered division rings provide models for ordered structures in quantum settings, enhancing the study of positive representations and order-preserving maps in Hopf algebras associated with quantum groups.

Related Concepts

Coalgebras and duality

A coalgebra over a field k is a vector space C equipped with a linear map \Delta: C \to C \otimes C, called the comultiplication, and a linear map \varepsilon: C \to k, called the counit, satisfying coassociativity and counit axioms. The coassociativity condition is expressed as (\Delta \otimes \mathrm{id}_C) \circ \Delta = (\mathrm{id}_C \otimes \Delta) \circ \Delta, ensuring that the comultiplication can be iterated unambiguously, while the counit axioms are (\varepsilon \otimes \mathrm{id}_C) \circ \Delta = \mathrm{id}_C = (\mathrm{id}_C \otimes \varepsilon) \circ \Delta, which provide a notion of "scalar" extraction dual to the unit in algebras. Such structures are coassociative coalgebras, and their duality with associative algebras arises naturally: the dual of a coassociative coalgebra is an associative algebra, with multiplication induced by the comultiplication via \langle ab, c \rangle = \langle a \otimes b, \Delta(c) \rangle. In the finite-dimensional case, this duality is perfect and contravariant: every finite-dimensional associative algebra is the dual of a unique finite-dimensional , and vice versa, preserving the unital and associative structures through the . For infinite-dimensional settings, the duality is more subtle and typically realized through topological spaces, where continuous or completions ensure well-defined operations, or via the finite dual , which embeds finite-dimensional subcoalgebras into the structure. This extends the finite-dimensional correspondence to broader classes, such as topological associative algebras dual to topological , though the infinite-dimensional may not capture the full structure without additional . Dual associative algebras represent a specific instance of this interplay, where the dual directly yields an associative algebra structure. To handle computations with comultiplication, especially in infinite sums, Sweedler notation is employed: for c \in C, one writes \Delta(c) = \sum c_{(1)} \otimes c_{(2)}, suppressing the explicit summation index while iterating coassociativity as \sum c_{(1)} \otimes \Delta(c_{(2)}) = \sum \Delta(c_{(1)}) \otimes c_{(2)}. This notation simplifies expressions in theory, dual to the Einstein summation convention in tensor algebras, and is particularly useful for defining comodules and corepresentations. A key property links in s to that in algebras: cocommutative s, where \Delta = \tau \circ \Delta with \tau the twist map \tau(a \otimes b) = b \otimes a, are to commutative associative algebras, as the induced on the satisfies ab = ba. An illustrative example is the group k^G for a G, with basis \{\delta_g \mid g \in G\}, comultiplication \Delta(\delta_g) = \sum_{h \in G} \delta_h \otimes \delta_{h^{-1}g}, and counit \varepsilon(\delta_g) = \delta_{g,e}, where e is the of G; this is the of the k[G], where in k[G] corresponds to to the group in the . The Milnor-Moore theorem provides a profound between coalgebras, their primitive elements, and : for a connected graded cocommutative H over a of characteristic zero, H is isomorphic to the universal enveloping algebra of the formed by its primitive elements \{x \in H \mid \Delta(x) = x \otimes 1 + 1 \otimes x\}, with the Lie bracket induced by the in H. This result underscores how primitive elements in coalgebras generate Lie structures dual to the enveloping process in associative algebras, linking the two via graded connectivity and characteristic zero assumptions.

Hopf algebras from representations

In the context of , Hopf algebras extend the structures of group algebras and universal enveloping algebras by incorporating both multiplicative and comultiplicative operations that facilitate the study of modules as representations. A Hopf algebra H over a k is fundamentally a , meaning it is an associative algebra equipped with a coassociative coalgebra structure (\Delta, \epsilon) satisfying the compatibility conditions \Delta(ab) = \Delta(a)\Delta(b) and \epsilon(ab) = \epsilon(a)\epsilon(b) for all a, b \in H, where \Delta: H \to H \otimes H is the and \epsilon: H \to k is the counit. This bialgebra framework allows representations of H to be viewed as modules over the algebra, while the coalgebra aspect enables the construction of tensor product representations via \Delta, mirroring how group representations extend to tensor products. The defining feature of a Hopf algebra is the antipode S: H \to H, a satisfying m (id \otimes S) \Delta = \eta \epsilon = m (S \otimes id) \Delta, where m: H \otimes H \to H is the multiplication and \eta: k \to H is the unit map. In , the antipode provides a for "inverting" group-like or elements, allowing the extraction of invariants and enabling the algebra to act as a "group" or "" in a generalized sense on modules. This structure is crucial for corepresentations and Hopf modules, where the antipode ensures that representations can be "twisted" or integrated over the algebra. A primary motivation for Hopf algebras arises from the , where the group algebra k[G] naturally forms a Hopf algebra with basis elements g \in G as group-like, satisfying \Delta(g) = g \otimes g and S(g) = g^{-1}. This endows representations of G with a compatible tensor product structure, generalizing to infinite or quantum settings. Similarly, the universal enveloping algebra U(\mathfrak{L}) of a \mathfrak{L} over k becomes a Hopf algebra by declaring primitive elements x \in \mathfrak{L} to satisfy \Delta(x) = x \otimes 1 + 1 \otimes x and S(x) = -x, allowing infinitesimal actions on modules to extend multiplicatively. Finite-dimensional noncommutative examples of Hopf algebras, such as the Taft algebras, illustrate the richness of this structure beyond commutative or cocommutative cases; for instance, the Taft algebra of dimension nq (with q a primitive nth root of unity, n > 1) has an antipode of order $2n and serves as a building block for classifying low-dimensional Hopf algebras in . These algebras arise in contexts and provide explicit models for studying indecomposable representations.

Lie algebras from associative structures

Given an associative algebra A over a k, the [a, b] = ab - ba for a, b \in A defines a bilinear, skew-symmetric on the underlying of A, thereby endowing A with the structure of a over k. This construction applies similarly to ideals of A, yielding Lie subalgebras. The for this follows directly from the associativity of the product in A: [[a, b], c] + [[b, c], a] + [[c, a], b] = 0, which expands to (ab)c - (ba)c - c(ab) + c(ba) + (bc)a - (cb)a - a(bc) + a(cb) + (ca)b - (ac)b - b(ca) + b(ac) and cancels pairwise due to (xy)z = x(yz) for all x, y, z \in A. This commutator bracket defines a functor from the category of associative algebras over k to the category of Lie algebras over k, sending an associative algebra A to the Lie algebra \mathrm{Lie}(A) with the same underlying vector space and the commutator as bracket; on morphisms, it acts as the identity on the underlying spaces. A key theorem states that every Lie algebra L over a field of characteristic zero embeds as a Lie subalgebra into an associative algebra, specifically via the canonical inclusion L \hookrightarrow U(L)^-, where U(L) is the universal enveloping algebra of L and U(L)^- is U(L) with the commutator bracket. The Whitehead lemmas provide foundational results on the representation theory of semisimple Lie algebras arising in this way: for a finite-dimensional semisimple Lie algebra \mathfrak{g} over a field of characteristic zero and any finite-dimensional \mathfrak{g}-module V, the cohomology groups H^1(\mathfrak{g}, V) = 0 and H^2(\mathfrak{g}, V) = 0. These vanishing results imply important trace conditions, such as the invariance of trace forms in representations and the fact that traces of adjoint operators vanish appropriately, ensuring that the Killing form B(x, y) = \mathrm{tr}(\mathrm{ad}_x \mathrm{ad}_y) is nondegenerate on \mathfrak{g}. An important application is to derivations: the space \mathrm{Der}(A) of k-linear derivations of A (maps d: A \to A satisfying d(ab) = d(a)b + a d(b)) forms a of \mathrm{gl}(A), the of all endomorphisms of A under the , with [d_1, d_2] = d_1 \circ d_2 - d_2 \circ d_1. Inner derivations, given by [a, \cdot] = ad_a for a \in A, form an in \mathrm{Der}(A).