Associative algebra

In mathematics, an associative algebra is an algebraic structure consisting of a vector space, or more generally a module, equipped with a bilinear and associative multiplication operation. It serves as a fundamental concept in abstract algebra, bridging the theories of rings and modules, and providing the essential framework for studying linear transformations, polynomials, and various other mathematical objects.

Definition

Let $R$ be a fixed commutative ring, which is most commonly a field $K$. An associative algebra over $R$ (or an $R$-algebra) is an $R$-module $A$ equipped with an $R$-bilinear map $\cdot : A \times A \to A$, called multiplication, which satisfies the associative law:
$(x \cdot y) \cdot z = x \cdot (y \cdot z)$ for all $x, y, z \in A$.

If the algebra contains an identity element for multiplication—an element $1$ such that $1 \cdot x = x \cdot 1 = x$ for all $x \in A$—it is called a unital associative algebra. In modern literature, the term "associative algebra" usually implies the existence of a unit unless explicitly stated otherwise.

Equivalently, an associative algebra over a field $K$ can be defined as a ring $A$ that is also a $K$-vector space, such that the ring multiplication and scalar multiplication are compatible. This compatibility is expressed by the condition $c(x \cdot y) = (cx) \cdot y = x \cdot (cy)$ for all $c \in K$ and $x, y \in A$.

Examples

Several fundamental mathematical structures naturally form associative algebras:

• Matrix algebras: The set of $n \times n$ matrices over a field $K$, denoted $M_n(K)$, with standard matrix addition and multiplication, forms a unital associative algebra over $K$.
• Polynomial rings: The ring of polynomials $K[x_1, \dots, x_n]$ over a field $K$ is a commutative associative algebra over $K$.
• Group algebras: Given a group $G$ and a field $K$, the group algebra $K[G]$ consists of formal finite linear combinations of elements of $G$ with coefficients in $K$. The multiplication is defined by extending the group operation linearly.
• Endomorphism algebras: For any vector space $V$ over $K$, the set of all linear transformations from $V$ to $V$, denoted $\text{End}(V)$, forms an associative algebra under the addition of functions and the composition of functions.
• Quaternions: The quaternions $\mathbb{H}$ form a 4-dimensional, non-commutative associative algebra over the real numbers $\mathbb{R}$.

Subalgebras, Ideals, and Quotients

A subset $B$ of an $R$-algebra $A$ is a subalgebra if it is closed under addition, scalar multiplication, and the algebra multiplication, and contains the multiplicative identity in the unital case.

An ideal of an associative algebra is a ring ideal that is also an $R$-submodule. Specifically, a two-sided ideal $I$ is an $R$-submodule such that $AI \subseteq I$ and $IA \subseteq I$. One can also define left and right ideals analogously.

The quotient of an associative algebra $A$ by a two-sided ideal $I$, denoted $A/I$, naturally inherits the structure of an associative algebra. The elements of $A/I$ are cosets of the form $x + I$, and the multiplication is well-defined by $(x + I)(y + I) = xy + I$.

Homomorphisms

An algebra homomorphism between two $R$-algebras $A$ and $B$ is an $R$-module homomorphism $f: A \to B$ that also preserves multiplication, meaning $f(x \cdot y) = f(x) \cdot f(y)$ for all $x, y \in A$. If the algebras are unital, a unital homomorphism additionally requires $f(1_A) = 1_B$.

The kernel of an algebra homomorphism is a two-sided ideal of the domain algebra. The First Isomorphism Theorem holds for associative algebras: if $f: A \to B$ is a surjective algebra homomorphism, then the quotient algebra $A/\ker(f)$ is isomorphic to $B$ as an $R$-algebra.

Free Associative Algebras

Given a set $X$ and a commutative ring $R$, the free associative algebra over $R$ generated by $X$, often denoted $R\langle X \rangle$, is the algebra of non-commutative polynomials in the variables from $X$. Its elements are finite $R$-linear combinations of finite words (strings) formed by elements of $X$, with multiplication given by the concatenation of words.

The free associative algebra satisfies a universal property: any set map from $X$ to an $R$-algebra $A$ extends uniquely to an $R$-algebra homomorphism from $R\langle X \rangle$ to $A$. The tensor algebra $T(V)$ of an $R$-module $V$ is a direct generalization of this concept, where the set $X$ is replaced by a module, and the algebra is constructed as the direct sum of the tensor powers of $V$.

Representations and Modules

The study of an associative algebra $A$ is deeply connected to the study of its modules. A left module over an associative algebra $A$ (also referred to as a representation of $A$) is an $R$-module $M$ equipped with an $R$-bilinear map $A \times M \to M$ that satisfies associativity and unit conditions.

Equivalently, a left $A$-module structure on $M$ is given by an algebra homomorphism from $A$ to the endomorphism algebra $\text{End}_R(M)$. The representation theory of associative algebras generalizes the representation theory of groups and Lie algebras, providing tools to analyze the structure of the algebra through its actions on vector spaces or modules.