Associative algebra

Intuition [Beginner]

An associative algebra is a vector space where vectors can also be multiplied.

Numbers are the first example: you can add them, scale them, and multiply them. Matrices are a richer example. You can add matrices, multiply them by numbers, and multiply two matrices together.

The word associative means grouping does not change a product. If you multiply three objects, doing the first two first or the last two first gives the same final answer.

Visual [Beginner]

The two bracketings of a three-term product land at the same result.

Associativity is about grouping, not about swapping order.

Worked example [Beginner]

Use ordinary two-by-two matrices. Let

$$A=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),B=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right).$$

You can add them:

$$A+B=\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right).$$

You can scale $B$ by 3. You can also multiply $A$ and $B$, and the result is again a two-by-two matrix.

What this tells us: an associative algebra combines vector-space operations with an internal product.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $K$ be a field 01.01.01. An associative unital $K$-algebra is a $K$-vector space $A$ 01.01.03 equipped with a bilinear multiplication map

$$m:A\times A\to A,m(a,b)=ab,$$

and an element $1_A\in A$ such that

$$(ab)c=a(bc),1_{A}a=a=a{1}_A,$$

for all $a,b,c\in A$.

Equivalently, an associative unital $K$-algebra is a ring $A$ together with a homomorphism $K\to Z(A)$ into the center of $A$. The vector-space scalar action is recovered from $r\cdot a=(r{1}_A)a$ ^{[Lang §III §1]}.

An algebra homomorphism $\varphi :A\to B$ is a $K$-linear map preserving multiplication and unit.

Key theorem with proof [Intermediate+]

Theorem (commutator Lie algebra). Let $A$ be an associative $K$-algebra. Define

$$[a,b]=ab-ba.$$

Then this bracket is bilinear, alternating, and satisfies the Jacobi identity

$$[a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0.$$

Proof. Bilinearity follows from bilinearity of multiplication and subtraction. Alternating follows from $[a,a]=aa-aa=0$.

For Jacobi, expand:

$$[a,[b,c]]=a(bc-cb)-(bc-cb)a=abc-acb-bca+cba.$$

Similarly,

$$[b,[c,a]]=bca-bac-cab+acb,$$

and

$$[c,[a,b]]=cab-cba-abc+bac.$$

Adding the three displayed expressions, every word cancels with its negative. Hence the Jacobi sum is zero. $\square$

Bridge. The construction here builds toward 03.01.03 (ideal in an algebra), where the same data is upgraded, and the symmetry side is taken up in 03.01.05 (quotient algebra). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

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Advanced results [Master]

Associative algebras over $K$ form a category with unital algebra homomorphisms. Tensor algebra 03.01.04 is the free object generated by a vector space: linear maps $V\to A$ into an associative algebra extend uniquely to algebra homomorphisms $T(V)\to A$.

Every associative algebra has a Lie algebra obtained by the commutator bracket. This construction is functorial and explains the appearance of matrix Lie algebras inside associative matrix algebras 03.04.01. Conversely, the universal enveloping algebra of a Lie algebra is an associative algebra built to encode Lie brackets as commutators.

Two-sided ideals are precisely the subspaces that can be collapsed while preserving multiplication. Quotient algebras 03.01.05 and Clifford algebras 03.09.02 use this mechanism to impose relations.

Synthesis. This construction generalises the pattern fixed in 01.01.01 (field), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Proposition. The kernel of an algebra homomorphism $\varphi :A\to B$ is a two-sided ideal of $A$.

The kernel is a vector subspace because $\varphi$ is linear. If $x\in \ker \varphi$ and $a\in A$, then

$$\varphi (ax)=\varphi (a)\varphi (x)=\varphi (a)0=0,\varphi (xa)=\varphi (x)\varphi (a)=0\varphi (a)=0.$$

Thus $ax$ and $xa$ lie in $\ker \varphi$. This is exactly the two-sided absorption condition.

Proposition. A unital algebra homomorphism preserves invertible elements.

If $a\in A$ has inverse $a^{-1}$, then

$$\varphi (a)\varphi (a^{-1})=\varphi (a{a}^{-1})=\varphi (1_A)=1_B,$$

and similarly $\varphi (a^{-1})\varphi (a)=1_B$. Hence $\varphi (a)$ is invertible with inverse $\varphi (a^{-1})$.

Connections [Master]

• Tensor algebra 03.01.04 is the free associative algebra on a vector space.

• Ideals in an algebra 03.01.03 are the subspaces compatible with multiplication.

• Quotient algebra 03.01.05 imposes algebraic relations by collapsing a two-sided ideal.

• Lie algebra 03.04.01 can be obtained from an associative algebra by the commutator bracket.

Historical & philosophical context [Master]

Associative algebras emerged from matrices, polynomial algebras, group algebras, and linear operators. Wedderburn's structure theory made finite-dimensional associative algebras central objects in early twentieth-century algebra ^{[Wedderburn 1908]}.

Bourbaki's algebra volumes present algebras through rings, modules, tensor products, and universal constructions, matching the modern categorical treatment used by tensor and quotient algebras ^{[Bourbaki Algebra I Ch. III]}.

Bibliography [Master]

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