Math 202: Abstract Algebras

This is the review post for Part $\mathcal{A}$ : abstract algebras.

Throughout the Math 202 sequence and on the qualifying exam, the term “algebra” by default refers to a complex vector space $\mathcal{A}$ of positive, finite dimension equipped with a unital, associative, bilinear multiplication $\mathcal{A} \times \mathcal{A} \to \mathcal{A}$ denoted $(A,B) \mapsto AB$ and an antilinear, antimultiplicative, involution conjugation $\mathcal{A} \to \mathcal{A}$ denoted $A \mapsto A^*.$

The fundamental example is $\mathcal{A}=\mathbb{C}.$ Two further examples are $\mathbb{C}^n$ with coordinatewise multiplication and conjugation, and $\mathbb{C}^{n \times n}$ with matrix multiplication and Hermitian conjugation. Both coincide with $\mathbb{C}$ for $n=1.$ Thus, we have an example of a commutative algebra of every dimension, and for algebras of square dimension bigger than one we also have a noncommutative example.

Inside a given algebra $\mathcal{A}$ there are various classes of special elements. The broadest of these is the class of normal elements, which consists of those $A \in \mathcal{A}$ which commute with their conjugate, $A^*A=AA^*.$ Two subclasses thereof are selfadjoint elements which are equal to their conjugate, $X^*=X$ , and unitary elements which satisfy $U^*U=I.$ Normal elements which are both selfadjoint and unitary are called reflections and they satisfy $R^2=I.$ Selfadjoint elements which can be factored as $N=A^*A$ are called nonnegative, and selfadjoint elements such that $P^2=P$ are called projections. Two projections which satisfy $PQ=0$ are said to be orthogonal.

Theorem A1. Every $A \in \mathcal{A}$ can be written $A=X+iY$ with $X,Y$ selfadjoint. Furthermore, this decomposition is unique and $A$ is normal if and only if $X$ and $Y$ commute.

Theorem A2. An algebra $\mathcal{A}$ is commutative if and only if all its elements are normal.

Theorem A3. Any set of nonzero pairwise orthogonal projections in $\mathcal{A}$ is linearly independent.

The category $\mathbf{Alg}$ has algebras as objects. Morphisms are linear transformations $\Phi \colon \mathcal{A} \to \mathcal{B}$ such that

$\Phi(I_\mathcal{A}) = I_\mathcal{B},\ \Phi(AB)=\Phi(A)\Phi(B),\ \Phi(A^*)=\Phi(A)^*,$

and these are called algebra homomorphisms. Two algebras $\mathcal{A}$ and $\mathcal{B}$ are said to be isomorphic if and only if there exists a bijective homomorphism between them.

Theorem A4. If $\mathcal{A}$ is a one-dimensional algebra then there exists a unique isomorphism $\mathcal{A} \to \mathbb{C}.$

Theorem A5. If $\mathcal{A}$ is a two-dimensional algebra then it is commutative.

Given any algebra homomorphism $\Phi \colon \mathcal{A} \to \mathcal{B},$ the set

$\mathrm{Im}(\Phi) = \{\Phi(A) \colon A \in \mathcal{A}\}$

is a subspace of $\mathcal{B}$ which contains $I_\mathcal{B}$ and is closed under multiplication and conjugation, i.e. it is a subalgebra of $\mathcal{B}.$ The set

$\mathrm{Ker}(\Phi) =\{K \in \mathcal{A} \colon \Phi(K)=0_\mathcal{B}\}$

is a subspace of $\mathcal{A}$ which does not necessarily contain $I_\mathcal{A}.$ However, $\mathrm{Ker}(\Phi)$ is closed under multiplication and conjugation, and in fact for every $K \in \mathrm{Ker}(\Phi)$ we have $\{AK,KA\} \subset \mathrm{Ker}(\Phi)$ for all $A \in \mathcal{A}$ , i.e. $\mathrm{Ker}(\Phi)$ is an ideal in $\mathcal{A}.$

Like the collection of subspaces of a given vector space, we can partially order the set of subalgebras of a given algebra $\mathcal{A}$ by set-theoretic inclusion. This poset has a unique minimal element given by $\mathbb{C}I=\{\alpha I \colon \alpha \in \mathbb{C}\}$ and a unique maximal element of the poset of subalgebras of $\mathcal{A}$ is $\mathcal{A}$ itself. Given two subalgebras $\mathcal{B}$ and $\mathcal{C}$ of $\mathcal{A}$ , there is a unique maximal subalgebra contained in both of them, namely $\mathcal{B} \cap \mathcal{C}$ . There is also a unique minimal subalgebra containing both $\mathcal{B}$ and $\mathcal{C}$ , namely the intersection of all subalgebras containing $\mathcal{B} \cap \mathcal{C}$ . This is called the algebra generated by $\mathcal{B} \cup \mathcal{C}$ and it is denoted $\mathrm{Alg}(\mathcal{B} \cup \mathcal{C}).$

Theorem A6. We have $\mathrm{Alg}(\mathcal{B} \cup \mathcal{C})=\mathrm{Span}\{A_1 \dots A_m \colon m \geq 0,\ A_i \in \mathcal{B} \cup \mathcal{C}\}.$

In fact, for an arbitrary subset $S \subseteq \mathcal{A}$ there is a unique minimal algebra $\mathrm{Alg}(S)$ containing $S$ , namely the intersection of all subalgebras containing it, and this is called the subalgebra generated by $S$ .

Theorem A7. We have $\mathrm{Alg}(S) = \mathrm{Span}\{A_1 \dots A_m \colon m \geq 0,\ A_i \in S \cup S^*\},$ where $S^*=\{A^* \colon A \in S\}.$

An important subalgebra of a given algebra $\mathcal{A}$ is its center,

$Z(\mathcal{A)}=\{Z \in \mathcal{A} \colon ZA=AZ \text{ for all }A \in \mathcal{A}\}.$

We have $Z(\mathcal{A})=\mathcal{A}$ if and only if $\mathcal{A}$ is commutative, and $\dim Z(\mathcal{A})$ can be viewed as a measure of how (non)commutative $\mathcal{A}$ is.

In fact, subalgebras always come in pairs: given $\mathcal{B} \leq \mathcal{A}$ the corresponding centralizer is

$Z(\mathcal{B},\mathcal{A})=\{A \in \mathcal{A} \colon AB=BA \text{ for all }B \in \mathcal{B}\}.$

Theorem A8. If $\mathcal{B}$ and $\mathcal{C}$ are subalgebras of $\mathcal{A}$ such that $\mathcal{C} \subseteq \mathcal{B},$ then their centralizers are subalgebras such that $Z(\mathcal{B},\mathcal{A}) \subseteq Z(\mathcal{C},\mathcal{A}).$

One may also consider the set of commutative subalgebras of a given algebra $\mathcal{A}$ ordered by inclusion. The intersection of any family of commutative subalgebras is again a commutative subalgebra. However, for an arbitrary set $X \subset \mathcal{A}$ there may not exist a commutative subalgebra containing it; this occurs for example if $X=\{A\}$ consists of a single non-normal element.

Theorem A9. A set $X \subseteq \mathcal{A}$ is contained in a commutative subalgebra of $\mathcal{A}$ if and only if $\mathrm{Alg}(X)$ is commutative.

The unique minimal element of the poset of commutative subalgebras of $\mathcal{A}$ is still $\mathbb{C}I$ , but there may be many maximal commutative subalgebras $\mathcal{B} \subseteq \mathcal{A}$ having the property that they are not contained in any strictly larger commutative subalgebra.

Theorem A10. A commutative subalgebra $\mathcal{B} \subseteq \mathcal{A}$ is a maximal commutative subalgebra if and only if $Z(\mathcal{B},\mathcal{A})=\mathcal{B}.$

A natural question is whether we can define a scalar product on a given algebra $\mathcal{A}$ which is compatible with its multiplication and conjugation. Given any scalar product on $\mathcal{A}$ , we can normalize it so that $\langle I,I \rangle =1.$ A scalar product so normalized is said to have the left-Frobenius property if

$\langle AB,C\rangle = \langle B,A^*C\rangle \quad\text{for all }A,B,C \in \mathcal{A}.$

We say that it has the right-Frobenius property if

$\langle AB,C\rangle = \langle A,CB^*\rangle \quad\text{for all }A,B,C \in \mathcal{A}.$

A scalar product satisfying $\langle I,I\rangle=1$ which has both the left-Frobenius property and the right-Frobenius property is called a Frobenius scalar product.

The existence of Frobenius scalar products on $\mathcal{A}$ is intimately related to the existence of special linear functionals on $\mathcal{A}$ . The reason for this is that given any linear functional $\varphi \colon \mathcal{A} \to \mathcal{C}$ we can define a corresponding Hermitian form on $\mathcal{A}$ by

$\langle A,B \rangle_\varphi := \varphi(A^*B),$

and this form has the left-Frobenius property.

A faithful state is a linear functional $\sigma \colon \mathcal{A} to \mathbb{C}$ satisfying $\sigma(I)=1$ and $\sigma(A^*A) \geq 0$ with equality if and only if $A=0_\mathcal{A}.$

Theorem A.10. Faithful states on $\mathcal{A}$ are in bijection with left-Frobenius scalar products on $\mathcal{A}.$

A trace is a linear functional $\tau \colon \mathcal{A} \to \mathcal{C}$ which satisfies $\tau(AB)=\tau(BA).$

Theorem A.11. Faithful tracial states on $\mathcal{A}$ are in bijection with Frobenius scalar products on $\mathcal{A}.$

An algebra $\mathcal{A}$ equipped with a Frobenius scalar product is called a von Neumann algebra. Equivalently, a von Neumann algebra is an algebra $\mathcal{A}$ together with a faithful tracial state.

Math 202: Abstract Algebras

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