Math 202B: Lecture 1

Definition 1.1. An algebra is a complex vector space $\mathcal{A}$ of positive dimension equipped with an, associative, bilinear, unital multiplication, and an antilinear, antimultiplicative, involutive conjugation.

Let us unpack this definition.

Vector space structure. First of all, $\mathcal{A}$ is a complex vector space. We will denote vectors in this space by uppercase Roman letters

$A,B,C,\dots$

and use lowercase Greek letters

$\alpha,\beta,\gamma,\dots$

for scalars in $\mathbb{C}$ .

Multiplication. Multiplication is a map

$\mathcal{A}\times\mathcal{A}\to\mathcal{A}$

whose values are denoted by concatenating its arguments,

$(A,B)\mapsto AB$ .

Associativity means that the symbol $ABC$ is unambiguous, because its two possible interpretations coincide:

$(AB)C=A(BC)$ .

We do not assume that multiplication is commutative – there may exist elements $A,B\in\mathcal{A}$ such that $AB\neq BA$ . When no such elements exist, we say that $\mathcal{A}$ is a commutative algebra.

Bilinearity means that multiplication interacts with the vector space structure according to the rule

$(\alpha_1A_1+\alpha_2A_2)(\beta_1B_1+\beta_2B_2)=\alpha_1\beta_1A_1B_1+\alpha_1\beta_2A_1B_2+\alpha_2\beta_1A_2B_1]+\alpha_2\beta_2A_2B_2.$

Problem 1.1 Let $0_\mathcal{A}$ denote the zero vector in $\mathcal{A}$ . Prove that for all $A\in\mathcal{A}$ we have

$A0_\mathcal{A}=0_\mathcal{A}A=0_\mathcal{A}.$

Unital means that there exists an element $I\in\mathcal{A}$ such that $IA=AI=A$ for all $A\in\mathcal{A}$ . Any such element is called a multiplicative unit. Since $\dim\mathcal{A}>0$ , the multiplicative unit is distinct from the additive unit $0_\mathcal{A}$ . Moreover, the multiplicative unit is unique.

Problem 1.2 Let $I,J\in\mathcal{A}$ be multiplicative units. Prove that $I=J$ .

Henceforth, we write $I_\mathcal{A}$ for the unique multiplicative unit in $\mathcal{A}$ . When no confusion is possible, we simply write $I$ .

An element $A\in\mathcal{A}$ is said to be invertible if there exists $B\in\mathcal{A}$ such that $AB=BA=I_\mathcal{A}$ . When this holds, we say that $B$ is the inverse of $A$ , and write $B=A^{-1}$ and $A=B^{-1}$ .

Problem 1.3 Suppose $A,B,C\in\mathcal{A}$ satisfy $AB=BA=I$ and $AC=CA=I$ . Prove that $B=C$ .

Multiplication in $\mathcal{A}$ can be concrete and numerical. Let

${E_x:x\in X}$

be a vector space basis of $\mathcal{A}$ indexed by a nonempty set $X$ . Any elements $A,B\in\mathcal{A}$ can be written as linear combinations

$A=\sum_{x\in X}\alpha_xE_x,\qquad B=\sum_{y\in X}\beta_yE_y$

with all but finitely many terms equal to $0_\mathcal{A}.$ By bilinearity,

$AB=\sum_{x,y\in X}\alpha_x\beta_yE_xE_y.$

Each product of basis vectors can be expanded as

$E_xE_y=\sum_{z\in X}\gamma_{xyz}E_z$

for uniquely determined scalars $\gamma_{xyz}\in\mathbb{C}$ . Thus multiplication in $\mathcal{A}$ is completely determined by the scalars

$\gamma_{xyz}, \quad x,y,z\in X$ ,

which are called the connection coefficients or structure constants of the basis $\{E_x \colon x \in X\}.$ From a computational perspective, it is desirable to choose a basis for which many of these coefficients vanish. This idea underlies Strassen’s algorithm for matrix multiplication.

An element $P \in \mathcal{P}$ is said to be idempotent if $P^2=P.$ This is equivalent to saying that the coefficients in the expansion

$P=\sum\limits_{x \in X} \pi_x E_x$

satisfy

$\sum\limits_{x,y \in X} \pi_x\pi_y\gamma_{xyz}=\pi_z$

for each $z \in X.$

Problem 1.4 Prove that a two-dimensional algebra must be commutative.

Conjugation. Conjugation is a function $\mathcal{A}\to\mathcal{A}$ denoted by $A\mapsto A^*$ . Antilinearity means that conjugation and the vector space structure interact according to the rule

$(\alpha A+\beta B)^*=\overline{\alpha}A^+\overline{\beta}B^*.$

Antimultiplicativity means that conjugation and multiplication interact according to the rule

$(AB)^*=B^*A^*.$

Involutive means that conjugation is two-periodic,

$(A^*)^*=A.$

Let $I(\mathcal{A})$ denote the set of invertible elements in $\mathcal{A}.$

Problem 1.5 Prove that:

$I(\mathcal{A})$ is a group under multiplication.
$A$ is invertible if and only if $A^*$ is invertible.
$(A^*)^{-1}=(A^{-1})^*$ .

Problem 1.6 Let $\{E_x:x\in X\}$ be a vector space basis of $\mathcal{A}$ . Prove that $\{E_x^*:x\in X\}$ is also a vector space basis of $\mathcal{A}$ .

Special classes of elements. There are three special classes of elements in $\mathcal{A}$ .

An element $X\in\mathcal{A}$ is said to be selfadjoint if $X^*=X.$ The set $S(\mathcal{A})$ of selfadjoint elements in $\mathcal{A}$ forms a real vector space.

An element $U\in\mathcal{A}$ is said to be unitary if it is invertible and $U^{-1}=U^*.$ The set of unitary elements is denoted $U(\mathcal{A})$ and is called the unitary group of $\mathcal{A}.$

Problem 1.7 Prove that $U(\mathcal{A})$ is a subgroup of $I(\mathcal{A}).$

An element $A\in\mathcal{A}$ is said to be normal if it commutes with its conjugate: $A^*A=AA^*.$

Problem 1.8 Prove that every $A\in\mathcal{A}$ can be uniquely expressed in the form $A=X+iY,$ where $X,Y\in S(\mathcal{A}).$ The selfadjoint elements $X$ and $Y$ are called the real and imaginary parts of $A$ , respectively.

Problem 1.9 Prove that $A\in\mathcal{A}$ is normal if and only if its real and imaginary parts commute.

Problem 1.10 Prove that $\mathcal{A}$ is commutative if and only if all its elements are normal.

Math 202B: Lecture 1

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