Algebras and Representations MATH 3193

115 Pages·2011·0.651 MB·English

by  Andrew Hubery

Algebras and Representations
MATH 3193
Andrew Hubery
[email protected]
An algebra is a set A which at the same time has the structure of a ring and a
vector space in a compatible way. Thus you can add and multiply elements of
A, as well as multiply by a scalar. One example is the set of matrices of size n
over a field.
A representation is an action of an algebra on a vector space, similar to how
matrices of size n act on an n-dimensional vector space. It is often the case
thatinformationaboutthealgebracanbededucedfromknowingenoughabout
its representations. An analogy might be that one can begin to understand a
complicated function by computing its derivatives, or more generally a surface
by computing its tangent planes.
There are lots of interesting examples of algebras, with applications to mathe-
matics and physics. In this course we will introduce some of these algebras, as
well as some of the general theory of algebras and their representations.
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Contents
I Algebras 4
1 Quaternions 4
2 Algebras 9
3 Homomorphisms and Subalgebras 16
4 Ideals 21
5 Quotient Algebras 25
6 Presentations of Algebras 28
II Representations 35
7 Representations 35
8 Modules 40
9 Homomorphisms and Submodules 43
10 Quotient Modules 47
11 More on homomorphism spaces 49
III Two Examples 52
12 Modules over the polynomial algebra 52
13 The Weyl Algebra 54
IV Semisimple Algebras 56
14 Semisimple Modules and Schur’s Lemma 56
15 Wedderburn’s Structure Theorem 59
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16 Maschke’s Theorem 62
17 The Jacobson Radical 65
18 Clifford Algebras 69
19 The Jordan-H¨older Theorem 76
20 Division algebras and Frobenius’s Theorem 80
V Some More Examples 86
21 K[x]/(x2)-modules 86
22 Symmetric Groups (non-examinable) 87
VI Appendix 91
A Review of Some Linear Algebra 91
B Rotations 106
C Presentations of Groups (non-examinable) 111
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Part I
Algebras
1 Quaternions
1.1 Complex Numbers
SincetheworkofWessel(1799)andArgand(1806)wethinkofcomplexnumbers
as formal expressions
z =x+yi with x,y ∈R,
which we can add and multiply by expanding out, substituting i2 = −1, and
collecting terms. In other words, we have a two-dimensional real vector space
C with basis {1,i}, on which we have defined a multiplication
C×C→C
satisfying the following properties for all a,b,c∈C and λ∈R:
Associative a(bc)=(ab)c.
Unital there exists 1∈C with 1a=a=a1.
Bilinear a(b+λc)=ab+λac and (+λb)c=ac+λbc.
Commutative ab=ba.
Complex numbers have wonderful properties, for example:
• The conjugate of z = x + yi is z¯ = x − yi, and its absolute value is
p
|z| = x2+y2. Thus |z¯| = |z| and zz¯= |z|2. Also, |zw| = |z||w| for all
complex numbers z,w.
• Every non-zero complex number z =x+yi has an inverse
z−1 =(x−yi)/(x2+y2)=z¯/|z|2,
so they form a field.
• Rotations of the plane correspond to multiplication by complex numbers
of absolute value 1.
1.2 Quaternions
Trying to find a way to represent rotations in three dimensions, Sir William
RowanHamiltoninventedthequaternionsin1843. Heneededfourrealnumbers,
not three, and also had to drop commutativity.
Quaternions are expressions a+bi+cj+dk with a,b,c,d ∈ R. They add and
subtract in the obvious way, and multiply by first expanding out (being careful
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with the ordering), making the following substitutions
i2 =−1 ij =k ik =−j
ji=−k j2 =−1 jk =i
ki=j kj =−i k2 =−1
and then collecting terms. For example
(2+3i)(i−4j)=2i−8j+3i2−12ij =−3+2i−8j−12k.
The set of all quaternions is denoted H.
In other words we have a four-dimensional real vector space H with basis
{1,i,j,k}, on which we have defined a multiplication
H×H→H
satisfying the following properties for all a,b,c∈H and λ∈R:
Associative a(bc)=(ab)c.
Unital there exists 1∈H with 1a=a=a1.
Bilinear a(b+λc)=ab+λac and (a+λb)c=ac+λbc.
Remark. Multiplication of quaternions is not commutative:
ij =k but ji=−k, so ij 6=ji.
1.3 Basic Definitions
Let q =a+bi+cj+dk be a quaternion.
We say that the real part of q is a, and the imaginary part is bi+cj+dk.
A pure quaternion is one whose real part is zero.
The conjugate of q is q¯=a−bi−cj−dk. Thus if a is a real number and p is
a pure quaternion, then the conjugate of a+p is a−p.
√
The absolute value of q is |q|= a2+b2+c2+d2. Note that |q¯|=|q|.
1.4 Elementary properties
1. If p is a pure quaternion then p2 =−|p|2. For,
(bi+cj+dk)(bi+cj+dk)
=b2i2+c2j2+d2k2+bc(ij+ji)+bd(ik+ki)+cd(jk+kj)
=−(b2+c2+d2).
2. Conjugation is a linear map, so
p+λq =p¯+λq¯ for all p,q ∈H and λ∈R.
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3. Conjugation satisfies
q¯¯=q and pq =q¯p¯ for all p,q ∈H.
The first property is clear, so we just need to prove the second. Since
multiplication is bilinear and conjugation is linear, we can reduce to the
case when p,q ∈ {1,i,j,k}. If p = 1 or q = 1, then the result is clear, so
we may assume that p,q ∈{i,j,k}.
If p = q, then p2 = −1 = p¯2. Otherwise p and q are distinct elements of
{i,j,k}. Letr bethethirdelement. Thenp¯=−p,andsimilarlyforq and
r. Using the multiplication rules we see that pq =±r and qp=∓r, so
q¯p¯=(−q)(−p)=qp=∓r =±r¯=pq
as required.
4. We have
qq¯=q¯q =|q|2.
For,writeq =a+pwitha∈Randpapurequaternion,sothatq¯=a−p.
Then ap=pa, so
qq¯=(a+p)(a−p)=a2+pa−ap−p2 =a2−p2 =(a−p)(a+p)=q¯q.
Using p2 =−|p|2 we get
qq¯=a2−p2 =a2+|p|2 =|q|2.
5. For any two quaternions p and q we have
|pq|=|p||q|.
For
|pq|2 =pqpq =pqq¯p¯=p|q|2p¯=|q|2pp¯=|q|2|p|2.
1.5 Lemma
Any non-zero quaternion has a multiplicative inverse; that is, H is a division
algebra.
Proof. The inverse of q is q−1 =q¯/|q|2.
1.6 Lemma
Every quaternion can be written in the form
q =r(cid:0)cos(1θ)+sin(1θ)n(cid:1)
2 2
where r,θ ∈ R with r = |q| ≥ 0 and θ ∈ [0,2π], and n is a pure quaternion of
absolute value 1.
The use of 1θ is traditional; the reason will become clear later.
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Proof. If q = 0, then this is clear; otherwise q/|q| is a quaternion of absolute
value 1.
Now let q have absolute value 1 and write q = a+p with a ∈ R and p a pure
quaternion. Then
1=|q|2 =a2+|p|2,
so we can write a = cos(1θ) for some unique θ ∈ [0,2π], whence |p| = sin(1θ).
2 2
Finally, if θ = 0,2π, then p = 0 so we can take n to be arbitrary; otherwise
n=p/|p| is a pure quaternion of absolute value 1.
1.7 Lemma
We may identify the set of pure quaternions P = {bi+cj +dk : b,c,d ∈ R}
with R3 such that i,j,k correspond respectively to the standard basis vectors
e ,e ,e . We equip R3 with the usual dot product and cross product. Then
1 2 3
pq =−p·q+p×q ∈H for all p,q ∈P.
Note that the dot product of two elements of P is in R, and the cross product
is in P, so the sum makes sense in H.
Proof. Each operation is bilinear, so it suffices to check this for p,q ∈ {i,j,k}.
1.8 Theorem
If q is a quaternion of absolute value 1, then the linear transformation
R : P →P, R (p):=qpq−1
q q
is a rotation. Explicitly, if q = cos(1θ) + sin(1θ)n, then R = R is the
2 2 q n,θ
rotation about axis n through angle θ.
Two quaternions q,q0 of absolute value 1 give the same rotation if and only if
q0 =±q.
Proof. Recall that an ordered basis (f ,f ,f ) of R3 is called a right-handed
1 2 3
orthonormal basis provided that
f ·f =δ and f =f ×f .
i j ij 3 1 2
Also, if n∈R3 has length 1, then the rotation about axis n through angle θ is
the linear map
n7→n, u7→cos(θ)u+sin(θ)v, v 7→−sin(θ)u+cos(θ)v
where (n,u,v) is any right-handed orthonormal basis.
Now let q be a quaternion of absolute value 1. By Lemma 1.6 we can write
q = cos(1θ)+sin(1θ)n with θ ∈ [0,2π] and n a pure quaternion of absolute
2 2
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value 1. Let (n,u,v) be a right-handed orthonormal basis for P. The previous
lemma tells us that
nu=−n·u+n×u=v and un=−u·n+u×n=−n×u=−v.
Similarly
uv =n=−vu and vn=u=−nv.
For simplicity set c := cos(1θ) and s := sin(1θ). Now, since n2 = −|n|2 = −1
2 2
we have
qnq−1 =(c+sn)n(c−sn)=(c+sn)(cn−sn2)=(c+sn)(cn+s)
=c2n+cs+csn2+s2n=(cs−cs)+(c2+s2)n=n.
Also
quq−1 =(c+sn)u(c−sn)=(c+sn)(cu−sun)=(c+sn)(cu+sv)
=c2u+csv+csnu+s2nv =(c2−s2)u+2csv =cos(θ)u+sin(θ)v
and hence
qvq−1 =qnuq−1 =qnq−1quq−1 =cos(θ)nu+sin(θ)nv =−sin(θ)u+cos(θ)v.
This is the rotation claimed.
Any particular rotation occurs in exactly two ways, as the rotation about axis
n through angle θ, and as the rotation about −n through angle 2π −θ. The
latter corresponds to the quaternion
cos(π− 1θ)+sin(π− 1θ)(−n)=−cos(1θ)−sin(1θ)n=−q.
2 2 2 2
1.9 Remarks (non-examinable)
The set of quaternions of absolute value 1 form a group under multiplication,
denotedSp(1),anditisnothardtoseethatthemapq 7→R fromtheprevious
q
theorem defines a surjective group homomorphism R: Sp(1)→SO(3,R) to the
groupofrotationsofR3. Thisgrouphomomorphismisadouble cover, meaning
that there are precisely two elements of Sp(1) mapping to each rotation in
SO(3,R).
In fact, we can say more. A quaternion q =a+bi+cj+dk has absolute value
1 precisely when a2+b2+c2+d2 =1, so Sp(1) can be thought of as a 3-sphere
S3 ={(a,b,c,d)∈R4 :a2+b2+c2+d2 =1}.
SimilarlySO(3,R)⊂M (R)∼=R9. Thereforebothofthesesetshaveaninduced
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topology on them, and both the multiplication and inversion maps are contin-
uous, so they are topological groups. In this set-up the group homomorphism
Sp(1)→SO(3,R) is also continuous.
Let us fix a pure quaternion n of absolute value 1. Then, as θ increases from 0
to 2π, we get a sequence of rotations starting and ending at the identity. The
sequenceofquaternions,however,startsat1butendsat−1. Wethereforeonly
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get 4π-periodicity for the quaternions. This is relevant in quantum mechanics
for ‘spin 1/2’ particles like electrons.
We can visualise this by rotating a book held in a hand: a 2π rotation returns
the book to its original position, but a 4π rotation is needed to return both the
book and the hand to their original positions.
You can read about the ‘quaternion machine’ in J. Conway and R. Guy, The
book of numbers.
2 Algebras
2.1 Definition
Fix a base field K, for example R or C.
An algebra over K, or K-algebra, consists of a K-vector space A together
with a multiplication
A×A→A, (a,b)7→ab,
satisfying the following properties for all a,b,c∈A and λ∈K:
Associative a(bc)=(ab)c.
Unital there exists 1∈A such that 1a=a=a1.
Bilinear a(b+λc)=ab+λ(ac) and (a+λb)c=ac+λ(bc).
Aside. If you have seen the definition of a ring, then you will see that the
distributivity axiom has been replaced by the stronger bilinearity axiom. We
can do this since our algebra is a priori a vector space.
AnalternativedescriptionwouldthereforebethatAisbothavectorspaceand
a ring, and that these structures are compatible in the sense that scalars can
always be brought to the front, so a(λb)=λ(ab).
2.2 Remarks
1. In the literature, the algebras we consider might be called unital, as-
sociative algebras. There are other types: Banach algebras are usually
non-unital; Lie algebras and Jordan algberas are non-associative.
2. Recall that a vector space V is finite dimensional if it has a finite basis.
Not all of our algebras will be finite dimensional.
3. There is a very rich theory of commutative algebras, where one assumes
thatthemultiplicationiscommutative, soab=baforalla,b∈A. Thisis
relatedto,amongstotherthings,algebraicgeometryandalgebraicnumber
theory. Inthiscoursewewillbeconcernedwithgeneral,non-commutative,
algebras.
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2.3 Examples
1. K is a 1-dimensional algebra over itself.
2. C is a 2-dimensional R-algebra with basis {1,i} as a vector space over R.
(It is also a 1-dimensional C-algebra, as in Example 1.)
3. H is a 4-dimensional R-algebra with basis {1,i,j,k}. Even though it
containsacopyofC,forexamplewithbasis{1,i},itcannotbeconsidered
as a C-algebra since i does not commute with j and k.
2.4 Division algebras
A division algebra is a non-zero algebra A in which every non-zero element
has a multiplicative inverse; that is, for all a 6= 0 there exists a−1 such that
aa−1 =1=a−1a.
2.5 Lemma
A division algebra has no zero divisors: ab=0 implies a=0 or b=0.
Proof. If ab=0 and a6=0, then 0=a−1(ab)=(a−1a)b=1b=b, so b=0.
2.6 Example
A field is the same as a commutative division algebra. The quaternions form a
division algebra, but are not commutative so do not form a field.
2.7 Characteristic
The characteristic of a field K is the smallest positive integer n such that
n·1=0∈K. If no such n exists, we define the characteristic to be zero.
Examples.
1. The fields Q, R and C all have characteristic zero.
2. If p∈Z is a prime, then F =Z is a field of characteristic p.
p p
3. The characteristic of a field is either zero or a prime number. For, if
char(K)=n and n=ab with 1<a,b<n, then ab=n=0∈K, whence
either a=0∈K or b=0∈K, contradicting the minimality of n.
2.8 Algebras given by a basis and structure coefficients
Oneimmediatequestionwhicharisesishowtodescribeanalgebra. Weshallgive
two answers to this question, both having their advantages and disadvantages.
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