Algebras and Representations MATH 3193 2016

75 Pages·2016·0.486 MB·English

by  Alison Parker, Oliver King

Algebras and Representations
MATH 3193
2016
Alison Parker, Oliver King
[email protected], [email protected]
after notes of
Andrew Hubery
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 that information about the algebra
can be deduced from knowing enough about 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 mathematics 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.
1
Contents
Chapter 1. Quaternions 5
1.1. Complex Numbers 5
1.2. Quaternions 5
1.3. Some Remarks (non-examinable) 9
Chapter 2. Algebras 10
2.1. Basic Definition 10
2.2. Division algebras 10
2.3. Characteristic of a Field 11
2.4. Algebras given by a basis and structure coefficients 11
2.5. Polynomials 12
2.6. Group algebras 13
2.7. Matrix algebras 13
2.8. Endomorphism algebras 14
2.9. Temperley-Lieb algebras 14
2.10. Direct Product of Algebras 15
Chapter 3. Homomorphisms and Subalgebras 16
3.1. Homomorphisms 16
3.2. Subalgebras 17
3.3. Intersections and generating sets 19
Chapter 4. Ideals 21
4.1. Ideals — Definition and Example 21
4.2. Sums and intersections of ideals 22
4.3. Products of ideals 23
Chapter 5. Quotient Algebras 24
5.1. Definition of Quotient Algebras 24
5.2. The Factor Lemma 25
Chapter 6. Presentations of Algebras 28
6.1. Free Algebras 28
6.2. Relations 30
Chapter 7. Representations 34
7.1. Basic Definition 34
2
CONTENTS 3
7.2. Examples 35
7.3. Representations of quotient algebras 36
7.4. Representations of group algebras 37
7.5. Equivalence 37
7.6. Direct product of representations 38
Chapter 8. Modules 39
8.1. Basic Definition 39
8.2. Direct product of modules 40
Chapter 9. Homomorphisms and Submodules 42
9.1. Homomorphisms 42
9.2. Endomorphism algebras of modules 42
9.3. Submodules 43
9.4. Submodules given via ideals 44
9.5. Restriction of scalars 45
Chapter 10. Quotient Modules and the First Isomorphism Theorem 46
10.1. Quotient Modules 46
10.2. First Isomorphism Theorem 46
10.3. Direct products again 47
10.4. Generating sets 47
Chapter 11. More on homomorphism spaces 48
11.1. The Opposite Algebra 48
11.2. Homomorphisms between direct products 49
Chapter 12. Semisimple Algebras 51
12.1. Semisimple Modules 51
12.2. Schur’s Lemma 52
12.3. Complements 52
Chapter 13. Wedderburn’s Structure Theorem 54
13.1. Semisimple algebras 54
13.2. Wedderburn’s Structure Theorem 55
Chapter 14. The Jacobson Radical 57
14.1. Definition of the Jacobson Radical 57
14.2. Nilpotent Ideals 58
14.3. Two key theorems 59
14.4. Examples 60
Chapter 15. Modules over the polynomial algebra 61
15.1. Submodules 61
15.2. Direct sums 62
15.3. Generalised eigenspaces 62
CONTENTS 4
15.4. The minimal polynomial 62
Appendix A. Rotations 64
A.1. Orthogonal Matrices 64
A.2. Rotations in 2-Space 65
A.3. Rotations in 3-space 66
A.4. Rotations in n-space 67
Appendix B. Review of Some Linear Algebra 68
B.1. Vector spaces 68
B.2. Matrices 69
B.3. Linear Combinations 69
B.4. The example of the vector space of functions KX 69
B.5. Subspaces 70
B.6. Sums and Intersections 70
B.7. Quotient Spaces 71
B.8. Linear Maps 71
B.9. Bases 73
B.10. Dimension 74
CHAPTER 1
Quaternions
1.1. Complex Numbers
Since the work of Wessel (1799) and Argand (1806) we think of complex numbers 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 (a+λb)c=ac+λbc.
Commutative ab=ba.
Complex numbers have wonderful properties, for example:
(cid:112)
• The conjugate of z =x+yi is z¯=x−yi, and its absolute value is |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.
• Rotationsoftheplanecorrespondtomultiplicationbycomplexnumbersofabsolutevalue
1.
1.2. Quaternions
Trying to find a way to represent rotations in three dimensions,
Sir William Rowan Hamilton invented the quaternions in 1843. He needed four real numbers,
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 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
5
1.2. QUATERNIONS 6
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.
Remark 1.2.1. This looks a bit like the multiplication rule for cross product except i×i=0
and not −1. So we can’t use the determinant trick to work out the product.
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 1.2.2. Multiplication of quaternions is not commutative:
ij =k but ji=−k, so ij (cid:54)=ji.
Some basic definitions: these are all analogous to the definitions for the complex numbers.
Definition 1.2.3. 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.
Theconjugate ofq isq¯=a−bi−cj−dk. Thusifaisarealnumberandpisapurequaternion,
then the conjugate of a+p is a−p.
√
The absolute value of q is |q|= a2+b2+c2+d2. Note that |q¯|=|q|.
Some 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.
(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}.
1.2. QUATERNIONS 7
If p=q, then p2 =−1=p¯2. Otherwise p and q are distinct elements of {i,j,k}. Let
r be the third element. Then p¯=−p, and similarly for q 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, write q = a+p with a ∈ R and p a pure quaternion, so that q¯ = a−p. Then
ap=pa,(as a is real), 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.
The first equality follows from property 4, the second from property 3, the third from
property4,thefourthas|q|2 isreal,andthefifthbyproperty4again. Finallytheanswer
follows by taking square roots.
Lemma 1.2.4. Any non-zero quaternion has a multiplicative inverse; that is, H is a division
algebra.
Proof. If q (cid:54)=0 then the inverse of q is q−1 =q¯/|q|2. (cid:3)
Lemma 1.2.5. 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.
2
Proof. If q = 0, then take r = 0 and θ and n arbitrary. Otherwise q/|q| is a quaternion of
absolute value 1. So it’s enough to prove it for quaternions of absolute value 1.
Nowletq haveabsolutevalue1andwriteq =a+pwitha∈Randpapurequaternion. Then
1=|q|2 =a2+|p|2,
sowecanwritea=cos(1θ)forsomeuniqueθ ∈[0,2π]. (Notethatasa2 ≤1wehave−1<a<1.
2
Hence |p| = sin(1θ). (Note as θ ∈ [0,2π] that this RHS is indeed non-negative.) Finally, if
2
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. We then have
p
q =a+p=cos(1θ)+ sin(1θ)=cos(1θ)+nsin(1θ)
2 |p| 2 2 2
1.2. QUATERNIONS 8
in the required form. (cid:3)
Lemma 1.2.6. 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
1 2 3
R3 with the usual dot product and cross product. Then
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}. This gives 9
possible cases and symmetry means we only need to check 3, namely i2, ij, and ji.
−i·i+i×i=−1+0=i2, −i·j+i×j =0+k =ij, −j·i+j×i=0−k =ji.
(cid:3)
The following theorem explains the reason for the 1θ in Lemma 1.2.5.
2
Theorem 1.2.7. 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 rotation about axis n
2 2 q n,θ
through angle θ.
Two quaternions q, q(cid:48) of absolute value 1 give the same rotation if and only if q(cid:48) =±q.
Proof. Recall that an ordered basis (f ,f ,f ) of R3 is called a right-handed orthonormal
1 2 3
basis provided that
f ·f =δ and f =f ×f .
i j ij 3 1 2
(NB: the “ortho” means right angled, the “normal” means length one and the “right-handed”
comes from the right-handed rule for cross product.) Also, if n ∈ R3 has length 1, then the
rotation about axis n through angle θ is the linear map
n(cid:55)→n, u(cid:55)→cos(θ)u+sin(θ)v, v (cid:55)→−sin(θ)u+cos(θ)v
where (n,u,v) is any right-handed orthonormal basis. With respect to this basis the matrix of the
rotation is:
 
1 0 0
 
0 cosθ sinθ
 
0 −sinθ cosθ
Now let q be a quaternion of absolute value 1. By Lemma 1.2.5 we can write q = cos(1θ)+
2
sin(1θ)nwithθ ∈[0,2π]andnapurequaternionofabsolutevalue1. Let(n,u,v)bearight-handed
2
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.
1.3. SOME REMARKS (NON-EXAMINABLE) 9
For simplicity set c:=cos(1θ) and s:=sin(1θ). Then q =c+s and q−1 =c−s. Now, since
2 2
n2 =−|n|2 =−1 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.
Similarly
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
using the double angle formula 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.
Anyparticularrotationoccursinexactlytwoways,astherotationaboutaxisnthroughangle
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. (cid:3)
2 2 2 2
1.3. Some Remarks (non-examinable)
The set of quaternions of absolute value 1 form a group under multiplication, denoted Sp(1),
anditisnothardtoseethatthemapq (cid:55)→R fromtheprevioustheoremdefinesasurjectivegroup
q
homomorphism R: Sp(1)→SO(3,R) to the group of rotations of R3. This group homomorphism
isadouble cover, meaningthattherearepreciselytwoelementsofSp(1)mappingtoeachrotation
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. Thereforebothofthesesetshaveaninducedtopologyonthem,
3
and both the multiplication and inversion maps are continuous, 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 sequence of quaternions, however,
starts at 1 but ends at −1. We therefore only 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.
Youcanreadaboutthe‘quaternionmachine’inJ.ConwayandR.Guy,Thebookofnumbers.
CHAPTER 2
Algebras
2.1. Basic Definition
Definition 2.1.1. Fix a base field K, for example R or C.
Analgebra over K,orK-algebra,consistsofaK-vectorspaceAtogetherwithamultiplication
A×A→A, (a,b)(cid:55)→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).
Remark2.1.2. Ifyouhaveseenthedefinitionofaring,thenyouwillseethatthedistributivity
axiom has been replaced by the stronger bilinearity axiom. We can do this since our algebra is a
priori a vector space.
An alternative description would therefore be that A is both a vector space and 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).
Remark 2.1.3. (1) In the literature, the algebras we consider might be called unital,
associative algebras. There are other types: Banach algebras are usually non-unital; Lie
algebras and Jordan algebras 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 that the mul-
tiplication is commutative, so ab = ba for all a,b ∈ A. This is related to, amongst
other things, algebraic geometry and algebraic number theory. In this course we will be
concerned with general, non-commutative, algebras.
Example 2.1.4. (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 contains a copy of
C, for example with basis {1,i}, it cannot be considered as a C-algebra since i does not
commute with j and k.
2.2. Division algebras
Definition 2.2.1. Adivision algebra isanon-zeroalgebraAinwhicheverynon-zeroelement
has a multiplicative inverse; that is, for all a(cid:54)=0 there exists a−1 such that aa−1 =1=a−1a.
10