Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 6 November 2012


An algebra is a vector space A over with a multiplication such that A is a ring with identity and such that for all a1,a2A and c,

(ca1)a2= a1(ca2)= c(a1a2). (1.1)

More precisely, an algebra is a vector space over with a multiplication that is associative, distributive, has an identity, and satisfies (1.1). Suppose that a1,a2,,an is a basis of A and that cijk are constants in such that

aiaj= k=1n cijk ak. (1.2)

It follows from (1.1) and the distributive property that the equations (1.2) for 1i,jn completely determine the multiplication in A. The cijk are called structure constants. The center of an algebra A is the subalgebra

Z(A)= { bAab= bafor allaA } .

A nonzero element pA such that pp=p is called an idempotent. Two idempotents p1,p2A are orthogonal if p1p2=p2p1=0. A minimal idempotent is an idempotent pA that cannot be written as a sum p=p1+p2 of orthogonal idempotents p1,p2A.

For each positive integer d we denote the algebra of d×d matrices with entries from and ordinary matrix multiplication by Md(). We denote the d×d identity matrix in Md() by Id. For a general algebra A, Md(A) denotes d×d matrices with entries in A. We denote the algebra of matrices of the form

( a00 0a0 00a ) ,aA,

by In(A). Note that In(A)A, as algebras. The trace, tr(a), of a matrix a=aij is the sum of the diagonal entries of a, tr(a)=iaii.

An algebra homomorphism of an algebra A into an algebra V is a -linear map f:AB such that for all a1,a2A,

f(1)=1, f(a1a2)=f(a1)f(a2). (1.3)

A representation of an algebra A is an algebra homomorphism

V:AMd ().

The dimension of the representation V is d. The image V(A) of the representation V is a finite dimensional algebra of d×d matrices which we call the algebra of the representation V. It is a subalgebra of Md(). A faithful representation is a representation which is injective. In this case the algebra V(A) is called a faithful realization of A and AV(A) The character of the representation V of A is the function χV:A given by

χV(a)= tr(V(a)). (1.4)

An anti-representation of an algebra A is a -linear map V:AMd() such that for all a1,a2A,

V(1)=Id, V(a1a2)= V(a2) V(a1).

As before the dimension of the anti-representation is d and the image, V(A), of the anti-representation is an algebra of matrices called the algebra of the anti-representation.

The group algebra G of a group G is the algebra of formal finite linear combinations of elements of G where the multiplication is given by the linear extension of the multiplication in G. The elements of G constitute a basis of G. A representation of the group G is a representation of its group algebra.

Let A be an algebra. An A-module is a vector space V with an A action A×VV such that for all a,a1,a2A, v,v1,v2V, and c1,c2,

1v = v, a1(a2v) = (a1a2)v, (a1+a2)v = a1v+a2v, a ( c1v1+ c2v2 ) = c1(av1)+ c2(av2). (1.5)

An A-module homomorphism is a -linear map f:VV between A-modules V and V such that for all aA and vV,

f(av)=af(v). (1.6)

An A-module isomorphism is a bijective A-module homomorphism.

By condition 3 of (1.5) the action of aA on V is a linear transformation V(a) of V. If we specify a basis B of V then the linear transformation V(a) can be written as a d×d matrix, where dimV=d. In this way we associate to every element of A a d×d matrix. This gives a representation of A which we shall also denote by V.

Conversely, if T is a d dimensional representation of A and V is a d dimensional vector space with basis B then we can define the action of an element a in A by the action of the linear transformation on V determined by the matrix T(a) so that for all vV,


In this way V becomes an A-module. Thus the notion of A A-module is equivalent to the notion of representation. When we view the A-module we are focusing on the vector space and when we view the representation we are focusing on the linear transformations (matrices).

Let V be an A-module with basis B and let B be another basis of V and denote the change of basis matrix by P. Let aA and let V(a), V(a) be the matrices, with respect to the bases B and B respectively, of the linear transformation on V induced by a. Then by elementary linear algebra we have that

V(a)=PV (a)P-1. (1.7)

This leads us to the following definition. Two d dimensional representations V and V of an algebra A are equivalent if there exists an invertible d×d matrix P such that (1.7) holds for all aA. Isomorphic modules define equivalent representations.

The direct sum V1V2 of two A-modules V1 and V2 is the A-module of all pairs (v1,v2), v1V1 and v2V2, with the A action given by

a(v1,v2)= (av1,av2),

for all aA. The direct sum V1V2 of two representations V1 and V2 of A is the representation V of A given by

V(a)= ( V1(a) 0 0 V2(a) ) . (1.8)

Direct sums of n>2 representations or A-modules are defined analogously. We denote VVV, n factors, by Vn. Note that the algebra of the representation Vn, Vn(A), is In(V(A)).

An A-invariant subspace of an A-module V is a subspace V of V such that

{ ava A,vV } =AVV.

An A-invariant subspace of V is just a submodule of V. Note that the intersection VV of any two invariant subspaces V, V of V is also an invariant subspace of V.

An A-module with no submodules is a simple module. An irreducible representation is a representation that is not equivalent to a representation of the form

V(a)= ( V(a) * 0 * ) , (1.9)

where V is also representation of A. If V, V are invariant subspaces of a representation V and V is irreducible then VV is either equal to 0 or V. A completely decomposable representation is a representation that is equivalent to a direct sum of irreducible representations. An algebra A is called completely decomposable if every representation of A is completely decomposable.

The centralizer of an algebra A of d×d matrices is the algebra A of d×d matrices a such that for all matrices aA,

aa=a a. (1.10)

The centralizer of a representation V of an algebra A is the algebra V(A).


1. Let A be an algebra of d×d matrices. Since all matrices in A commute with all elements of A,



In(A)= Mn(A)and Mn(A)= In(A).


In(A) =In(A).

2. Schur's lemma. Let W1 and W2 be irreducible representations of A of dimensions d1 and d2. If B is a d1×d2 matrix such that

W1(a)B=B W2(a), for allaA,

then either

  1. W1W2 and B=0, or
  2. W1W2 and if W1=W2 then B=cId1 for some c.


B determines a linear transformation B:W1W2. Since Ba=aB for all aA we have that

B(aw1)=Baw1 =aBw1=aB(w1) ,

for all aA and w1W1 Thus B is an A-module homomorphism. kerB and imB are submodules of W1 and W2 respectively and are therefore either 0 or equal to W1 or W2 respectively. If kerB=W1 or imB=0 then B=0. In the remaining case B is a bijection, and thus an isomorphism between W1 and W2 In this case we have that d1=d2. Thus the matrix B is square and invertible.

Now suppose that W1=W2 and let c be an eigenvalue of B. Then the matrix cId1-B is such that W1(a) (cId1-B) =(cId1-B) W1(a) for all aA. The argument in the preceding paragraph shows that cId1-B is either invertible or 0. But if c is an eigenvalue of B then det(cId1-B)=0. Thus cId1-B=0.

3. Suppose that V is a completely decomposable representation of an algebra A and that VλWλmλ where the W λ are nonisomorphic irreducible representations of A. Schur's lemma shows that the A -homomorphisms from W λ to V form a vector space Hom A (Wλ,V) m λ . The multiplicity of the irreducible representation W λ un V is m λ =dim Hom A (Wλ,V) .

4. Suppose that V is a completely decomposable representation of an algebra A and that V λ W λ m λ where the W λ are nonisomorphic irreducible representations of A and let dim W λ = d λ . Then V (A) i W λ m λ (A) λ I m λ (Wλ(A)) λ W λ (A) . If we view elements of λ I m λ W λ (A) as block diagonal matrices with m λ blocks of size d λ × d λ for each λ , then by using Ex 1 and Schur's lemma we get that V(A) λ Imλ (Wλ(A)) = λ M m λ (Wλ(A)) = λ M m λ ( Idλ ()) .

5. Let V be an A -module and let p be an idempotent of A. Then pV is a subspace of V and the action of p on V is a projection from V to pV. If p 1 , p 2 A are orthogonal idempotents of A then p 1 V and p 2 V are mutually orthogonal subspaces of V, since if p 1 v= p 2 v' for some v,v'V then p 1 v= p 1 p 1 v= p 1 p 2 v'=0. So V= p 1 V p 2 V.

6. Let p be an idempotent in A and suppose that for every aA,pap=kp for some constant k. If p is not minimal then p= p 1 + p 2 , where p 1 , p 2 A are idempotents such that p 1 p 2 = p 2 p 1 =0. Then p 1 =p p 1 p=kp for some constant k. This implies that p 1 = p 1 p 1 =k p 1 p 1 =k p 1 , giving that either k=1 or p 1 =0. So p is minimal.

7. Let A be a finite dimensional algebra and suppose that zA is an idempotent of A. If z is not minimal then z= p 1 + p 2 where p 1 and p 2 are orthogonal idempotents of A. If any idempotent in this sum is not minimal we can decompose it into a sum of orthogonal idempotents. We continue this process until we have decomposed z as a sum of minimal orthogonal idempotents. At any particular stage in this process z is expresed as a sum of orthogonal idempotents, z= i p i . So zA= i p i A. None of the spaces p i A is 0 since p i = p i .1 p i A and the spacers p i A are all mutually orthogonal. Thus, since zA is finite dimensional it will only take a finite number of steps to decompose z into minimal idempotents. A partition of unity is a decomposition of 1 into minimal orthogonal idempotents.

Notes and References

This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.

page history