Marathe K. Topics in Physical Mathematics

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8 1Algebra

particular, if V itself is irreducible, then we have V

= V . The module V

called the highest weight module and the corresponding representation is

called the highest weight representation. A highest weight vector always

exists for a ﬁnite-dimensional representation of a semi-simple Lie algebra. It

plays a fundamental role in the theory of such representations. In the following

examplewedescribealltheirreduciblerepresentations of the simple complex

Lie algebra sl(2, C) as highest weight representations.

Example 1.4 The Lie algebra sl(2, C) consists of 2-by-2 complex matrices

with trace zero. A standard basis for it is given by the elements h, e, f deﬁned

h :=



0 −1



,e:=





,f:=





The commutators of the basis elements are given by

[h, e]=2e, [h, f]=−2f, [e, f]=h.

We note that the basis and commutators are valid for sl(2,K) for any ﬁeld

K. The Cartan subalgebra is one dimensional and is generated by h.The

following theorem gives complete information about the ﬁnite dimensional

irreducible representations of the Lie algebra sl(2, C).

Theorem 1.2 For each n ∈ N there exists a unique (up to isomorphism)

irreducible representation ρ

of sl(2, C) on a complex vector space V

of di-

mension n. There exists a basis {v

,...,v

n−1

} of V

consisting of eigenvec-

tors of ρ

(h) with the vacuum vector (highest weight vector) v

satisfying the

following properties:

1. h.v

=(n − 1 −2i)v

2. e.v

=(n −i)v

i−1

3. f.v

= v

i+1

wherewehaveputρ

(x)v = x.v, x ∈ sl(2, C),v ∈ V , v

−1

=0=v

and where

0 ≤ i ≤ n − 1.

1.2.1 Graded Algebras

Graded algebraic structures appear naturally in many mathematical and

physical theories. We shall restrict our considerations only to Z-andZ

gradings. The most basic such structure is that of a graded vector space

which we now describe. Let V be a vector space. We say that V is Z-graded

(resp., Z

-graded)ifV is the direct sum of vector subspaces V

, indexed by

the integers (resp., integers mod 2), i.e.

V =



i∈Z

(resp., V = V

⊕ V

1.2 Algebras 9

The elements of V

are said to be homogeneous of degree i.Inthecaseof

-grading it is customary to call the elements of V

(resp. V

) even (resp.

odd). If V and W are two Z-graded vector spaces, a linear transformation

f : V → W is said to be graded of degree k if f (V

) ⊂ W

i+k

, ∀i ∈ Z. If

V and W are Z

-graded, then a linear map f : V → W is said to be even if

f(V

) ⊂ W

,i∈ Z

,andodd if f (V

) ⊂ W

i+1

,i∈ Z

.AnalgebraA is said

to be Z-graded if A is Z-graded as a vector space; i.e.

A =



i∈Z

and A

⊂ A

i+j

, ∀i, j ∈ Z. An ideal I ⊂ A is called a homogeneous ideal

I =



i∈Z

(I ∩ A

Other algebraic structures (such as Lie, commutative, etc.) have their

graded counterparts. It was Hermann Grassmann (1809–1877) who

ﬁrst deﬁned the structure of an exterior algebra associated to a ﬁnite-

dimensional vector space. Grassmann’s work was well ahead of his time and

did not receive recognition for a long time. In the preface to his 1862 book he

wrote: [T]here will come a time when these ideas, perhaps in a new form, will

enter into contemporary developments. Indeed, Grassmann’s expectation has

come to fruition and his work has found many applications in mathematics

and physics. An example of a Z-graded algebra is given by the exterior algebra

of diﬀerential forms Λ(M )ofamanifoldM if we deﬁne Λ

(M)=0fori<0.

It is called the Grassmann algebra of the manifold M. (We discuss mani-

folds and their associated structures in Chapter 3.) The exterior diﬀerential

d is a graded linear transformation of degree 1 of Λ(M). The transformation

d is nilpotent of order 2, i.e., it satisﬁes the condition d

= 0. A graded

algebra together with a diﬀerential d (i.e. a graded linear transformation d

of degree 1 satisfying the Leibniz rule and d

= 0) is called a diﬀerential

graded algebra or DGA. Thus the Grassmann algebra is an example of

a DGA. Similarly, we can deﬁne a diﬀerential graded Lie algebra,or

DGLA. The DGA is a special case of the A

∞

algebra, also known as strong

homotopy associative algebra. Similarly, the DGLA is a special case of an

inﬁnite-dimensional algebra called the L

∞

-algebra. These algebras appear in

open-closed string ﬁeld theory.

The graded or quantum dimension of a Z-graded vector space V is

deﬁned by

dim

V =



i∈Z

(dim(V

)) ,

where q is a formal variable.

If A and B are graded algebras, their graded tensor product, denoted

⊗B, is the tensor product A ⊗ B of A and B as vector spaces with the

product deﬁned on the homogeneous elements by

10 1Algebra

(u ⊗v)(x ⊗ y)=(−1)

ux ⊗vy, v ∈ B

,x∈ A

A similar deﬁnition can be given for Z

-graded algebras. In the physics liter-

ature a Z

-graded algebra is referred to as a superalgebra. For example,

a(complex)Lie superalgebra is a Z

-graded vector space g = g

⊕g

with

a product satisfying the superanticommutativity and super Jacobi identity.

An important example of a Lie superalgebra is the space of all linear en-

domorphisms of a Z

-graded vector space; an important example of graded

algebras is provided by Vertex operator algebras,orVOA, introduced

in [139]. This deﬁnition of vertex operator algebras was motivated by the

Conway–Norton conjectures about properties of the monster sporadic group.

There are now a number of variants of these algebras, including the vertex

algebras and the closely related Lie algebras now called the Borcherds

algebras introduced by RichardEwenBorcherdsin his proof of the

Conway-Norton conjectures. (Borcherds received a Fields Medal at the ICM

1998 in Berlin for his work in algebra, the theory of automorphic forms and

mathematical physics.) These vertex algebras are closely related to algebras

arising in conformal ﬁeld theory and string theory. Vertex operators arise in 2-

dimensional Euclidean conformal ﬁeld theory from ﬁeld insertions at marked

points on a Riemann surface. A deﬁnition of VOA is obtained by isolating the

mathematical properties related to interactions represented by operator prod-

uct expansion in chiral CFT. An algebraic description of such interactions and

their products codify these new graded algebras. These inﬁnite-dimensional

symmetry algebras were ﬁrst introduced in [37]. Borcherds deﬁnition of ver-

tex algebras is closely related to these. These algebras are also expected to

play a role in the study of the geometric Langlands conjecture. We discuss

the VOA in more detail in the last section.

1.3 Kac–Moody Algebras

Kac–Moody algebras were discovered independently in 1968 by Victor Kac

and Robert Moody. These algebras generalize the notion of ﬁnite-dimensional

semi-simple Lie algebras. Many properties related to the structure of Lie alge-

bras and their representations have counterparts in the theory of Kac–Moody

algebras. In fact, they are usually deﬁned via a generalization of the Cartan

matrix associated to a Lie algebra. We restrict ourselves to a special class

of Kac–Moody algebras that arise in many physical and mathematical situa-

tions, such as conformal ﬁeld theory, exactly solvable statistical models, and

combinatorial identities (e.g., Rogers–Ramanujan and Macdonald identities).

They are called aﬃne algebras.

Let g be a Lie algebra with a symmetric bilinear form ·, ·.Thisform

is said to be g-invariant (or simply invariant)if

1.3 Kac–Moody Algebras 11

[x, y],z = x, [y, z], ∀x, y, z ∈ g. (1.9)

In the rest of this section we assume that a symmetric, g-invariant, bilinear

form on g is given. Let t be a formal variable and let K[t, t

−1

] be the algebra

of Laurent polynomials in t. Then the vector space g

deﬁned by

:= g ⊕ K[t, t

−1

] (1.10)

can be given a Lie algebra structure by deﬁning the product

[x ⊕t

,y⊕t

]:=[x, y] ⊕ t

m+n

, for x, y ∈ g,m,n∈ Z. (1.11)

When the base ﬁeld K = C, the algebra g

can be regarded as the Lie algebra

of polynomial maps of the unit circle into g. For this reason the algebra g

called the loop algebra of g.Theaﬃne Kac–Moody algebra,orsimply

the aﬃne algebra, is a central extension of g

.Letc be another formal

variable. It is called the central charge in physics literature. Using the one-

dimensional space Kc we obtain a central extension g

t,c

of g

deﬁned by

t,c

:= g ⊕ K[t, t

−1

] ⊕Kc , (1.12)

where c is taken as a central element. The product on g

t,c

is deﬁned by

extending the product on g

deﬁned in (1.11) as follows:

[x ⊕ t

,y⊕t

]:=[x, y] ⊕ t

m+n

+ x, ymδ

m,−n

c, for x, y ∈ g . (1.13)

The choice of c as a central element means that c commutes with every

element of the algebra g

t,c

. The algebra g

t,c

, with product deﬁned by extend-

ing (1.13) by linearity, is called the aﬃne Lie algebra, or simply the aﬃne

algebra associated to the Lie algebra g with central charge c and the given

invariant bilinear form. Similarly, we can deﬁne the aﬃne algebra g

√

t,c

letting m, n take half-integer values in the deﬁnition (1.13).

Example 1.5 (Virasoro algebra) Applying the construction of example (1.3)

to the associative algebra K[t, t

−1

], we obtain the Lie algebra of its derivations

d(K[t, t

−1

]). It can be shown that this Lie algebra has a basis d

, n ∈ Z,

deﬁned by

(f(t)) := −t

n+1



(t), ∀f ∈ K[t, t

−1

], (1.14)

where f



(t) is the derivative of the Laurent polynomial f(t). The product of

the basis elements is given by the formula

]=(m − n)d

m+n

, ∀m, n ∈ Z. (1.15)

The algebra K[t, t

−1

] has a unique nontrivial one-dimensional central ex-

tension (up to isomorphism). It is called the Virasoro algebra and is de-

noted by v. It is generated by the central element (also called the central

charge C) and the elements L

corresponding to the elements d

deﬁned

12 1Algebra

above. The product in v is given by

]=(m − n)L

m+n

− m)δ

m,−n

C, ∀m, n ∈ Z. (1.16)

The constant 1/12 is chosen for physical reasons. It can be replaced by an

arbitrary complex constant giving an isomorphic algebra. We note that the

central term is zero when m = −1, 0, 1. Thus the central extension is trivial

when restricted to the subalgebra of v generated by the elements L

−1

,and

. This subalgebra is denoted by p and it can be shown to be isomorphic to

the Lie algebra sl(2,K) of matrices of order 2 and trace zero. The Virasoro

algebra is an example of an inﬁnite-dimensional Lie algebra. It was deﬁned

by physicists in their study of conformal ﬁeld theory for the case K = C.

Gelfand and Fuchs have shown that the Virasoro algebra can be realized as a

central extension of the algebra of polynomial vector ﬁelds on the unit circle

by identifying the derivation d

used in our deﬁnition with a vector ﬁeld

on S

An inﬁnite-dimensional algebra does not, in general, have a highest weight

representation. However, the Virasoro algebra does admit such modules. We

now describe its construction. Let U denote the universal enveloping algebra

of the Virasoro algebra v deﬁned by the relations (1.16)andlet(h, c) ∈ C

be a given pair of complex numbers. Let I be the left ideal in U generated

by L

− hι, C − cι, L

,i ∈ N,whereι denotes the identity. Let V (h, c)be

the quotient of U by the ideal I and let v ∈ V (h, c) be the class of the

identity (i.e., v = ι + I). Then it can be shown that V (h, c)isahighest

weight module for v with highest weight vector v and highest weight (h, c)

satisfying the following conditions:

1. L

v = hv, Cv = cv, L

v =0,i∈ N;

2. the set of vectors L

...L

v,wherei

,...,i

is a decreasing sequence of

negative integers, generates V (h, c).

Any v-module V satisfying the above two conditions with a nonzero vector v

deﬁnes a highest weight module with highest weight (h, c). A highest weight

module is called a Verma module if the vectors in condition 2 form a basis

of V . It can be shown that this is the case for the module V (h, c) constructed

above. Thus V (h, c) is a Verma module for the Virasoro algebra v.Moreover,

every highest weight module is a quotient of V (h, c) by a submodule with the

same highest weight (h, c). Verma modules have many other interesting prop-

erties. For example, their homomorphisms are closely related to the invariant

diﬀerential operators on homogeneous manifolds obtained as quotients of Lie

groups by their subgroups. Verma modules were deﬁned by D. N. Verma

his study of representations of semi-simple Lie algebras over forty years ago.

Example 1.6 (Heisenberg algebras) In classical mechanics the state of

aparticleattimet is given by its position and momentum vectors (q, p)

My long-time friend whose enthusiasm and interest in mathematics is still strong.

1.3 Kac–Moody Algebras 13

in Euclidean space. The evolution of the system is governed by Hamilton’s

equations (see Chapter 3). Heisenberg’s fundamental idea for quantization of

such a system was to take the components of q and p to be operators on a

Hilbert space satisfying the canonical commutation relations

]=0, [p

]=−iδ

j,m

, 1 ≤ j, m ≤ n,

where n is the dimension of the Euclidean space. Heisenberg’s uncertainty

principle is closely related to the noncommuting of the position and its con-

jugate momentum. If −i.1 is replaced by a basis vector c taken as a central

element, then we obtain a (2n +1)-dimensional real Lie algebra with basis

,c. This algebra is denoted by h

and is called a Heisenberg algebra

with central element c. This Heisenberg algebra is isomorphic to an algebra

of upper triangular matrices. If we deﬁne an index set I to be the set of the

ﬁrst n natural numbers, then the algebra h

can be called the (I,c)-Heisenberg

algebra. This deﬁnition can be generalized to arbitrary index set I.IfI is in-

ﬁnite we get an inﬁnite-dimensional Heisenberg algebra. Another example of

an inﬁnite-dimensional Heisenberg algebra is given by a Lie algebra with a

basis a

, n ∈ Z, together with the central element b. The product of the basis

elements is given by

]=mδ

m,−n

b, [a

,b]=0, ∀m, n ∈ Z. (1.17)

This Heisenberg algebra is also referred to as the oscillator algebra.It

arises as an algebra of operators on the bosonic Fock space F. The space

F is deﬁned as the space of complex polynomials in inﬁnitely many variables

,i∈ N. The operators on F deﬁned by

∂

∂z

−n

:= nz

, ∀n ∈ N, and a

= id,

together with the central element b generate an algebra isomorphic to the

Heisenberg algebra. It is also possible to generate a copy of the Virasoro al-

gebra by a suitable combination of the operators deﬁned above.

The original Heisenberg commutation relations were applied successfully in

producing the spectrum of the quantum harmonic oscillator (see Appendix

B) and the hydrogen atom, even though the Hilbert space on which the

operators act was not speciﬁed. The following theorem explains why this did

not cause any problems.

Theorem 1.3 (Stone–von Neumann theorem) Fix a nonzero scalar with

which the central element c acts. Then there is a unique irreducible represen-

tation of the Heisenberg algebra h

The experimental agreement of the spectra resulted in the early acceptance

of quantum mechanics based on canonical quantization. It is important to

note that canonical quantization does not apply to all classical systems and

14 1Algebra

there is no method of quantizing a general classical mechanical system at this

time.

1.4 Cliﬀord Algebras

Important examples of graded and superalgebras are furnished by the Cliﬀord

algebras of pseudo-inner product spaces. In deﬁning these algebras, William

Kingdon Cliﬀord (1845–1879) generalized Grassmann’s work on exterior al-

gebras. Cliﬀord algebras arise in many physical applications. In particular,

their relation to the Pauli spin matrices and the Dirac operator on spinorial

spaces and to the Dirac γ matrices is well known. Let V be a real vector

space and g : V × V → R a bilinear symmetric map. A Cliﬀord map for

(V,g)isapair(A, φ), where A is an associative algebra with unit 1

(often

denoted simply by 1) and φ : V → A a linear map satisfying the condition

φ(u)φ(v)+φ(v)φ(u)=2g(u, v)1

, ∀u, v ∈ V.

Such pairs are the objects of a category whose morphisms are, for two objects

(A, φ), (B,ψ), the algebra homomorphisms h : A → B such that ψ = h ◦ φ;

i.e., the following diagram commutes:





This category has a universal initial object (C(V,g),γ). The algebra C(V,g),

or simply C(V ) is called the Cliﬀord algebra of (V,g). In other words given

any Cliﬀord map (A, φ), there is a unique algebra homomorphism Φ : C(V ) →

A such that φ = Φ ◦ γ, i.e. the following diagram commutes.

VC(V )





There are several ways of constructing a model for C(V ). For example, let

T (V )=



r≥0

(V )

and let J be the two-sided ideal generated by elements of the type

1.4 Cliﬀord Algebras 15

u ⊗v + v ⊗ u −2g(u, v)1

T (V )

, ∀u, v ∈ V.

We note that every element of J can be written as a ﬁnite sum



⊗ (v

⊗ v

− g(v

)1) ⊗ μ

where λ

,μ

∈ T (V )andv

∈ V . Then we deﬁne C(V )tobethequotient

T (V )/J and γ = π ◦ ι,whereι : V → T (V ) is the canonical injection and

π : T (V ) → T (V )/J is the canonical projection. C(V ) is generated by the

elements of γ(V ). The tensor multiplication on T (V ) induces a multiplication

on the quotient algebra C(V ) that is uniquely determined by the products

γ(u)γ(v)=u ⊗ v + J, ∀u, v ∈ V.

We often write γ(v) ∈ C(V )assimplyv.ThenanelementofC(V )can

be written as a ﬁnite sum of products of elements v ∈ V .Inparticular,

= v ⊗ v + J = g(v, v)1

C(V )

and, by polarization,

uv + vu =2g(u, v)1

C(V )

, (1.18)

which shows that γ is a Cliﬀord map. We observe that the tensor algebra

T (V )isZ-graded but the ideal J is not homogeneous. Therefore, the Z-

grading does not pass to the Cliﬀord algebra. However, there exists a natural

-grading on C(V ), which makes it into a superalgebra. This Z

-grading is

deﬁned as follows. Let C

(V )(resp.,C

(V )) be the vector subspace of C(V )

generated by the products of an even (resp., odd) number of elements of V .

Then

C(V )=C

(V ) ⊕C

(V )

and the elements of C

(V )(resp.C

(V )) are the even (resp. odd) elements

of C(V ). We observe that, when g =0,C(V ) reduces to the exterior algebra

Λ(V )ofV . An important result is given by the following theorem.

Theorem 1.4 Let (V, g), (V



) be two pseudo-inner product spaces. Then

there exists a natural Cliﬀord algebra isomorphism

C(V ⊕ V



,g⊕g



)

∼

C(V,g)

⊗C(V



where

⊗ is the graded tensor product of Z

-graded algebras.

Proof: Let γ : V → C(V ),γ



: V



→ C(V



) be the canonical inclusion map

and deﬁne

˜γ : V ⊕ V



→ C(V )

⊗C(V



)

˜γ(v, v



)=γ(v) ⊗ 1+1⊗γ



It is easy to check that ˜γ(v, v



)

=[g(v, v)+g



)]1 and hence by the

universal property of Cliﬀord algebras ˜γ induces an algebra homomorphism

16 1Algebra

ψ : C(V ⊕ V



) → C(V )

⊗C(V



It can be shown that ψ is a Cliﬀord algebra isomorphism (see M. Karoubi

[216] for further details). 

Inﬁnite-dimensional Cliﬀord algebras have been studied and applied to the

problem of describing the quantization of gauge ﬁelds and BRST cohomology

in [236]. The structure of ﬁnite-dimensional Cliﬀord algebras is well known

and has numerous applications in mathematics as well as in physics. We now

describe these algebras in some detail.

Recall that a subspace U ⊂ V is called g-isotropic or simply isotropic if

=0. Factoring out the maximal isotropic subspace of V , we can restrict

ourselves to the case when g is non-degenerate (i.e., g(x, y)=0, ∀x ∈ V

implies y = 0). In the ﬁnite-dimensional case a pseudo-inner product space

(V,g), with pseudo-metric or pseudo-inner product g of signature

(k, n−k), is isometric to (R

), where g

is the canonical pseudo-metric on

of index n −k.WedenotebyC(k, n −k) the Cliﬀord algebra C(R

Let {e

| i =1,...,n} be a g

-orthonormal basis of R

. To emphasize the

Cliﬀord algebra aspect we denote the images γ(e

)byγ

;thisisalsoinaccord

with the notation used in the physical literature. Then the Cliﬀord algebra

C(k, n −k) has a basis consisting of 1 and the products

···γ

, 1 ≤ i

< ···<i

≤ n,

where r =1,...,n and hence has dimension 2

. The element γ

···γ

usually denoted by γ

n+1

.Fromequation(1.18) it follows that the product is

subject to the relations

=1,i=1,...,k , γ

= −1,j= k +1,...,n (1.19)

and

= −γ

, ∀ i = j. (1.20)

Conversely, the product is completely determined by these relations. In phys-

ical applications one often starts with matrices satisfying the product rela-

tions (1.19) to construct various Cliﬀord algebras. In particular, starting with

the Pauli spin matrices one can construct various Cliﬀord algebras as is

shown below. Recall that the Pauli spin matrices are the complex 2 ×2ma-

trices





,σ



0 −i

i 0



,σ



0 −1



Taking γ

= σ

,γ

= iσ

,weobservethatI, γ

,γ

= −σ

form a basis

of M

(R). The resulting algebra is isomorphic to the Cliﬀord algebra C(1, 1).

If on the other hand we choose γ

= σ

,γ

= σ

,thenI, γ

,γ

−iσ

is a basis of M

(R). The resulting algebra is isomorphic to the Cliﬀord

algebra C(2, 0). Thus, we have two non-isomorphic Cliﬀord algebras with the

same underlying vector space. Taking γ

= iσ

,γ

= iσ

, we obtain the

1.4 Cliﬀord Algebras 17

Cliﬀord algebra C(0, 2) with basis I, γ

,γ

= −iσ

.Itiseasytoverify

that C(0, 2)

∼

H. Consider now the case C(m, n)wherem + n =2p. Then

(γ

2p+1

)

=(−1)

n+p

1. The following theorem is useful for the construction of

Cliﬀord algebras.

Theorem 1.5 Let m + n =2p; then we have the isomorphisms

C(m, n) ⊗ C(m



)

∼

C(m + m



,n+ n



) if (−1)

n+p

=1, (1.21)

C(m, n) ⊗ C(m



)

∼

C(m + n



,n+ m



) if (− 1)

n+p

= −1. (1.22)

Proof: Consider the set

B = {γ

⊗ 1



,γ

2p+1

⊗ γ

},i=1,...,m+ n, j =1,...,m



+ n



Observe that

(γ

⊗ 1



)

=(1⊗ 1



)g(e

(γ

2p+1

⊗ γ

)

=(−1)

n+p

(1 ⊗1



)g(e

)

and that diﬀerent elements of B anticommute. The number of elements of

B whose square is −1 ⊗ 1



is n + n



(resp., n + m



)if(−1)

n+p

=1(resp.,

(−1)

n+p

= −1). If (−1)

n+p

= 1, consider the map

φ : R

n+m

⊕ R



→ C(m, n) ⊗ C(m



)

such that

φ(v, v



)=v ⊗ 1+γ

2p+1

⊗ v



By reasoning similar to that in the proof of Theorem (1.4)itcanbeshown

that φ induces an isomorphism of C(m+m



,n+n



)withC(m, n)⊗C(m



This proves (1.21). Similar reasoning, in the case that (−1)

n+p

= −1, estab-

lishes the isomorphism indicated in (1.22). 

The theorem (1.5) and the structure of matrix algebras lead to the iso-

morphism

C(p + n, q + n)

∼

C(p, q) ⊗ M

(R)

∼

(C(p, q)), (1.23)

where M

(A) is the algebra of k ×k-matrices with coeﬃcients in the algebra

A.Inparticularwehave

C(n, n)

∼

(R), (1.24)

C(p, q)

∼

C(p − q, 0) ⊗ M

(R), if p>q (1.25)

and

C(p, q)

∼

C(0,q− p) ⊗M

(R), if p<q. (1.26)

Example 1.7 In this example we consider the important special case of the

Cliﬀord algebra C(p, q) where p =3and q =1. Our discussion is based on