Download Introduction to representation theory and more Lecture notes Mathematics in PDF only on Docsity!
Introduction to representation theory
Pavel Etingof, Oleg Golberg, Sebastian Hensel,
6.6 Adjoint functors...................................... 102 6.7 Abelian categories..................................... 103 6.8 Exact functors....................................... 104
7 Structure of finite dimensional algebras 106
7.1 Projective modules..................................... 106 7.2 Lifting of idempotents................................... 106 7.3 Projective covers...................................... 107 INTRODUCTION Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics and quantum field theory.
Representation theory was born in 1896 in the work of the German mathematician F. G. Frobenius. This work was triggered by a letter to Frobenius by R. Dedekind. In this letter Dedekind made the following observation: take the multiplication table of a finite group G and turn it into a matrix XG by replacing every entry g of this table by a variable xg. Then the determinant of XG factors into a product of irreducible polynomials in {xg }, each of which occurs with multiplicity equal to its degree. Dedekind checked this surprising fact in a few special cases, but could not prove it in general. So he gave this problem to Frobenius. In order to find a solution of this problem (which we will explain below), Frobenius created representation theory of finite groups. 1
The present lecture notes arose from a representation theory course given by the first author to the remaining six authors in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students, and its extended version given by the first author to MIT undergraduate math students in the Fall of 2008. The lectures are supplemented by many problems and exercises, which contain a lot of additional material; the more difficult exercises are provided with hints.
The notes cover a number of standard topics in representation theory of groups, Lie algebras, and quivers. We mostly follow [FH], with the exception of the sections discussing quivers, which follow [BGP]. We also recommend the comprehensive textbook [CR]. The notes should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.
Acknowledgements. The authors are grateful to the Clay Mathematics Institute for hosting the first version of this course. The first author is very indebted to Victor Ostrik for helping him prepare this course, and thanks Josh Nichols-Barrer and Thomas Lam for helping run the course in 2004 and for useful comments. He is also very grateful to Darij Grinberg for very careful reading of the text, for many useful comments and corrections, and for suggesting the Exercises in Sections 1.10, 2.3, 3.5, 4.9, 4.26, and 6.8.
(^1) For more on the history of representation theory, see [Cu].
1 Basic notions of representation theory
1.1 What is representation theory?
In technical terms, representation theory studies representations of associative algebras. Its general content can be very briefly summarized as follows.
An associative algebra over a field k is a vector space A over k equipped with an associative bilinear multiplication a, b 7 → ab, a, b ∈ A. We will always consider associative algebras with unit, i.e., with an element 1 such that 1 · a = a · 1 = a for all a ∈ A. A basic example of an associative algebra is the algebra EndV of linear operators from a vector space V to itself. Other important examples include algebras defined by generators and relations, such as group algebras and universal enveloping algebras of Lie algebras.
A representation of an associative algebra A (also called a left A-module) is a vector space V equipped with a homomorphism ρ : A → EndV , i.e., a linear map preserving the multiplication and unit.
A subrepresentation of a representation V is a subspace U ⊂ V which is invariant under all operators ρ(a), a ∈ A. Also, if V 1 , V 2 are two representations of A then the direct sum V 1 ⊕ V 2 has an obvious structure of a representation of A.
A nonzero representation V of A is said to be irreducible if its only subrepresentations are 0 and V itself, and indecomposable if it cannot be written as a direct sum of two nonzero subrepresentations. Obviously, irreducible implies indecomposable, but not vice versa.
Typical problems of representation theory are as follows:
- Classify irreducible representations of a given algebra A.
- Classify indecomposable representations of A.
- Do 1 and 2 restricting to finite dimensional representations. As mentioned above, the algebra A is often given to us by generators and relations. For example, the universal enveloping algebra U of the Lie algebra sl(2) is generated by h, e, f with defining relations he − eh = 2e, hf − f h = − 2 f, ef − f e = h. (1)
This means that the problem of finding, say, N -dimensional representations of A reduces to solving a bunch of nonlinear algebraic equations with respect to a bunch of unknown N by N matrices, for example system (1) with respect to unknown matrices h, e, f.
It is really striking that such, at first glance hopelessly complicated, systems of equations can in fact be solved completely by methods of representation theory! For example, we will prove the following theorem.
Theorem 1.1. Let k = C be the field of complex numbers. Then:
(i) The algebra U has exactly one irreducible representation Vd of each dimension, up to equiv- alence; this representation is realized in the space of homogeneous polynomials of two variables x, y of degree d − 1 , and defined by the formulas
ρ(h) = x
∂x − y
∂y , ρ(e) = x
∂y , ρ(f ) = y
∂x
(ii) Any indecomposable finite dimensional representation of U is irreducible. That is, any finite
1.2 Algebras
Let us now begin a systematic discussion of representation theory.
Let k be a field. Unless stated otherwise, we will always assume that k is algebraically closed, i.e., any nonconstant polynomial with coefficients in k has a root in k. The main example is the field of complex numbers C, but we will also consider fields of characteristic p, such as the algebraic closure Fp of the finite field Fp of p elements.
Definition 1.3. An associative algebra over k is a vector space A over k together with a bilinear map A × A → A, (a, b) 7 → ab, such that (ab)c = a(bc).
Definition 1.4. A unit in an associative algebra A is an element 1 ∈ A such that 1a = a1 = a.
Proposition 1.5. If a unit exists, it is unique.
Proof. Let 1, 1 ′^ be two units. Then 1 = 11′^ = 1′.
From now on, by an algebra A we will mean an associative algebra with a unit. We will also assume that A 6 = 0.
Example 1.6. Here are some examples of algebras over k:
- A = k.
- A = k[x 1 , ..., xn] – the algebra of polynomials in variables x 1 , ..., xn.
- A = EndV – the algebra of endomorphisms of a vector space V over k (i.e., linear maps, or operators, from V to itself). The multiplication is given by composition of operators.
- The free algebra A = k〈x 1 , ..., xn〉. A basis of this algebra consists of words in letters x 1 , ..., xn, and multiplication in this basis is simply concatenation of words.
- The group algebra A = k[G] of a group G. Its basis is {ag, g ∈ G}, with multiplication law agah = agh.
Definition 1.7. An algebra A is commutative if ab = ba for all a, b ∈ A.
For instance, in the above examples, A is commutative in cases 1 and 2, but not commutative in cases 3 (if dim V > 1), and 4 (if n > 1). In case 5, A is commutative if and only if G is commutative.
Definition 1.8. A homomorphism of algebras f : A → B is a linear map such that f (xy) = f (x)f (y) for all x, y ∈ A, and f (1) = 1.
1.3 Representations
Definition 1.9. A representation of an algebra A (also called a left A-module) is a vector space V together with a homomorphism of algebras ρ : A → EndV.
Similarly, a right A-module is a space V equipped with an antihomomorphism ρ : A → EndV ; i.e., ρ satisfies ρ(ab) = ρ(b)ρ(a) and ρ(1) = 1.
The usual abbreviated notation for ρ(a)v is av for a left module and va for the right module. Then the property that ρ is an (anti)homomorphism can be written as a kind of associativity law: (ab)v = a(bv) for left modules, and (va)b = v(ab) for right modules.
Here are some examples of representations.
Example 1.10. 1. V = 0.
- V = A, and ρ : A → EndA is defined as follows: ρ(a) is the operator of left multiplication by a, so that ρ(a)b = ab (the usual product). This representation is called the regular representation of A. Similarly, one can equip A with a structure of a right A-module by setting ρ(a)b := ba.
- A = k. Then a representation of A is simply a vector space over k.
- A = k〈x 1 , ..., xn〉. Then a representation of A is just a vector space V over k with a collection of arbitrary linear operators ρ(x 1 ), ..., ρ(xn) : V → V (explain why!).
Definition 1.11. A subrepresentation of a representation V of an algebra A is a subspace W ⊂ V which is invariant under all the operators ρ(a) : V → V , a ∈ A.
For instance, 0 and V are always subrepresentations.
Definition 1.12. A representation V 6 = 0 of A is irreducible (or simple) if the only subrepresenta- tions of V are 0 and V.
Definition 1.13. Let V 1 , V 2 be two representations of an algebra A. A homomorphism (or in- tertwining operator) φ : V 1 → V 2 is a linear operator which commutes with the action of A, i.e., φ(av) = aφ(v) for any v ∈ V 1. A homomorphism φ is said to be an isomorphism of representations if it is an isomorphism of vector spaces. The set (space) of all homomorphisms of representations V 1 → V 2 is denoted by HomA(V 1 , V 2 ).
Note that if a linear operator φ : V 1 → V 2 is an isomorphism of representations then so is the linear operator φ−^1 : V 2 → V 1 (check it!).
Two representations between which there exists an isomorphism are said to be isomorphic. For practical purposes, two isomorphic representations may be regarded as “the same”, although there could be subtleties related to the fact that an isomorphism between two representations, when it exists, is not unique.
Definition 1.14. Let V 1 , V 2 be representations of an algebra A. Then the space V 1 ⊕ V 2 has an obvious structure of a representation of A, given by a(v 1 ⊕ v 2 ) = av 1 ⊕ av 2.
Definition 1.15. A nonzero representation V of an algebra A is said to be indecomposable if it is not isomorphic to a direct sum of two nonzero representations.
It is obvious that an irreducible representation is indecomposable. On the other hand, we will see below that the converse statement is false in general.
One of the main problems of representation theory is to classify irreducible and indecomposable representations of a given algebra up to isomorphism. This problem is usually hard and often can be solved only partially (say, for finite dimensional representations). Below we will see a number of examples in which this problem is partially or fully solved for specific algebras.
We will now prove our first result – Schur’s lemma. Although it is very easy to prove, it is fundamental in the whole subject of representation theory.
Proposition 1.16. (Schur’s lemma) Let V 1 , V 2 be representations of an algebra A over any field F (which need not be algebraically closed). Let φ : V 1 → V 2 be a nonzero homomorphism of representations. Then:
(i) If V 1 is irreducible, φ is injective;
This example shows that an indecomposable representation of an algebra need not be irreducible.
- The group algebra A = k[G], where G is a group. A representation of A is the same thing as a representation of G, i.e., a vector space V together with a group homomorphism ρ : G → Aut(V ), whre Aut(V ) = GL(V ) denotes the group of invertible linear maps from the space V to itself.
Problem 1.20. Let V be a nonzero finite dimensional representation of an algebra A. Show that it has an irreducible subrepresentation. Then show by example that this does not always hold for infinite dimensional representations.
Problem 1.21. Let A be an algebra over a field k. The center Z(A) of A is the set of all elements z ∈ A which commute with all elements of A. For example, if A is commutative then Z(A) = A.
(a) Show that if V is an irreducible finite dimensional representation of A then any element z ∈ Z(A) acts in V by multiplication by some scalar χV (z). Show that χV : Z(A) → k is a homomorphism. It is called the central character of V.
(b) Show that if V is an indecomposable finite dimensional representation of A then for any z ∈ Z(A), the operator ρ(z) by which z acts in V has only one eigenvalue χV (z), equal to the scalar by which z acts on some irreducible subrepresentation of V. Thus χV : Z(A) → k is a homomorphism, which is again called the central character of V.
(c) Does ρ(z) in (b) have to be a scalar operator?
Problem 1.22. Let A be an associative algebra, and V a representation of A. By EndA(V ) one denotes the algebra of all homomorphisms of representations V → V. Show that EndA(A) = Aop, the algebra A with opposite multiplication.
Problem 1.23. Prove the following “Infinite dimensional Schur’s lemma” (due to Dixmier): Let A be an algebra over C and V be an irreducible representation of A with at most countable basis. Then any homomorphism of representations φ : V → V is a scalar operator.
Hint. By the usual Schur’s lemma, the algebra D := EndA(V ) is an algebra with division. Show that D is at most countably dimensional. Suppose φ is not a scalar, and consider the subfield C(φ) ⊂ D. Show that C(φ) is a transcendental extension of C. Derive from this that C(φ) is uncountably dimensional and obtain a contradiction.
1.4 Ideals
A left ideal of an algebra A is a subspace I ⊆ A such that aI ⊆ I for all a ∈ A. Similarly, a right ideal of an algebra A is a subspace I ⊆ A such that Ia ⊆ I for all a ∈ A. A two-sided ideal is a subspace that is both a left and a right ideal.
Left ideals are the same as subrepresentations of the regular representation A. Right ideals are the same as subrepresentations of the regular representation of the opposite algebra Aop.
Below are some examples of ideals:
- If A is any algebra, 0 and A are two-sided ideals. An algebra A is called simple if 0 and A are its only two-sided ideals.
- If φ : A → B is a homomorphism of algebras, then ker φ is a two-sided ideal of A.
- If S is any subset of an algebra A, then the two-sided ideal generated by S is denoted 〈S〉 and is the span of elements of the form asb, where a, b ∈ A and s ∈ S. Similarly we can define 〈S〉ℓ = span{as} and 〈S〉r = span{sb}, the left, respectively right, ideal generated by S.
1.5 Quotients
Let A be an algebra and I a two-sided ideal in A. Then A/I is the set of (additive) cosets of I. Let π : A → A/I be the quotient map. We can define multiplication in A/I by π(a) · π(b) := π(ab). This is well defined because if π(a) = π(a′) then
π(a′b) = π(ab + (a′^ − a)b) = π(ab) + π((a′^ − a)b) = π(ab)
because (a′^ − a)b ∈ Ib ⊆ I = ker π, as I is a right ideal; similarly, if π(b) = π(b′) then
π(ab′) = π(ab + a(b′^ − b)) = π(ab) + π(a(b′^ − b)) = π(ab)
because a(b′^ − b) ∈ aI ⊆ I = ker π, as I is also a left ideal. Thus, A/I is an algebra.
Similarly, if V is a representation of A, and W ⊂ V is a subrepresentation, then V /W is also a representation. Indeed, let π : V → V /W be the quotient map, and set ρV /W (a)π(x) := π(ρV (a)x).
Above we noted that left ideals of A are subrepresentations of the regular representation of A, and vice versa. Thus, if I is a left ideal in A, then A/I is a representation of A.
Problem 1.24. Let A = k[x 1 , ..., xn] and I 6 = A be any ideal in A containing all homogeneous polynomials of degree ≥ N. Show that A/I is an indecomposable representation of A.
Problem 1.25. Let V 6 = 0 be a representation of A. We say that a vector v ∈ V is cyclic if it generates V , i.e., Av = V. A representation admitting a cyclic vector is said to be cyclic. Show that
(a) V is irreducible if and only if all nonzero vectors of V are cyclic. (b) V is cyclic if and only if it is isomorphic to A/I, where I is a left ideal in A. (c) Give an example of an indecomposable representation which is not cyclic. Hint. Let A = C[x, y]/I 2 , where I 2 is the ideal spanned by homogeneous polynomials of degree ≥ 2 (so A has a basis 1 , x, y). Let V = A∗^ be the space of linear functionals on A, with the action of A given by (ρ(a)f )(b) = f (ba). Show that V provides such an example.
1.6 Algebras defined by generators and relations
If f 1 ,... , fm are elements of the free algebra k〈x 1 ,... , xn〉, we say that the algebra A := k〈x 1 ,... , xn〉/〈{f 1 ,... , fm}〉 is generated by x 1 ,... , xn with defining relations f 1 = 0,... , fm =
1.7 Examples of algebras
- The Weyl algebra, k〈x, y〉/〈yx − xy − 1 〉.
- The q-Weyl algebra, generated by x, x−^1 , y, y−^1 with defining relations yx = qxy and xx−^1 = x−^1 x = yy−^1 = y−^1 y = 1.
Proposition. (i) A basis for the Weyl algebra A is {xiyj^ , i, j ≥ 0 }. (ii) A basis for the q-Weyl algebra Aq is {xiyj^ , i, j ∈ Z}.
Hint. Show that xp^ and yp^ are central elements. (c) Find all irreducible finite dimensional representations of A. Hint. Let V be an irreducible finite dimensional representation of A, and v be an eigenvector of y in V. Show that {v, xv, x^2 v, ..., xp−^1 v} is a basis of V.
Problem 1.27. Let q be a nonzero complex number, and A be the q-Weyl algebra over C generated by x±^1 and y±^1 with defining relations xx−^1 = x−^1 x = 1, yy−^1 = y−^1 y = 1, and xy = qyx.
(a) What is the center of A for different q? If q is not a root of unity, what are the two-sided ideals in A?
(b) For which q does this algebra have finite dimensional representations? Hint. Use determinants. (c) Find all finite dimensional irreducible representations of A for such q. Hint. This is similar to part (c) of the previous problem.
1.8 Quivers
Definition 1.28. A quiver Q is a directed graph, possibly with self-loops and/or multiple edges between two vertices.
Example 1.29.
O O
We denote the set of vertices of the quiver Q as I, and the set of edges as E. For an edge h ∈ E, let h′, h′′^ denote the source and target of h, respectively:
h′^ h
/ / (^) h•′′
Definition 1.30. A representation of a quiver Q is an assignment to each vertex i ∈ I of a vector space Vi and to each edge h ∈ E of a linear map xh : Vh′ −→ Vh′′.
It turns out that the theory of representations of quivers is a part of the theory of representations of algebras in the sense that for each quiver Q, there exists a certain algebra PQ, called the path algebra of Q, such that a representation of the quiver Q is “the same” as a representation of the algebra PQ. We shall first define the path algebra of a quiver and then justify our claim that representations of these two objects are “the same”.
Definition 1.31. The path algebra PQ of a quiver Q is the algebra whose basis is formed by oriented paths in Q, including the trivial paths pi, i ∈ I, corresponding to the vertices of Q, and multiplication is concatenation of paths: ab is the path obtained by first tracing b and then a. If two paths cannot be concatenated, the product is defined to be zero.
Remark 1.32. It is easy to see that for a finite quiver
i∈I
pi = 1, so PQ is an algebra with unit.
Problem 1.33. Show that the algebra PQ is generated by pi for i ∈ I and ah for h ∈ E with the defining relations:
- p^2 i = pi, pipj = 0 for i 6 = j
- ahph′^ = ah, ahpj = 0 for j 6 = h′
- ph′′ ah = ah, piah = 0 for i 6 = h′′
We now justify our statement that a representation of a quiver is the same thing as a represen- tation of the path algebra of a quiver.
Let V be a representation of the path algebra PQ. From this representation, we can construct a representation of Q as follows: let Vi = piV, and for any edge h, let xh = ah|ph′ V : ph′^ V −→ ph′′^ V be the operator corresponding to the one-edge path h.
Similarly, let (Vi, xh) be a representation of a quiver Q. From this representation, we can construct a representation of the path algebra PQ: let V =
i Vi, let^ pi^ :^ V^ →^ Vi^ →^ V^ be the projection onto Vi, and for any path p = h 1 ...hm let ap = xh 1 ...xhm : Vh′ m → Vh′′ 1 be the composition of the operators corresponding to the edges occurring in p (and the action of this operator on the other Vi is zero).
It is clear that the above assignments V 7 → (piV) and (Vi) 7 →
i Vi^ are inverses of each other. Thus, we have a bijection between isomorphism classes of representations of the algebra PQ and of the quiver Q.
Remark 1.34. In practice, it is generally easier to consider a representation of a quiver as in Definition 1.30.
We lastly define several previous concepts in the context of quivers representations.
Definition 1.35. A subrepresentation of a representation (Vi, xh) of a quiver Q is a representation (Wi, x′ h) where Wi ⊆ Vi for all i ∈ I and where xh(Wh′ ) ⊆ Wh′′ and x′ h = xh|Wh′ : Wh′ −→ Wh′′ for all h ∈ E.
Definition 1.36. The direct sum of two representations (Vi, xh) and (Wi, yh) is the representation (Vi ⊕ Wi, xh ⊕ yh).
As with representations of algebras, a nonzero representation (Vi) of a quiver Q is said to be irreducible if its only subrepresentations are (0) and (Vi) itself, and indecomposable if it is not isomorphic to a direct sum of two nonzero representations.
Definition 1.37. Let (Vi, xh) and (Wi, yh) be representations of the quiver Q. A homomorphism ϕ : (Vi) −→ (Wi) of quiver representations is a collection of maps ϕi : Vi −→ Wi such that yh ◦ ϕh′ = ϕh′′ ◦ xh for all h ∈ E.
Problem 1.38. Let A be a Z+-graded algebra, i.e., A = ⊕n≥ 0 A[n], and A[n] · A[m] ⊂ A[n + m]. If A[n] is finite dimensional, it is useful to consider the Hilbert series hA(t) =
dim A[n]tn^ (the generating function of dimensions of A[n]). Often this series converges to a rational function, and the answer is written in the form of such function. For example, if A = k[x] and deg(xn) = n then
hA(t) = 1 + t + t^2 + ... + tn^ + ... =
1 − t
Find the Hilbert series of: (a) A = k[x 1 , ..., xm] (where the grading is by degree of polynomials);
- The Heisenberg Lie algebra H of matrices
0 0 0 0 0 ∗
It has the basis
x =
(^) y =
(^) c =
with relations [y, x] = c and [y, c] = [x, c] = 0.
- The algebra aff(1) of matrices ( ∗ ∗ 0 0 ) Its basis consists of X = ( 1 00 0 ) and Y = ( 0 10 0 ), with [X, Y ] = Y.
- so(n), the space of skew-symmetric n × n matrices, with [a, b] = ab − ba.
Exercise. Show that Example 1 is a special case of Example 5 (for n = 3).
Definition 1.42. Let g 1 , g 2 be Lie algebras. A homomorphism ϕ : g 1 −→ g 2 of Lie algebras is a linear map such that ϕ([a, b]) = [ϕ(a), ϕ(b)].
Definition 1.43. A representation of a Lie algebra g is a vector space V with a homomorphism of Lie algebras ρ : g −→ End V.
Example 1.44. Some examples of representations of Lie algebras are:
- V = 0.
- Any vector space V with ρ = 0 (the trivial representation).
- The adjoint representation V = g with ρ(a)(b) := [a, b]. That this is a representation follows from Equation (2). Thus, the meaning of the Jacobi identity is that it is equivalent to the existence of the adjoint representation.
It turns out that a representation of a Lie algebra g is the same thing as a representation of a certain associative algebra U(g). Thus, as with quivers, we can view the theory of representations of Lie algebras as a part of the theory of representations of associative algebras.
Definition 1.45. Let g be a Lie algebra with basis xi and [ , ] defined by [xi, xj ] =
k c k ij xk. The universal enveloping algebra U(g) is the associative algebra generated by the xi’s with the defining relations xixj − xj xi =
k c
k ij xk.
Remark. This is not a very good definition since it depends on the choice of a basis. Later we will give an equivalent definition which will be basis-independent.
Exercise. Explain why a representation of a Lie algebra is the same thing as a representation of its universal enveloping algebra.
Example 1.46. The associative algebra U(sl(2)) is the algebra generated by e, f , h with relations
he − eh = 2e hf − f h = − 2 f ef − f e = h.
Example 1.47. The algebra U(H), where H is the Heisenberg Lie algebra, is the algebra generated by x, y, c with the relations
yx − xy = c yc − cy = 0 xc − cx = 0.
Note that the Weyl algebra is the quotient of U(H) by the relation c = 1.
1.10 Tensor products
In this subsection we recall the notion of tensor product of vector spaces, which will be extensively used below.
Definition 1.48. The tensor product V ⊗W of vector spaces V and W over a field k is the quotient of the space V ∗ W whose basis is given by formal symbols v ⊗ w, v ∈ V , w ∈ W , by the subspace spanned by the elements
(v 1 + v 2 ) ⊗ w − v 1 ⊗ w − v 2 ⊗ w, v ⊗ (w 1 + w 2 ) − v ⊗ w 1 − v ⊗ w 2 , av ⊗ w − a(v ⊗ w), v ⊗ aw − a(v ⊗ w),
where v ∈ V, w ∈ W, a ∈ k.
Exercise. Show that V ⊗ W can be equivalently defined as the quotient of the free abelian group V • W generated by v ⊗ w, v ∈ V, w ∈ W by the subgroup generated by
(v 1 + v 2 ) ⊗ w − v 1 ⊗ w − v 2 ⊗ w, v ⊗ (w 1 + w 2 ) − v ⊗ w 1 − v ⊗ w 2 , av ⊗ w − v ⊗ aw,
where v ∈ V, w ∈ W, a ∈ k.
The elements v ⊗ w ∈ V ⊗ W , for v ∈ V, w ∈ W are called pure tensors. Note that in general, there are elements of V ⊗ W which are not pure tensors.
This allows one to define the tensor product of any number of vector spaces, V 1 ⊗ ... ⊗ Vn. Note that this tensor product is associative, in the sense that (V 1 ⊗ V 2 ) ⊗ V 3 can be naturally identified with V 1 ⊗ (V 2 ⊗ V 3 ).
In particular, people often consider tensor products of the form V ⊗n^ = V ⊗ ... ⊗ V (n times) for a given vector space V , and, more generally, E := V ⊗n^ ⊗ (V ∗)⊗m. This space is called the space of tensors of type (m, n) on V. For instance, tensors of type (0, 1) are vectors, of type (1, 0) - linear functionals (covectors), of type (1, 1) - linear operators, of type (2, 0) - bilinear forms, of type (2, 1)
If V is finite dimensional with basis ei, i = 1, ..., N , and ei^ is the dual basis of V ∗, then a basis of E is the set of vectors ei 1 ⊗ ... ⊗ ein ⊗ ej^1 ⊗ ... ⊗ ejm^ ,
and a typical element of E is
∑^ N
i 1 ,...,in,j 1 ,...,jm=
T (^) ji 11 ...j...inm ei 1 ⊗ ... ⊗ ein ⊗ ej^1 ⊗ ... ⊗ ejm^ ,
where T is a multidimensional table of numbers.
Physicists define a tensor as a collection of such multidimensional tables TB attached to every basis B in V , which change according to a certain rule when the basis B is changed. Here it is important to distinguish upper and lower indices, since lower indices of T correspond to V and upper ones to V ∗. The physicists don’t write the sum sign, but remember that one should sum over indices that repeat twice - once as an upper index and once as lower. This convention is called the Einstein summation, and it also stipulates that if an index appears once, then there is no summation over it, while no index is supposed to appear more than once as an upper index or more than once as a lower index.
One can also define the tensor product of linear maps. Namely, if A : V → V ′^ and B : W → W ′ are linear maps, then one can define the linear map A ⊗ B : V ⊗ W → V ′^ ⊗ W ′^ given by the formula (A ⊗ B)(v ⊗ w) = Av ⊗ Bw (check that this is well defined!)
Similarly, if C is another k-algebra, and if the left B-module structure on W is part of a (B, C)- bimodule structure, then V ⊗B W becomes a right C-module by (v ⊗B w) c = v ⊗B wc for any c ∈ C, v ∈ V and w ∈ W.
If V is an (A, B)-bimodule and W is a (B, C)-bimodule, then these two structures on V ⊗B W can be combined into one (A, C)-bimodule structure on V ⊗B W.
(a) Let A, B, C, D be four algebras. Let V be an (A, B)-bimodule, W be a (B, C)-bimodule, and X a (C, D)-bimodule. Prove that (V ⊗B W ) ⊗C X ∼= V ⊗B (W ⊗C X) as (A, D)-bimodules. The isomorphism (from left to right) is given by (v ⊗B w) ⊗C x 7 → v ⊗B (w ⊗C x) for all v ∈ V , w ∈ W and x ∈ X.
(b) If A, B, C are three algebras, and if V is an (A, B)-bimodule and W an (A, C)-bimodule, then the vector space HomA (V, W ) (the space of all left A-linear homomorphisms from V to W ) canonically becomes a (B, C)-bimodule by setting (bf ) (v) = f (vb) for all b ∈ B, f ∈ HomA (V, W ) and v ∈ V and (f c) (v) = f (v) c for all c ∈ C, f ∈ HomA (V, W ) and v ∈ V.
Let A, B, C, D be four algebras. Let V be a (B, A)-bimodule, W be a (C, B)-bimodule, and X a (C, D)-bimodule. Prove that HomB (V, HomC (W, X)) ∼= HomC (W ⊗B V, X) as (A, D)-bimodules. The isomorphism (from left to right) is given by f 7 → (w ⊗B v 7 → f (v) w) for all v ∈ V , w ∈ W and f ∈ HomB (V, HomC (W, X)).
1.11 The tensor algebra
The notion of tensor product allows us to give more conceptual (i.e., coordinate free) definitions of the free algebra, polynomial algebra, exterior algebra, and universal enveloping algebra of a Lie algebra.
Namely, given a vector space V , define its tensor algebra T V over a field k to be T V = ⊕n≥ 0 V ⊗n, with multiplication defined by a · b := a ⊗ b, a ∈ V ⊗n, b ∈ V ⊗m. Observe that a choice of a basis x 1 , ..., xN in V defines an isomorphism of T V with the free algebra k < x 1 , ..., xn >.
Also, one can make the following definition.
Definition 1.50. (i) The symmetric algebra SV of V is the quotient of T V by the ideal generated by v ⊗ w − w ⊗ v, v, w ∈ V.
(ii) The exterior algebra ∧V of V is the quotient of T V by the ideal generated by v ⊗ v, v ∈ V. (iii) If V is a Lie algebra, the universal enveloping algebra U(V ) of V is the quotient of T V by the ideal generated by v ⊗ w − w ⊗ v − [v, w], v, w ∈ V.
It is easy to see that a choice of a basis x 1 , ..., xN in V identifies SV with the polynomial algebra k[x 1 , ..., xN ], ∧V with the exterior algebra ∧k(x 1 , ..., xN ), and the universal enveloping algebra U(V ) with one defined previously.
Also, it is easy to see that we have decompositions SV = ⊕n≥ 0 SnV , ∧V = ⊕n≥ 0 ∧n^ V.
1.12 Hilbert’s third problem
Problem 1.51. It is known that if A and B are two polygons of the same area then A can be cut by finitely many straight cuts into pieces from which one can make B. David Hilbert asked in 1900 whether it is true for polyhedra in 3 dimensions. In particular, is it true for a cube and a regular tetrahedron of the same volume?
The answer is “no”, as was found by Dehn in 1901. The proof is very beautiful. Namely, to any polyhedron A let us attach its “Dehn invariant” D(A) in V = R ⊗ (R/Q) (the tensor product of Q-vector spaces). Namely,
D(A) =
a
l(a) ⊗
β(a) π
where a runs over edges of A, and l(a), β(a) are the length of a and the angle at a.
(a) Show that if you cut A into B and C by a straight cut, then D(A) = D(B) + D(C). (b) Show that α = arccos(1/3)/π is not a rational number. Hint. Assume that α = 2m/n, for integers m, n. Deduce that roots of the equation x+x−^1 = 2/ 3 are roots of unity of degree n. Conclude that xk^ + x−k^ has denominator 3 k^ and get a contradiction.
(c) Using (a) and (b), show that the answer to Hilbert’s question is negative. (Compute the Dehn invariant of the regular tetrahedron and the cube).
1.13 Tensor products and duals of representations of Lie algebras
Definition 1.52. The tensor product of two representations V, W of a Lie algebra g is the space V ⊗ W with ρV ⊗W (x) = ρV (x) ⊗ Id + Id ⊗ ρW (x).
Definition 1.53. The dual representation V ∗^ to a representation V of a Lie algebra g is the dual space V ∗^ to V with ρV ∗^ (x) = −ρV (x)∗.
It is easy to check that these are indeed representations.
Problem 1.54. Let V, W, U be finite dimensional representations of a Lie algebra g. Show that the space Homg(V ⊗ W, U ) is isomorphic to Homg(V, U ⊗ W ∗). (Here Homg := HomU (g)).
1.14 Representations of sl(2)
This subsection is devoted to the representation theory of sl(2), which is of central importance in many areas of mathematics. It is useful to study this topic by solving the following sequence of exercises, which every mathematician should do, in one form or another.
Problem 1.55. According to the above, a representation of sl(2) is just a vector space V with a triple of operators E, F, H such that HE − EH = 2E, HF − F H = − 2 F, EF − F E = H (the corresponding map ρ is given by ρ(e) = E, ρ(f ) = F , ρ(h) = H).
Let V be a finite dimensional representation of sl(2) (the ground field in this problem is C). (a) Take eigenvalues of H and pick one with the biggest real part. Call it λ. Let V¯ (λ) be the generalized eigenspace corresponding to λ. Show that E| (^) V¯ (λ) = 0.
(b) Let W be any representation of sl(2) and w ∈ W be a nonzero vector such that Ew = 0. For any k > 0 find a polynomial Pk(x) of degree k such that EkF kw = Pk(H)w. (First compute EF kw, then use induction in k).
(c) Let v ∈ V¯ (λ) be a generalized eigenvector of H with eigenvalue λ. Show that there exists N > 0 such that F N^ v = 0.
(d) Show that H is diagonalizable on V¯ (λ). (Take N to be such that F N^ = 0 on V¯ (λ), and compute EN^ F N^ v, v ∈ V¯ (λ), by (b). Use the fact that Pk(x) does not have multiple roots).
