Basic Notions of Representation Theory

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 �⊃ 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 δ : 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 δ(a), a � A. Also, if V1 , V2 are two representations of A then the direct sum V 1 � V2 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: 1. Classify irreducible representations of a given algebra A. 2. Classify indecomposable representations of A. 3. 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 = −2f, 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 V d 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 δ(h) = x � � −y , �x �y δ(e) = x � , �y δ(f ) = y � . �x (ii) Any indecomposable finite dimensional representation of U is irreducible. That is, any finite dimensional representation of U is a direct sum of irreducible representations. As another example consider the representation theory of quivers. A quiver is a finite oriented graph Q. A representation of Q over a field k is an assignment of a k-vector space Vi to every vertex i of Q, and of a linear operator A h : Vi ⊃ Vj to every directed edge h going from i to j (loops and multiple edges are allowed). We will show that a representation of a quiver Q is the same thing as a representation of a certain algebra P Q called the path algebra of Q. Thus one may ask: what are the indecomposable finite dimensional representations of Q? More specifically, let us say that Q is of finite type if it has finitely many indecomposable representations. We will prove the following striking theorem, proved by P. Gabriel about 35 years ago: Theorem 1.2. The finite type property of Q does not depend on the orientation of edges. The connected graphs that yield quivers of finite type are given by the following list: • An : �−−� · · · �−−� • Dn : �−−� · · · �−− | � • E6 : �−−�−−�−−�−−� �| • E7 : �−−�−−�−−�−−�−−� | � • E8 : �−−�−−�−−�−−�− | −�−− � The graphs listed in the theorem are called (simply laced) Dynkin diagrams. These graphs arise in a multitude of classification problems in mathematics, such as classification of simple Lie algebras, singularities, platonic solids, reflection groups, etc. In fact, if we needed to make contact with an alien civilization and show them how sophisticated our civilization is, perhaps showing them Dynkin diagrams would be the best choice! As a final example consider the representation theory of finite groups, which is one of the most fascinating chapters of representation theory. In this theory, one considers representations of the group algebra A = C[G] of a finite group G – the algebra with basis a g , g � G and multiplication law ag ah = agh . We will show that any finite dimensional representation of A is a direct sum of irreducible representations, i.e., the notions of an irreducible and indecomposable representation are the same for A (Maschke’s theorem). Another striking result discussed below is the Frobenius divisibility theorem: the dimension of any irreducible representation of A divides the order of G. Finally, we will show how to use representation theory of finite groups to prove Burnside’s theorem: any finite group of order pa q b , where p, q are primes, is solvable. Note that this theorem does not mention representations, which are used only in its proof; a purely group-theoretical proof of this theorem (not using representations) exists but is much more difficult! 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) �⊃ 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. � units. Then 1 = 11� = 1� . Proof. Let 1, 1� be two From now on, by an algebra A we will mean an associative algebra with a unit. We will also assume that A = 0. Example 1.6. Here are some examples of algebras over k: 1. A = k. 2. A = k[x1 , ..., xn ] – the algebra of polynomials in variables x 1 , ..., xn . 3. 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. 4. The free algebra A = k◦x1 , ..., xn �. A basis of this algebra consists of words in letters x1 , ..., xn , and multiplication in this basis is simply concatenation of words. 5. The group algebra A = k[G] of a group G. Its basis is {a g , g � G}, with multiplication law ag ah = 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 δ : A ⊃ EndV . Similarly, a right A-module is a space V equipped with an antihomomorphism δ : A ⊃ EndV ; i.e., δ satisfies δ(ab) = δ(b)δ(a) and δ(1) = 1. The usual abbreviated notation for δ(a)v is av for a left module and va for the right module. Then the property that δ 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. 2. V = A, and δ : A ⊃ EndA is defined as follows: δ(a) is the operator of left multiplication by a, so that δ(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 δ(a)b := ba. 3. A = k. Then a representation of A is simply a vector space over k. 4. A = k◦x1 , ..., xn �. Then a representation of A is just a vector space V over k with a collection of arbitrary linear operators δ(x1 ), ..., δ(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 δ(a) : V ⊃ V , a � A. � For instance, 0 and V are always subrepresentations. Definition 1.12. A representation V = 0 of A is irreducible (or simple) if the only subrepresenta­ tions of V are 0 and V . Definition 1.13. Let V1 , V2 be two representations of an algebra A. A homomorphism (or in­ tertwining operator) θ : V1 ⊃ V2 is a linear operator which commutes with the action of A, i.e., θ(av) = aθ(v) for any v � V1 . A homomorphism θ is said to be an isomorphism of representations if it is an isomorphism of vector spaces. The set (space) of all homomorphisms of representations V1 ⊃ V2 is denoted by HomA (V1 , V2 ). Note that if a linear operator θ : V1 ⊃ V2 is an isomorphism of representations then so is the linear operator θ−1 : V2 ⊃ V1 (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 V1 , V2 be representations of an algebra A. Then the space V 1 � V2 has an obvious structure of a representation of A, given by a(v 1 � v2 ) = av1 � av2 . 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 , V2 be representations of an algebra A over any field F (which need not be algebraically closed). Let θ : V 1 ⊃ V2 be a nonzero homomorphism of representations. Then: (i) If V1 is irreducible, θ is injective; (ii) If V2 is irreducible, θ is surjective. Thus, if both V1 and V2 are irreducible, θ is an isomorphism. Proof. (i) The kernel K of θ is a subrepresentation of V 1 . Since θ = 0, this subrepresentation cannot be V1 . So by irreducibility of V1 we have K = 0. (ii) The image I of θ is a subrepresentation of V 2 . Since θ = 0, this subrepresentation cannot be 0. So by irreducibility of V2 we have I = V2 . Corollary 1.17. (Schur’s lemma for algebraically closed fields) Let V be a finite dimensional irreducible representation of an algebra A over an algebraically closed field k, and θ : V ⊃ V is an intertwining operator. Then θ = ∂ · Id for some ∂ � k (a scalar operator). Remark. Note that this Corollary is false over the field of real numbers: it suffices to take A = C (regarded as an R-algebra), and V = A. Proof. Let ∂ be an eigenvalue of θ (a root of the characteristic polynomial of θ). It exists since k is an algebraically closed field. Then the operator θ − ∂Id is an intertwining operator V ⊃ V , which is not an isomorphism (since its determinant is zero). Thus by Proposition 1.16 this operator is zero, hence the result. Corollary 1.18. Let A be a commutative algebra. Then every irreducible finite dimensional rep­ resentation V of A is 1-dimensional. Remark. Note that a 1-dimensional representation of any algebra is automatically irreducible. Proof. Let V be irreducible. For any element a � A, the operator δ(a) : V ⊃ V is an intertwining operator. Indeed, δ(a)δ(b)v = δ(ab)v = δ(ba)v = δ(b)δ(a)v (the second equality is true since the algebra is commutative). Thus, by Schur’s lemma, δ(a) is a scalar operator for any a � A. Hence every subspace of V is a subrepresentation. But V is irreducible, so 0 and V are the only subspaces of V . This means that dim V = 1 (since V = ⇒ 0). Example 1.19. 1. A = k. Since representations of A are simply vector spaces, V = A is the only irreducible and the only indecomposable representation. 2. A = k[x]. Since this algebra is commutative, the irreducible representations of A are its 1-dimensional representations. As we discussed above, they are defined by a single operator δ(x). In the 1-dimensional case, this is just a number from k. So all the irreducible representations of A are V� = k, ∂ � k, in which the action of A defined by δ(x) = ∂. Clearly, these representations are pairwise non-isomorphic. The classification of indecomposable representations of k[x] is more interesting. To obtain it, recall that any linear operator on a finite dimensional vector space V can be brought to Jordan normal form. More specifically, recall that the Jordan block J �,n is the operator on k n which in the standard basis is given by the formulas J �,n ei = ∂ei + ei−1 for i > 1, and J�,n e1 = ∂e1 . Then for any linear operator B : V ⊃ V there exists a basis of V such that the matrix of B in this basis is a direct sum of Jordan blocks. This implies that all the indecomposable representations of A are V�,n = k n , ∂ � k, with δ(x) = J�,n . The fact that these representations are indecomposable and pairwise non-isomorphic follows from the Jordan normal form theorem (which in particular says that the Jordan normal form of an operator is unique up to permutation of blocks). This example shows that an indecomposable representation of an algebra need not be irreducible. 3. 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 δ : 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 ν V (z). Show that νV : 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 δ(z) by which z acts in V has only one eigenvalue ν V (z), equal to the scalar by which z acts on some irreducible subrepresentation of V . Thus ν V : Z(A) ⊃ k is a homomorphism, which is again called the central character of V . (c) Does δ(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 End A (V ) one denotes the algebra of all homomorphisms of representations V ⊃ V . Show that End A (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 θ : 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 θ is not a scalar, and consider the subfield C(θ) → D. Show that C(θ) is a transcendental extension of C. Derive from this that C(θ) 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 A op . 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 θ : A ⊃ B is a homomorphism of algebras, then ker θ 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�φ = 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 β : A ⊃ A/I be the quotient map. We can define multiplication in A/I by β(a) · β(b) := β(ab). This is well defined because if β(a) = β(a � ) then β(a� b) = β(ab + (a� − a)b) = β(ab) + β((a� − a)b) = β(ab) because (a� − a)b � Ib ∧ I = ker β, as I is a right ideal; similarly, if β(b) = β(b � ) then β(ab� ) = β(ab + a(b� − b)) = β(ab) + β(a(b� − b)) = β(ab) because a(b� − b) � aI ∧ I = ker β, 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 β : V ⊃ V /W be the quotient map, and set δ V /W (a)β(x) := β(δV (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[x1 , ..., xn ] and I = 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 ⇒= 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]/I2 , where I2 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 (δ(a)f )(b) = f (ba). Show that V provides such an example. 1.6 Algebras defined by generators and relations If f1 , . . . , fm are elements of the free algebra k◦x 1 , . . . , xn �, we say that the algebra A := k◦x1 , . . . , xn �/◦{f1 , . . . , fm }� is generated by x1 , . . . , xn with defining relations f1 = 0, . . . , fm = 0. 1.7 Examples of algebras 1. The Weyl algebra, k◦x, y�/◦yx − xy − 1�. 2. 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 {x i y j , i, j ⊂ 0}. (ii) A basis for the q-Weyl algebra A q is {xi y j , i, j � Z}. Proof. (i) First let us show that the elements x i y j are a spanning set for A. To do this, note that any word in x, y can be ordered to have all the x on the left of the y, at the cost of interchanging some x and y. Since yx − xy = 1, this will lead to error terms, but these terms will be sums of monomials that have a smaller number of letters x, y than the original word. Therefore, continuing this process, we can order everything and represent any word as a linear combination of x i y j . The proof that xi y j are linearly independent is based on representation theory. Namely, let a be a variable, and E = ta k[a][t, t−1 ] (here ta is just a formal symbol, so really E = k[a][t, t −1 ]). Then E d(ta+n ) df a+n−1 ). is a representation of A with action given by xf = tf and ⎨ yf =i jdt (where dt := (a + n)t Suppose now that we have a nontrivial linear relation cij x y = 0. Then the operator L= cij ti � d dt �j Qj (t) � d dt �j � acts by zero in E. Let us write L as L= r � j=0 , where Qr ⇒= 0. Then we have a Lt = r � j =0 Qj (t)a(a − 1)...(a − j + 1)ta−j . ⎨r This must be zero, so we have j=0 Qj (t)a(a − 1)...(a − j + 1)t−j = 0 in k[a][t, t−1 ]. Taking the leading term in a, we get Qr (t) = 0, a contradiction. (ii) Any word in x, y, x−1 , y −1 can be ordered at the cost of multiplying it by a power of q. This easily implies both the spanning property and the linear independence. Remark. The proof of (i) shows that the Weyl algebra A can be viewed as the algebra of p olynomial differential operators in one variable t. The proof of (i) also brings up the notion of a faithful representation. Definition. A representation δ : A ⊃ End V is faithful if δ is injective. For example, k[t] is a faithful representation of the Weyl algebra, if k has characteristic zero (check it!), but not in characteristic p, where (d/dt) p Q = 0 for any polynomial Q. However, the representation E = ta k[a][t, t−1 ], as we’ve seen, is faithful in any characteristic. Problem 1.26. Let A be the Weyl algebra, generated by two elements x, y with the relation yx − xy − 1 = 0. (a) If chark = 0, what are the finite dimensional representations of A? What are the two-sided ideals in A? Hint. For the first question, use the fact that for two square matrices B, C, Tr(BC) = Tr(CB). For the second question, show that any nonzero two-sided ideal in A contains a nonzero polynomial in x, and use this to characterize this ideal. Suppose for the rest of the problem that chark = p. (b) What is the center of A? Hint. Show that xp and y p 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, x2 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. �•� � • • • 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 P Q , 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 p i , 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 pi = 1, so PQ is an algebra with unit. i�I Problem 1.33. Show that the algebra P Q is generated by pi for i � I and ah for h � E with the defining relations: 1. p2i = pi , pi pj = 0 for i ⇒= j 2. ah ph⊗ = ah , ah pj = 0 for j ⇒= h� 3. ph⊗⊗ ah = ah , pi ah = 0 for i ⇒= 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 P Q . From this representation, we can construct a representation of Q as follows: let V i = pi V, 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 P Q : let V = i Vi , let pi : V ⊃ Vi ⊃ V be the projection onto Vi , and for any path p = h1 ...hm let ap = xh1 ...xhm : Vh⊗m ⊃ Vh1⊗⊗ 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 �⊃ (p i V) and (Vi ) �⊃ i Vi are inverses of each other. Thus, we have a bijection between isomorphism classes of representations of the algebra P Q 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 (V i , 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 (V i , xh ) and (Wi , yh ) is the representation (Vi � Wi , xh � yh ). As with representations of algebras, a nonzero representation (V i ) of a quiver Q is said to be irreducible if its only subrepresentations are (0) and (V i ) 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 � : (Vi ) −⊃ (Wi ) of quiver representations is a collection of maps � i : Vi −⊃ Wi such that yh ∞ �h⊗ = �h⊗⊗ ∞ 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 n+ m]. If A[n] is finite dimensional, it is useful to consider the Hilbert series h A (t) = dim A[n]t (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(x n ) = n then hA (t) = 1 + t + t2 + ... + tn + ... = 1 1−t Find the Hilbert series of: (a) A = k[x1 , ..., xm ] (where the grading is by degree of polynomials); (b) A = k < x1 , ..., xm > (the grading is by length of words); (c) A is the exterior (=Grassmann) algebra √ k [x1 , ..., xm ], generated over some field k by x1 , ..., xm with the defining relations xi xj + xj xi = 0 and x2i = 0 for all i, j (the grading is by degree). (d) A is the path algebra PQ of a quiver Q (the grading is defined by deg(p i ) = 0, deg(ah ) = 1). Hint. The closed answer is written in terms of the adjacency matrix M Q of Q. 1.9 Lie algebras Let g be a vector space over a field k, and let [ , ] : g × g −⊃ g be a skew-symmetric bilinear map. (That is, [a, a] = 0, and hence [a, b] = −[b, a]). Definition 1.39. (g, [ , ]) is a Lie algebra if [ , ] satisfies the Jacobi identity � � � � � � [a, b] , c + [b, c] , a + [c, a] , b = 0. (2) Example 1.40. Some examples of Lie algebras are: 1. Any space g with [ , ] = 0 (abelian Lie algebra). 2. Any associative algebra A with [a, b] = ab − ba . 3. Any subspace U of an associative algebra A such that [a, b] � U for all a, b � U . 4. The space Der(A) of derivations of an algebra A, i.e. linear maps D : A ⊃ A which satisfy the Leibniz rule: D(ab) = D(a)b + aD(b). Remark 1.41. Derivations are important because they are the “infinitesimal version” of automor­ phisms (i.e., isomorphisms onto itself). For example, assume that g(t) is a differentiable family of automorphisms of a finite dimensional algebra A over R or C parametrized by t � (−φ, φ) such that g(0) = Id. Then D := g � (0) : A ⊃ A is a derivation (check it!). Conversely, if D : A ⊃ A is a derivation, then etD is a 1-parameter family of automorphisms (give a proof!). This provides a motivation for the notion of a Lie algebra. Namely, we see that Lie algebras arise as spaces of infinitesimal automorphisms (=derivations) of associative algebras. In fact, they similarly arise as spaces of derivations of any kind of linear algebraic structures, such as Lie algebras, Hopf algebras, etc., and for this reason play a very important role in algebra. Here are a few more concrete examples of Lie algebras: 1. R3 with [u, v] = u × v, the cross-product of u and v. 2. sl(n), the set of n × n matrices with trace 0. For example, sl(2) has the basis � � � � 0 1 0 0 e= f= 0 0 1 0 with relations [h, e] = 2e, [h, f ] = −2f, [e, f ] = h. h= � 1 0 0 −1 � 3. The Heisenberg Lie It has the basis ⎧ 0 x = �0 0 algebra H of matrices � 0 0 0 1⎝ 0 0 �0 ⊕ ⊕� 00⊕ 000 ⎧ � 0 1 0 y = �0 0 0⎝ 0 0 0 ⎧ � 0 0 1 c = �0 0 0 ⎝ 0 0 0 with relations [y, x] = c and [y, c] = [x, c] = 0. 4. The algebra aff(1) of matrices ( ⊕0 ⊕0 ) Its basis consists of X = ( 01 00 ) and Y = ( 00 01 ), with [X, Y ] = Y . 5. 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 g1 , g2 be Lie algebras. A homomorphism � : g1 −⊃ g2 of Lie algebras is a linear map such that �([a, b]) = [�(a), �(b)]. Definition 1.43. A representation of a Lie algebra g is a vector space V with a homomorphism of Lie algebras δ : g −⊃ End V . Example 1.44. Some examples of representations of Lie algebras are: 1. V = 0. 2. Any vector space V with δ = 0 (the trivial representation). 3. The adjoint representation V = g with δ(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 x i and [ , ] defined by [xi , xj ] = k c xk . The universal enveloping algebra⎨U(g) is the associative algebra generated by the x i ’s with the defining relations xi xj − xj xi = k ckij 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 = −2f 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 (v1 + v2 ) � w − v1 � w − v2 � w, v � (w1 + w2 ) − v � w1 − v � w2 , 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 (v1 + v2 ) � w − v1 � w − v2 � w, v � (w1 + w2 ) − v � w1 − v � w2 , 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 � V2 ) � V3 can be naturally identified with V1 � (V2 � V3 ). 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) - algebra structures, etc. If V is finite dimensional with basis e i , i = 1, ..., N , and ei is the dual basis of V ⊕ , then a basis of E is the set of vectors ei1 � ... � ein � ej1 � ... � ejm , and a typical element of E is N � i1 ,...,in ,j1 ,...,jm =1 ...in e � ... � ein � ej1 � ... � ejm , Tji11...j m i1 where T is a multidimensional table of numbers. Physicists define a tensor as a collection of such multidimensional tables T B 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!) The most important properties of tensor products are summarized in the following problem. Problem 1.49. (a) Let U be any k-vector space. Construct a natural bijection between bilinear maps V × W ⊃ U and linear maps V � W ⊃ U . (b) Show that if {vi } is a basis of V and {wj } is a basis of W then {vi � wj } is a basis of V � W. (c) Construct a natural isomorphism V ⊕ � W ⊃ Hom(V, W ) in the case when V is finite dimensional (“natural” means that the isomorphism is defined without choosing bases). (d) Let V be a vector space over a field k. Let S n V be the quotient of V �n (n-fold tensor product of V ) by the subspace spanned by the tensors T − s(T ) where T � V �n , and s is some transposition. Also let √n V be the quotient of V �n by the subspace spanned by the tensors T such that s(T ) = T for some transposition s. These spaces are called the n-th symmetric, respectively exterior, power of V . If {vi } is a basis of V , can you construct a basis of S n V, √n V ? If dimV = m, what are their dimensions? (e) If k has characteristic zero, find a natural identification of S n V with the space of T � V �n such that T = sT for all transpositions s, and of √ n V with the space of T � V �n such that T = −sT for all transpositions s. (f) Let A : V ⊃ W be a linear operator. Then we have an operator A �n : V �n ⊃ W �n , and its symmetric and exterior powers S n A : S n V ⊃ S n W , √n A : √n V ⊃ √n W which are defined in an obvious way. Suppose V = W and has dimension N , and assume that the eigenvalues of A are ∂1 , ..., ∂N . Find T r(S n A), T r(√n A). (g) Show that √N A = det(A)Id, and use this equality to give a one-line proof of the fact that det(AB) = det(A) det(B). Remark. Note that a similar definition to the above can be used to define the tensor product V �A W , where A is any ring, V is a right A-module, and W is a left A-module. Namely, V � A W is the abelian group which is the quotient of the group V • W freely generated by formal symbols v � w, v � V , w � W , modulo the relations (v1 + v2 ) � w − v1 � w − v2 � w, v � (w1 + w2 ) − v � w1 − v � w2 , va � w − v � aw, a � A. Exercise. Throughout this exercise, we let k be an arbitrary field (not necessarily of charac­ teristic zero, and not necessarily algebraically closed). If A and B are two k-algebras, then an (A, B)-bimodule will mean a k-vector space V with both a left A-module structure and a right B-module structure which satisfy (av) b = a (vb) for any v � V , a � A and b � B. Note that both the notions of ”left A-module” and ”right Amodule” are particular cases of the notion of bimodules; namely, a left A-module is the same as an (A, k)-bimodule, and a right A-module is the same as a (k, A)-bimodule. Let B be a k-algebra, W a left B-module and V a right B-module. We denote by V � B W the k-vector space (V �k W ) / ◦vb � w − v � bw | v � V, w � W, b � B�. We denote the projection of a pure tensor v � w (with v � V and w � W ) onto the space V � B W by v �B w. (Note that this tensor product V �B W is the one defined in the Remark after Problem1.49.) If, additionally, A is another k-algebra, and if the right B-module structure on V is part of an (A, B)-bimodule structure, then V �B W becomes a left A-module by a (v �B w) = av �B w for any a � A, v � V and w � W . 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 �⊃ 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 � Hom A (V, W ) and v � V and (f c) (v) = f (v) c for all c � C, f � Hom A (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 �⊃ (w � B v �⊃ 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 x1 , ..., 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[x1 , ..., xN ], √V with the exterior algebra √k (x1 , ..., 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 S n V , √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, � α(a) D(A) = l(a) � , β a where a runs over edges of A, and l(a), α(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 ϕ = arccos(1/3)/β is not a rational number. Hint. Assume that ϕ = 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 x k + x−k has denominator 3k 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 δV �W (x) = δV (x) � Id + Id � δW (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 δV � (x) = −δV (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 Hom g (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 = −2F, EF − F E = H (the corresponding map δ is given by δ(e) = E, δ(f ) = F , δ(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 ∂. Let V¯ (∂) be the generalized eigenspace corresponding to ∂. Show that E| V̄ (�) = 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 E k F k w = Pk (H)w. (First compute EF k w, then use induction in k). (c) Let v � V¯ (∂) be a generalized eigenvector of H with eigenvalue ∂. Show that there exists N > 0 such that F N v = 0. (d) Show that H is diagonalizable on V¯ (∂). (Take N to be such that F N = 0 on V¯ (∂), and compute E N F N v, v � V¯ (∂), by (b). Use the fact that Pk (x) does not have multiple roots). (e) Let Nv be the smallest N satisfying (c). Show that ∂ = N v − 1. (f) Show that for each N > 0, there exists a unique up to isomorphism irreducible representation of sl(2) of dimension N . Compute the matrices E, F, H in this representation using a convenient basis. (For V finite dimensional irreducible take ∂ as in (a) and v � V (∂) an eigenvector of H. Show that v, F v, ..., F � v is a basis of V , and compute the matrices of the operators E, F, H in this basis.) Denote the ∂ + 1-dimensional irreducible representation from (f) by V � . Below you will show that any finite dimensional representation is a direct sum of V � . (g) Show that the operator C = EF + F E + H 2 /2 (the so-called Casimir operator) commutes with E, F, H and equals �(�+2) Id on V� . 2 Now it will be easy to prove the direct sum decomposition. Namely, assume the contrary, and let V be a reducible representation of the smallest dimension, which is not a direct sum of smaller representations. (h) Show that C has only one eigenvalue on V , namely �(�2+2) for some nonnegative integer ∂. (use that the generalized eigenspace decomposition of C must be a decomposition of representations). (i) Show that V has a subrepresentation W = V � such that V /W = nV� for some n (use (h) and the fact that V is the smallest which cannot be decomposed). (j) Deduce from (i) that the eigenspace V (∂) of H is n + 1-dimensional. If v 1 , ..., vn+1 is its basis, show that F j vi , 1 ∗ i ∗ n + 1, 0 ∗ j ∗ ∂ are linearly independent and therefore form a basis of V (establish that if F x = 0 and Hx = µx then Cx = µ(µ2−2) x and hence µ = −∂). (k) Define Wi = span(vi , F vi , ..., F � vi ). Show that Vi are subrepresentations of V and derive a contradiction with the fact that V cannot be decomposed. (l) (Jacobson-Morozov Lemma) Let V be a finite dimensional complex vector space and A : V ⊃ V a nilpotent operator. Show that there exists a unique, up to an isomorphism, representation of sl(2) on V such that E = A. (Use the classification of the representations and the Jordan normal form theorem) (m) (Clebsch-Gordan decomposition) Find the decomposition into irreducibles of the represen­ tation V� � Vµ of sl(2). Hint. For a finite dimensional representation V of sl(2) it is useful to introduce the character νV (x) = T r(exH ), x � C. Show that νV ∈W (x) = νV (x) + νW (x) and νV �W (x) = νV (x)νW (x). Then compute the character of V� and of V� �Vµ and derive the decomposition. This decomposition is of fundamental importance in quantum mechanics. (n) Let V = CM � CN , and A = JM (0) � IdN + IdM � JN (0), where Jn (0) is the Jordan block of size n with eigenvalue zero (i.e., J n (0)ei = ei−1 , i = 2, ..., n, and Jn (0)e1 = 0). Find the Jordan normal form of A using (l),(m). 1.15 Problems on Lie algebras Problem 1.56. (Lie’s Theorem) The commutant K(g) of a Lie algebra g is the linear span of elements [x, y], x, y � g. This is an ideal in g (i.e., it is a subrepresentation of the adjoint representation). A finite dimensional Lie algebra g over a field k is said to be solvable if there exists n such that K n (g) = 0. Prove the Lie theorem: if k = C and V is a finite dimensional irreducible representation of a solvable Lie algebra g then V is 1-dimensional. Hint. Prove the result by induction in dimension. By the induction assumption, K(g) has a common eigenvector v in V , that is there is a linear function ν : K(g) ⊃ C such that av = ν(a)v for any a � K(g). Show that g preserves common eigenspaces of K(g) (for this you will need to show that ν([x, a]) = 0 for x � g and a � K(g). To prove this, consider the smallest vector subspace U containing v and invariant under x. This subspace is invariant under K(g) and any a � K(g) acts with trace dim(U )ν(a) in this subspace. In particular 0 = Tr([x, a]) = dim(U )ν([x, a]).). Problem 1.57. Classify irreducible finite dimensional representations of the two dimensional Lie algebra with basis X, Y and commutation relation [X, Y ] = Y . Consider the cases of zero and positive characteristic. Is the Lie theorem true in positive characteristic? Problem 1.58. (hard!) For any element x of a Lie algebra g let ad(x) denote the operator g ⊃ g, y �⊃ [x, y]. Consider the Lie algebra g n generated by two elements x, y with the defining relations ad(x)2 (y) = ad(y)n+1 (x) = 0. (a) Show that the Lie algebras g1 , g2 , g3 are finite dimensional and find their dimensions. (b) (harder!) Show that the Lie algebra g 4 has infinite dimension. Construct explicitly a basis of this algebra.