DELIGNE’S THEOREM ON TENSOR CATEGORIES DENE LEPINE Abstract. We look at and explain the concepts needed to understand the statement of Deligne’s Theorem on Tensor Categories, which states that an arbitrary tensor category is equivalent to the category of representations of some super group. Throughout the paper we assume a background in only basic undergraduate mathematics. Contents 1. Introduction 1 Acknowledgments 2 2. A Brief Introduction to Supermath 2 3. Superalgebras 5 4. Super Hopf Algebras 9 5. Representation Theory 14 6. Tensor Categories 15 7. H-Categories 19 8. Supergroups 22 References 24 1. Introduction Supermathematics is the study of algebraic structures (vector spaces, algebras, modules, groups, etc.) that have a decomposition into “even” and “odd” pieces. Besides the purely mathematical reason to study supermath, one of the main motivations for supermath is the theory of supersymmetry. Supersymmetry is how supermath acquired such a “super” name and is where supermath offers much of its application and derives its motivation. Supersymmetry is a field within partical physics which studies the nature of fermions and bosons. An overly simplified example of a super vector space is if you consider two particles, either fermions or bosons, their spin can be represented as elements of a super vector space. When we take these two particles and allow them to interact, their resulting spin will be exactly the tensor product of the spins of the individual particles. This is just a simple example but supersymmetry works with complex theorems and results using supermath. Deligne’sTheoremonTensorCategorieswasfirstpublishedin2002andwasaresultdrawn from Deligne’s past work. This theorem ties together multiple fields of math and gives an equivalence to more easily work with tensor categories. The goal of this paper is actually quitesimple, wewanttobeabletoclearlyunderstandandstateDeligne’sTheoremonTensor Categories. To do so, we will start with simple concepts in super math, representation theory 1 2 DENE LEPINE and category theory then build these up with clear examples. Then we will bring together the concepts and examples to fully understand Deligne’s Theorem. This paper is aimed at readers with at least a background of undergraduate level algebra. Specifically, we assume the reader has taken some courses on basic group theory and has seen many of the linear algebra concepts taught at the undergraduate level. Althought there are some concepts looked at in this paper that may be beyond the understanding of an undergraduate student, our hope is to properly explain and build up from known concepts so that our reader can conceptualize Delgine’s Theorem. Acknowledgments. This work was completed as part of the Co-op Work-Study Program at the University of Ottawa and was partially funded by the N.S.E.R.C. Discovery Grant of Professor Alistair Savage. Furthermore, I would like to thank Professor Alistair Savage for his understanding and guidance throughout this project. 2. A Brief Introduction to Supermath In this section, we look at the “super” analog of some of the mathematical structures that we already know and studied to certain extent. While doing so, we will develop the terms and examples that are common place while working in supermath. Both the terms and examples will be used throughout this paper as we build more advanced mathematical structures. The goal of this section is to give the reader a strong foundation in supermath so that we may build fluidly to our final theorem. Definition 2.1 (Super Vector Space). A super vector space (SVS) or Z -graded vector space 2 is a vector space, V, over a field, K, with a direct sum decomposition, V = V ⊕V . 0 1 When working with super vector spaces, we will view the subscripts of the subspaces as 0,1 ∈ Z . 2 Definition 2.2 (Homogeneous). Let V be a super vector space. An element of V is said to be homogeneous if it is an element of either V or V . 0 1 Definition 2.3 (Parity). The parity of nonzero homogeneous elements v ∈ V, denoted by p(v) = |v|, is defined as: (cid:40) 0 if v ∈ V , 0 p(v) = |v| = 1 if v ∈ V . 1 We call all elements of V even and V odd. 0 1 Definition 2.4 (Dimension of a SVS). Given a finite-dimensional super vector space, V = V ⊕V , where dimV = p and dimV = q then we say that V is of dimension p|q. 0 1 0 1 Definition 2.5 (Linear Transformation). Let V and W be vector spaces over K. A linear transformation from V to W is a map, f: V → W, such that for all x,y ∈ V and k ∈ K we have f(x)+f(y) = f(x+y) and f(kx) = kf(x). We let L(V,W) denote the set of all linear transformations V → W. Notice that L(V,W) is also a vector space under pointwise addition and scalar multiplication. DELIGNE’S THEOREM ON TENSOR CATEGORIES 3 Definition 2.6 (Dual Space). Let V be a vector space. We define the dual space, denoted V∗, to be the vector space L(V,K). Remark 2.7. Notice that if V is of dimension p with basis B = {b ,...,b }, then V∗ is 1 p also of dimension p with basis B∨ = {b∨ | b ∈ B}, where b∨(b ) = 1 if and only if b = b . i i i j i j Furthermore, if V is a SVS then V∗ is also a SVS where p(b∨) = p(b ). i i Definition 2.8 (Grade-Preserving/Grade-Reversing). A linear transformation, f: V → W, between super vector spaces is said to be grade-preserving or even if f(V ) ⊆ W for i = 0 i i and 1. It is called grade-reversing or odd if f(V ) ⊆ W . i 1−i Definition 2.9 (Transpose). Given a linear map f: V → W then the transpose map is f∗: W∗ → V∗ defined for all g ∈ W∗ via f∗(g) = g ◦f. Notice that while working with SVS p(f∗) = p(f). Definition 2.10 (Isomorphic). Two SVS, V and W, are isomorphic if there exists a grade- preserving bijective linear transformation between V and W. Such a map is called an ∼ isomorphism. If there exists such a transformation, then we write V = W. Example 2.11. Consider the SVS Kp|q = Kp ⊕Kq, for some field K, where Kp|q = Kp and 0 Kp|q = Kq. Recall that any vector space V of dimension p is isomorphic to Kp. Similarly, if 1 V is a super vector space with dimV = p|q then V ∼= Kp|q. Example 2.12. Let V and W be SVS. Notice L(V,W) itself is a super vector space with L(V,W) = L(V,W) ⊕L(V,W) defined by: 0 1 L(V,W) = {f: V → W | f grade-preserving}, 0 L(V,W) = {f: V → W | f grade-reversing}. 1 Example 2.13. Another example to consider is the vector space of m×n matrices, denoted M (K). We turn these matrices into super matrices by partitioning both the rows and m×n colums into two. For example, the following m×n matrix can be turned into a super matrix by   a ··· a a ··· a 1,1 1,r 1,r+1 1,n  a ··· a   ... ... ... ... ... ...   1...,1 ... 1...,n  →  aas,1 ······ aas,r aas,r+1 ······ aas,n  = (cid:20) XX00 XX01 (cid:21).  s+1,1 s+1,r s+1,r+1 s+1,n  10 11 am,1 ··· am,n  ... ... ... ... ... ...  a ··· a a ··· a m,1 m,r m,r+1 m,n We say that these matrices are of dimension s|p × r|q where s + p = m and r + q = n, denoted M (K). A matrix as above is defined to be even if X = X = 0, and odd if s|p×r|q 01 10 X = X = 0. Thus we have 00 11 M (K) = M (K) ⊕M (K) . s|p×r|q s|p×r|q 0 s|p×r|q 1 Definition 2.14 (Bilinear Map). A bilinear map, B: V ×W → U, is a function such that for all v,v ,v ∈ V, w,w ,w ∈ W and k ∈ K, we have 1 2 1 2 4 DENE LEPINE • B(v,w +w ) = B(v,w )+B(v,w ), 1 2 1 2 • B(v +v ,w) = B(v ,w)+B(v ,w), 1 2 1 2 • B(kv,w) = kB(v,w) = B(v,kw). Definition 2.15 (Free Vector Space). Given a set, S, the free vector space generated by S is F(S) = {φ: S → K | φ(s) = 0 for all except finitely many s ∈ S}. With the usual pointwise addition and scalar multiplication of functions this is, in fact, a (cid:80) vector space. We will write φ(s)s instead of φ(s). s∈S Definition 2.16 (Tensor Product). Let V and W be vector spaces over K. We define Z ⊆ F(V ×W) to be the subspace defined by  (cid:12)  (v +v ,w)−(v ,w)−(v ,w), (cid:12)  1 2 1 2   (cid:12)  Z =  (v,w1 +w2)−(v,w1)−(v,w2), (cid:12)(cid:12)v,v ,v ∈ V,w,w ,w ∈ W,k ∈ K. k(v,w)−(kv,w), (cid:12) 1 2 1 2    (cid:12)   k(v,w)−(v,kw) (cid:12)  The tensor product of V and W is the quotient space F(V × W)/Z, denoted by V ⊗ W. We write v ⊗ w for (v,w) + Z. The tensor product has the following properties for all v,v ,v ∈ V, w,w ,w ∈ W and k ∈ K: 1 2 1 2 (v +v )⊗w = (v ⊗w)+(v ⊗w), 1 2 1 2 v ⊗(w +w ) = (v ⊗w )+(v ⊗w ), and 1 2 1 2 kv ⊗w = v ⊗kw. Lastly and foremost, V ⊗ W has a universal mapping property. This property states that for all vector spaces U and all bilinear maps B: V × W → U there exists a unique linear map (cid:96): V ⊗W → U such that for all v ∈ V and w ∈ W, B(v,w) = (cid:96)(v ⊗w). Remark 2.17 (Dimension of V ⊗ W). Let V and W be finite-dimensional vector spaces with bases b = {b ,...,b } and B = {B ,...,B } respectively. The tensor product V ⊗W 1 p 1 q has basis b ⊗ B = {b ⊗ B | b ∈ b and B ∈ B}. Thus we know that if dim(V) = p and i j i j dim(W) = q then dim(V ⊗W) = pq. For more on this refer to [1, §13]. Definition 2.18 (Tensor Product of Super Vector Spaces). Given SVS V = V ⊕ V and 0 1 W = W ⊕W , the resulting V ⊗W is naturally a super vector space with the Z -grading 0 1 2 (V ⊗W) = (V ⊗W )⊕(V ⊗W ) and 0 0 0 1 1 (V ⊗W) = (V ⊗W )⊕(V ⊗W ). 1 0 1 1 0 Definition 2.19 (Superring). A superring or Z -graded ring is a ring, R, with the decom- 2 postion R = R ⊕R where the product map R×R → R is even. As with SVS, the elements 0 1 of R are called even and elements of R odd. 0 1 DELIGNE’S THEOREM ON TENSOR CATEGORIES 5 Definition 2.20 (Ring Homomorphism). A ring homomorphism is an even map f: R → S, with R and S superrings, such that for all r,s ∈ R f(r+s) = f(r)+f(s), f(r·s) = f(r)·f(s), and f(1 ) = 1 . R S Definition 2.21 (Discrete Supergroup). A discrete supergroup is a Z -graded group, (G,·), 2 where · is such that G ·G ⊆ G . A Z -graded group is such that G = G (cid:116)G , where (cid:116) i j i+j 2 0 1 is the disjoint union. 3. Superalgebras In this section, we start by defining the “super” structure on associative algebras and give multiple examples of such structures. We then proceed by defining and giving examples of other super structures directly relating to superalgebras, like super coalgebras and supermod- ules. To follow, we will bring some of these definitions together to define one of the most important super structures looked at in this section, a super bialgebra. Definition 3.1 (Superalgebra). An associative superalgebra over some field K is a triple (A,∇,η) where A is a SVS over K, ∇: A ⊗ A → A is an even associative linear map and η: K → A. Furthermore, we require that the following diagrams commute. A A⊗A A⊗A ∇ A ∇ A⊗A ∇ ∇ ∇⊗idA idA η⊗idA idA⊗η A⊗A A⊗A⊗A A idA⊗∇ Notice when we say that id ⊗ η: A → A ⊗ A the domain is actually A ⊗ K ∼= A. For A a,b ∈ A, we often write a·b or ab for ∇(a⊗b). Definition 3.2 (Supercommutative). A superalgebra, A, is said to be supercommutative if for all homogeneous a ,a ∈ A, we have 1 2 a a = (−1)|a1||a2|a a . 1 2 2 1 Example 3.3. Recall from Example 2.12 that L(V,W) forms a super vector space. Now consider L(V,V), which is also denoted End(V) or gl(V). We define the multiplication to be regular composition of maps. To confirm that the image of End(V) ⊗ End(V) is in i j End(V) , take f,g ∈ End(V) such that p(f) = i, p(g) = j and v ∈ V such that p(v) = k i+j then p((f ◦g)(v)) = p(f(g(v))) = i+j +k thus p(f ◦g) = i+j. Example 3.4. Consider the super vector space of supermatrices of dimension p|q × p|q over some field K, denoted M (K). To make this a superalgebra, we use usual matrix p|q multiplication and show that M (K) ·M (K) ⊆ M (K) , notice the following p|q i p|q j p|q i+j • If both X,Y ∈ M (K) then p|q 0 (cid:20) (cid:21) X Y 0 X ·Y = 00 00 , 0 X Y 11 11 thus X ·Y ∈ M (K) . p|q 0 6 DENE LEPINE • If X ∈ M (K) and Y ∈ M (K) then p|q 0 p|q 1 (cid:20) (cid:21) 0 X Y X ·Y = 00 01 , X Y 0 11 10 thus X ·Y ∈ M (K) . p|q 1 • If X is M (K) and Y is M (K) then p|q 1 p|q 0 (cid:20) (cid:21) 0 X Y X ·Y = 01 11 , X Y 0 10 00 thus X ·Y ∈ M (K) . p|q 1 • If X,Y ∈ M (K) then p|q 1 (cid:20) (cid:21) X Y 0 X ·Y = 01 10 , 0 X Y 10 01 thus X ·Y ∈ M (K) . p|q 0 Thus M (K) ·M (K) ⊆ M (K) as required. p|q i p|q j p|q i+j Remark 3.5. Let V be a SVS of dimension p|q over K. We can pick bases for V and V , 0 1 ∼ b = {b ,b ,...,b } and B = {B ,B ,...,B } respectively. We can show that End(V) = 1 2 p 1 2 q M (K) by F: L(V,V) → M (K) defined for any T ∈ gl(V) p|q p|q (cid:20) (cid:21) (cid:20) (cid:21) [T(b )] ··· [T(b )] [T(B )] ··· [T(B )] X X F(T) = 1 b p b 1 b q B = 00 01 , [T(b )] ··· [T(b )] [T(B )] ··· [T(B )] X X 1 B p B 1 B q B 10 11 where, given x ∈ V, [T(x)] denotes the column coordinate vector of T(x) with respect to Y the basis Y. Now notice that if T is an even transformation then this matrix will be a block diagonal matrix with X = X = [0] and if T is an odd transformation then the matrix 01 10 will be a block off-diagonal matrix with X = X = [0]. To show that this is, in fact, an 00 11 isomorphism is left to reader. Example 3.6 (Polynomial Superalgebra). A polynomial superalgebra over K is defined to be the algebra over K generated by p even variables, X , and q odd variables, Y , with relations i j X X = X X , X Y = Y X , and Y Y = −Y Y for all i and j. This algebra is denoted by i j j i i j j i i j j i K[X ,...,X | Y ,...,Y ]. An element of K[X ,...,X | Y ,...,Y ] is called a monomial if 1 p 1 q 1 p 1 q it is only a product of variables. Furthermore, we determine the parity of a monomial by the sum of variables parities in Z . 2 Example 3.7 (Rank 1 Clifford Algebra). The rank 1 Clifford algebra is a superalgebra with the underlying SVS C[x]/(x2 − 1) = C ⊕ Cx, where C is even and Cx is odd, and multiplication being the standard polynomial multiplication. With this multiplication it is easy to verify that (C[x]/(x2−1)) ·(C[x]/(x2−1)) ⊆ (C[x]/(x2−1)) is true for all i and i j i+j j. Definition 3.8 (Tensor of Algebras). Given A and B algebras then naturally A ⊗ B is a vector space. To define an algebra structure on A ⊗ B, we define multiplication of any a ⊗b ,a ⊗b ∈ A⊗B by 1 1 2 2 (a ⊗b )·(a ⊗b ) = a a ⊗b b . 1 1 2 2 1 2 1 2 DELIGNE’S THEOREM ON TENSOR CATEGORIES 7 Definition 3.9 (Tensor of Superalgebras). Analogously if A and B are both superalgebras then A ⊗ B is a SVS as seen in Definition 2.18. Furthermore, A ⊗ B will also be an superalgebra with multiplication defined, for all simple tensors a ⊗b ,a ⊗b ∈ A⊗B, by 1 1 2 2 (a ⊗b )·(a ⊗b ) = (−1)|a2|·|b1|(a a ⊗b b ). 1 1 2 2 1 2 1 2 Example 3.10 (Rank r Clifford Algebra). Using the rank 1 Clifford algebra from Example 3.7 we can construct a rank r Clifford algebra by forming the tensor product C[x ]/(x2−1)⊗ 1 1 C[x ]/(x2−1)⊗···⊗C[x ]/(x2−1) with multiplication as defined in Definition 3.9. While 2 2 r r workingwithrankr CliffordAlgebraswewritex for1⊗···⊗x ⊗···⊗1foralli ∈ {1,...,r}. i i Therankr Cliffordalgebrahasabasisgivenbymonomialsinthex . Theparity0component i isspannedbymonomialsofevendegreeandtheparity1componentisspannedbymonomials of odd degree. Notice that the rank r Clifford algebra is a supercommutative superalgebra since for all i,j ∈ {1,...,r} with i (cid:54)= j we have x x = (−1)x x . i j j i Definition 3.11 (Supermodule). Let (A,∇,η) be a superalgebra. A right supermodule or right Z -graded module, M, is a SVS together with an even linear map σ: M ⊗ A → M, 2 called an action, such that the following diagrams commute. M ⊗A σ M σ M ⊗A M σ M ⊗A idM idM⊗∇ σ⊗idA idM⊗η M ⊗A⊗A M These diagrams commute when we identify M ∼= M ⊗ K. A left supermodule is defined analogously. Example 3.12. Notice that Kp|q = Kp ⊕ Kq over M (K) with standard matrix multipli- p|q cation forms a right supermodule (or a left supermodule if we view Kp|q as column vectors). To insure that this is a module, notice that Kp|q is an abelian group under standard vector addition, 1 ∈ M (K) is just the identity matrix and distributivity is inherited from distribu- p|q tivity of matrix multiplication over addition. Lastly to insure that M (K) · Kp|q ⊆ Kp|q , p|q i j i+j consider the following cases: • If Y ∈ M (K) and X ∈ Kp|q then p|q 0 0 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) Y 0 X Y X Y ·X = 00 · 0 = 00 0 ∈ Kp|q. 0 Y 0 0 0 11 • If Y ∈ M (K) and X ∈ Kp|q then p|q 0 1 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) Y 0 0 0 Y ·X = 00 · = ∈ Kp|q. 0 Y X Y X 1 11 1 11 1 • If Y ∈ M (K) and X ∈ Kp|q then p|q 1 0 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) 0 Y X 0 Y ·X = 01 · 0 = ∈ Kp|q. Y 0 0 Y X 1 10 10 0 • If Y ∈ M (K) and X ∈ Kp|q then p|q 1 1 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) 0 Y 0 Y X Y ·X = 01 · = 01 1 ∈ Kp|q. Y 0 X 0 0 10 1 8 DENE LEPINE Concluding that in fact M (K) ·Kp|q ⊆ Kp|q for all i,j ∈ Z , as required. p|q i j i+j 2 Definition 3.13 (Super Coalgebra). A super coalgebra is a triple (C,∆,(cid:15)) where C is a SVS over K and ∆: C → C ⊗C and (cid:15): C → K are even morphisms such that the following diagrams commute. C ∆ C ⊗C C ⊗C idC⊗(cid:15) C (cid:15)⊗idC C ⊗C ∆ ∆⊗idC idC ∆ ∆ C ⊗C idC⊗∆ C ⊗C ⊗C C That is, for all c ∈ C, (id ⊗(cid:15))◦∆(c) = c = ((cid:15)⊗id )◦∆(c). We call ∆ the comultiplication C C and (cid:15) the counit. (cid:80) Remark 3.14. While working with coalgebras, we write ∆(c) = c ⊗ c and we will 1 2 sometimes write c = c (cid:15)(c ) = (cid:15)(c )c instead of (id ⊗(cid:15))◦∆(c) = c = ((cid:15)⊗id )◦∆(c). 1 2 1 2 C C Definition 3.15 (Super Bialgebra). A super bialgebra is a 5-tuple (B,∇,η,∆,(cid:15)) such that (B,∇,η) is an associative superalgebra over some field K, (B,∆,(cid:15)) is a super coalgebra over K, and the following diagrams are commutative. B ⊗B ∇ B ∆ B ⊗B ∆⊗∆ ∇⊗∇ B ⊗B ⊗B ⊗B idB⊗µ⊗idB B ⊗B ⊗B ⊗B In the above, µ: B ⊗B → B ⊗B is the flip operator. B ⊗B ∇ B K⊗K ∼= K η (cid:15)⊗(cid:15) (cid:15) η⊗η K⊗K ∼= K B ⊗B B ∆ B η (cid:15) K idK K Example 3.16. Recallthepolynomialsuperalgebra, K[X ,...,X | Y ,...,Y ], asdescribed 1 p 1 q in Example 3.6. We make this into a super bialgebra by defining (cid:15)(f) = f(0,...,0 | 0,...,0) for all f ∈ K[X ,...,X | Y ,...,Y ] and we define 1 p 1 q ∆(W) = W ⊗1+1⊗W where W ∈ {X ,...,X ,Y ,...,Y } and extend by linearity and multiplicatively. It is 1 p 1 q important to notice that the relation Y Y = −Y Y of the polynomial superalgebra implies i j j i that Y2 = 0. Thus for all Y ∈ {Y ,...,Y }, ∆(Y2) = 0. To verfiy this consider the following. i i 1 q i ∆(Y2) = ∆(Y )∆(Y ) = (Y ⊗1+1⊗Y )2 i i i i i = Y2 ⊗1+Y ⊗Y −Y ⊗Y +1⊗Y2 = 0 i i i i i i DELIGNE’S THEOREM ON TENSOR CATEGORIES 9 4. Super Hopf Algebras In this section, we take the super bialgebras previously defined and use them to define super Hopf algebras. Continuing, we start by first looking at some examples of both Z - 2 graded Hopf algebras and ungraded Hopf algebra. One example to pay special attention to is the group algebra example, since it will be the motivation for when we talk about supergroups. Lastly, we will look at and prove some important properties concerning the duals of superalgebras, super coalgebras, super bialgebras and super Hopf algebras. Definition 4.1 (Super Hopf Algebra). A super Hopf algebra is a 6-tuple (H,∇,η,∆,(cid:15),S) where (H,∇,η,∆,(cid:15)) is a super bialgebra and S is an even linear map S: H → H, called the antipode, such that the following diagram commutes. H ⊗H ∇ H ∇ H ⊗H (4.1) idH⊗S η◦(cid:15) S⊗idH H ⊗H H H ⊗H ∆ ∆ That is for all h ∈ H ∇◦(id ⊗S)◦∆(h) = η ◦(cid:15)(h) = ∇◦(S ⊗id )◦∆(h). H H Example 4.2. Continuing with the polynomial superalgebra from Example 3.16, we will define the antipode, S, of H = K[X ,...,X | Y ,...,Y ] so that it is a super Hopf al- 1 p 1 q gebra. We define S: H → H to be the unique algebra homomorphism such that for all h ∈ {X ,...,X ,Y ,...,Y } we have 1 p 1 q S(h) = −h. Now to verify that this antipode, S, satisfies the diagram in Definition 4.1, we can look at h ∈ {X ,...,X ,Y ,...,Y } and consider 1 p 1 q ∇(id ⊗S(∆(h))) = ∇(id ⊗S(h⊗1+1⊗h)) H H = ∇(h⊗1−1⊗h) = h−h = 0, η((cid:15)(h)) = η(0) = 0, ∇(S ⊗id (∆(h))) = ∇(S ⊗id (h⊗1+1⊗h)) H H = ∇(−h⊗1+1⊗h) = −h+h = 0. Thus for all h ∈ {X ,...,X ,Y ,...,Y }, we have ∇(id ⊗ S(∆(h))) = η((cid:15)(h)) = ∇(S ⊗ 1 p 1 q H id (∆(h))). Since {X ,...,X ,Y ,...,Y } generates H, it follows that (4.1) commutes. We H 1 p 1 q conclude that H is a super Hopf algebra. Example 4.3 (Tensor Algebra). Let V be a SVS over K. Thus T(V) = (cid:76)∞ V⊗k is a SVS k=0 where V⊗0 = K and the parity decomposition is as described in Definition 2.18. We call T(V) a tensor algebra. It will be a superalgebra where ∇: T(V)⊗T(V) → T(V) is defined for all (cid:78)k v ,(cid:78)(cid:96) w ∈ T(V) by i=1 i j=1 j (cid:32)(cid:32) (cid:33) (cid:32) (cid:33)(cid:33) k (cid:96) (cid:79) (cid:79) ∇ v ⊗ w = v ⊗···⊗v ⊗w ⊗···⊗w , i j 1 k 1 (cid:96) i=1 j=1 10 DENE LEPINE and extend by linearity. Continuing, we let ∆: T(V) → T(V) ⊗ T(V), (cid:15): T(V) → K and S: T(V) → T(V) be algebra homomorphisms. Now we define these maps such that T(V) is a super Hopf algebra. For all x ∈ V we define ∆(x) = x⊗1+1⊗x, ∆(1) = 1⊗1, (cid:15)(x) = 0, and S(x) = −x, and extend linearly and multiplicatively to higher tensor powers. With these definitions it is straightforward to verify that T(V) is a Hopf superalgebra. Example 4.4 (Symmetric Algebra). Consider the Hopf superalgebra, T(V), from Exam- ple 4.3. We call the quotient algebra T(V)/I the symmetric algebra, denoted S(V), where I is the ideal generated by {x ⊗ y − (−1)p(x)p(y)y ⊗ x | x,y ∈ V}. Now S(V) is inheritely an algebra, Furthermore, the maps ∆, (cid:15), ∇, η, and S as defined in Example 4.3 will induce maps on the quotient such that S(V) will also be a super Hopf algebra. Notice that S(V) is supercommutative since for all x,y ∈ V 0 = x⊗y −(−1)p(x)p(y)y ⊗x =⇒ x⊗y = (−1)p(x)p(y)y ⊗x. Example 4.5 (Group Algebra). Given a group G. The group algebra of G is the vector space (cid:40) (cid:41) (cid:88) K[G] = a g | a ∈ K ∀g ∈ G , g g g∈G with multiplication defined for all a ,b ∈ K and g,h ∈ G to be g h (a g)·(b h) = (a b )gh g h g h and extended by linearity. Now to make this an example of a Hopf algebra, we define ∆: K[G] → K[G]⊗K[G], (cid:15): K[G] → K, and S: K[G] → K[G] for all g ∈ G to be ∆(g) = g ⊗1+1⊗g, (cid:40) 1 if g = e , G (cid:15)(g) = 0 if g (cid:54)= e , G S(g) = g−1. Given these definitions, it is straightforward to verify that these maps satisfy the conditions of Definition 4.1. Thus K[G] is a Hopf algebra. Throughout the rest of this work, we will need some commonplace identifications and the following lemma will provide us with the proofs of such identifications. Lemma 4.6. (i) If A is finite-dimensional vector space, then ϕ: (A∗ ⊗ A∗) → (A ⊗ A)∗ defined for all f ⊗g ∈ A∗⊗A∗ and a⊗b ∈ A⊗A via ϕ(f ⊗g)(a⊗b) = f(a)g(b) is an isomorphism when extended by linearity. (ii) Similarly, the map ψ: K∗ → K defined for all f ∈ K∗ via ψ(f) = f(1) is an isomorphism.