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“On a Hopf Algebra in Graph Theory” ([5]) AN ALGEBRAIC RELATION ON GRAPHS KOLYA MALKIN, MATH 336 TERM PAPER The paper of S.K. Lando \On a Hopf Algebra in Graph Theory" ([5]) defines a Hopf algebra structure on the vector space with basis the set of finite graphs and introduces a quotient Hopf algebra in order to study a ring of Tutte-like graph invariants indirectly motivated from knot theory. The aim of the present paper is to provide background for and clarify the construction of these Hopf algebras and to introduce another algebraic structure which could be used to study related graph invariants. 1. Introduction 1.1. Some background from algebra. This section summarizes the facts and definitions from algebra which shall later be used. More details can be found in texts such as [1]. A diagram is a diagram showing objects (algebraic structures such as groups, algebras, etc.) and arrows indicating functions between objects. Concatenation of paths along the arrows indicates composition of functions, so a path from one object to another along the arrows corresponds to a function from the starting point of the path to its ending point. A diagram commutes (and is called a commutative diagram) if the functions along any two paths between two objects in the diagram are the same. For example, in the diagram f A / B ; ' χ / C g D one could get from A to D via B or via C. The diagram commutes if and only if χ ◦ f = g ◦ '. The Cartesian product of two sets A and B is defined to be A × B = f(a; b): a 2 A; b 2 Bg : A function m : A × A ! A can be viewed as a binary operation ∗ on A, so m(a1; a2) = a1 ∗ a2. 1.1.1. Groups. A group (G; m; h; e) is a set G together with functions m : G × G ! G and h : G ! G and an element e 2 G satisfying the following properties. We write m with multiplicative notation, ab = m(a; b), and write a−1 for h(a). (1) The operation is associative: g1(g2g3) = (g1g2)g3. (2) e is the identity element: ge = eg = g. (3) Each element has an inverse: gg−1 = g−1g = e. These properties can be restated in terms of diagrams. This will help to make clearer the definition of algebras below. (1) For all g1; g2; g3 2 G, m(g1; m(g2; g3)) = m(m(g1; g2); g3). That is, the following diagram commutes: m×id G × G × G / G × G : id×m m × / G G m G 1 (2) For all g 2 G, m(g; e) = m(e; g) = g. That is, the following diagram commutes: id×e / G JJ G × G : JJ JJ e×id JJ m JJ id J% × / G G m G (3) For all g 2 G, m(g; h(g)) = m(h(g); g) = e. That is, the following diagram commutes: id×h / G JJ G × G : JJ JJ h×id JJ m e JJ J% × / G G m G If the operation is commutative { g1g2 = g2g1 for all g1; g2 2 G { then the group is called an abelian group. By a slight abuse of notation, G often denotes the entire quadruple (G; m; h; e) which the group comprises. One writes 1G for the identity element of G. Let G and H be groups. A morphism of groups is a function ' : G ! H such that '(g1g2) = '(g1)'(g2) for all g1; g2 2 G. Note that the multiplication on the left side is the operation of G and the multiplication on the right side is the operation of H. It is trivial to derive from this definition that a morphism takes inverse elements to inverse elements and takes the identity to the identity. If a morphism G ! H is a bijection, it is called an isomorphism and the domain and codomain are ∼ isomorphic. The notation for this is G = H. A (right) group action of a group G on a set S is a function G × S ! S, written multiplicatively (g; s) 7! s · g, satisfying (1) s(ab) = (sa)b for all a; b 2 G and s 2 S; (2) s1G = s for all s 2 S. n n If G is a group and g 2 G, the order of g is the smallest positive integer n such that g = 1G.(g denotes gg : : : g, with n factors.) 1.1.2. Algebras. A unital associative algebra A (also, k-algebra) over a field k is a vector space over k with an operation m : A ! A, written with multiplicative notation, and an element e 2 A such that (1) m is associative. (2) e is an identity for m. (3) m is bilinear. That is, if c; d 2 k and u; v; w 2 A, then (cu+dv)w = c(uw) = d(vw) and w(cu+dv) = c(wu) + d(wv). There are many examples of unital associative algebras. The complex numbers are an algebra over the real numbers with the usual multiplication. However, R3 with the vector cross product does not satisfy properties (1) and (2) because it has no identity and is not associative. (For this reason, we call R3 an algebra over R, but not a unital associative one. Henceforth, \algebra" always refers to a unital associative algebra.) Note that the multiplication need not be commutative or invertible. A morphism of algebras is defined exactly like a morphism of groups: it is a linear map between the underlying vector spaces that also preserves the multiplication operation and multiplicative identity. 1.1.3. Direct sum. Let U and V be k-vector spaces. Their direct sum U ⊕ V is a k-vector space, equal, as a set, to U × V with operations defined in the following way: • If (u1; v1); (u2; v2) 2 U × V , then (u1; v1) + (u2; v2) = (u1 + u2; v1 + v2), • If (u; v) 2 U × V and c 2 k, then c(u; v) = (cu; cv). ∼ For example, for integers m and n, Rm ⊕ Rn = Rm+n. In general, the dimension of the direct sum of vector spaces is the sum of the dimensions of the direct summands. We may extend this definition to define the direct sum of algebras, defining the multiplication by (u1; v1)(u2; v2) = (u1u2; v1v2). 2 1.1.4. Tensor product. Let U and V be k-algebras. Their tensor product U ⊗ V is defined as follows. Let F be the k-vector space with basis U × V . Write u ⊗ v for the basis element (u; v). Such basis elements are called simple tensors. F is a very large vector space: its basis elements are all pairs of an element of U with an element of V . For example, if k = R and U = V = R1, then F does not even have countable dimension, having basis ∼ R1 × R1 = R2. We we would like to define addition and multiplication of simple tensors ina reasonable way: • Declare (u ⊗ w) + (v ⊗ w) = (u + v) ⊗ w and (w ⊗ u) + (w ⊗ v) = w ⊗ (u + v). • For c 2 k, declare c(u ⊗ v) = (cu) ⊗ v) = u ⊗ (cv). These relations give a smaller vector space which is the tensor product U ⊗ V . For example, consider again the case U = V = R1. With these relations, the basis of F is now generated by 1 ⊗ 1, since u ⊗ v = u(1 ⊗ v) = uv(1 ⊗ 1), and R1 ⊗ R1 is a one-dimensional vector space, isomorphic to R1! If U has basis fe1; : : : ; emg and V has basis ff1; : : : ; fng, then fei ⊗ fjg is a basis for U ⊗ V , which is, therefore, an mn-dimensional vector space. Thus, given two k-vector spaces or k-algebras of dimensions m and n, the tensor products produces another with dimension mn. 1.1.5. Coalgebras. Let us revisit the definition of a k-algebra. The condition that the multiplication map m should be bilinear can be restated. Instead of defining the map m : A × A ! A, we define a map m : A ⊗ A ! A and require m to be a morphism of vector spaces. It is clear from the definition of the tensor product that m defines a multiplication on A assuming the associativity and identity properties are satisfied. The associativity can also be restated: the following diagram must commute. m⊗id A ⊗ A ⊗ A / A ⊗ A id⊗m m ⊗ / A A m A Finally, we may view the identity element as a function e : C ! K and require the diagram id⊗e / A JJ A ⊗ A JJ JJ e⊗id JJ m JJ id J$ ⊗ / A A m A to commute. Some explanation is required for why the maps id ⊗ e and e ⊗ id in this diagram can be seen ∼ ∼ as maps A ! A ⊗ A. The reason is simply that A ⊗ k = A = k ⊗ A, which is obvious from the definition of the tensor product. (Notice that these diagrams are the same as in the ones for associativity and unit in the definition of a group above, but the symbol × has been replaced by the somewhat scarier symbol ⊗ to enforce the bilinearity.) From now on, use this new definition and say that (A; m; e) is a unital associative algebra if these conditions are satisfied. What happens if the arrows in these diagrams are reversed? This leads to the following definition. A counital coassociative coalgebra (A; µ, ϵ) over a field k is a vector space A over k, a morphism of vector spaces µ : A ! A ⊗ A and a morphism of vector spaces ϵ : A ! k such that the diagrams that follow commute.
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