October 21, 2009
1 The Category The Hopf Algebra 2 3
Properties of the category A The Category The Hopf Algebra This recap is my attempt to distill the precise conditions required in the exposition by Kremnizer & Szczesny [3]. Requirements for the algebra structure There is a zero (initial and terminal) object,. Ob(A) is an abelian monoid with the zero object as identity. Every morphism has a kernel. Every morphism has a cokernel. Hom(M, N), Ext 1 (M, N) < for all objects M and N. Iso(A), the collection of isomorphism classes in Ob(A), is a set. Categorical consequences There are exact sequences, subobjects, and quotient objects. The Grothendieck group K(A) exists.
The Category The Hopf Algebra More Properties of the category A Requirement for the cocommutative coalgebra structure Ob(A) is a unique factorization monoid, i.e., decomposition of objects satisfies a Krull-Schmidt theorem. Requirement for the cocommutative bialgebra structure I m not sure. Maybe nothing else?
Some Notation Outline The Category The Hopf Algebra For L, M, N Ob(A), let GL,N M denote the (necessarily finite) set of all short exact sequences of the form L M N Set gl,n M G = M L,N and am = Aut(M). Lemma g M L,N a L a N = { R M R = L, M/R = N} Proof. Aut(L) Aut(N) acts freely on GL,N M, and the quotient is canonically identified with the right-hand side.
The Category The Hopf Algebra The Ringel-Hall algebra H A The Vector Space, Product, and Unit H A = [L] [N] = [M] Iso(A) [M] Iso(A) Q[M] g M L,N a L a N [M] The multiplicative unit is [ ]. The Vector Space, Product, and Unit (alternative description) { } H A = f : Iso(A) Q supp(f ) < (f g)(m) = L M f (L)g(M/L) The multiplicative unit is δ [ ], the characteristic function of [ ].
The Cocommutative Coalgebra The Category The Hopf Algebra The Coproduct and Counit ɛ ([M]) = L N =M [L] [N] ɛ = δ,m The Coproduct and Counit ɛ (alternative description) (f )(L N) = f (L N) ɛ(f ) = f ([ ])
The Cocommutative Bialgebra The Category The Hopf Algebra Theorem The coproduct is an algebra map Theorem The counit ɛ is an algebra map It is very easy to check that is compatible with. H A is therefore a graded connected bialgebra. Kremnizer & Szczesny
The Hopf algebra Outline The Category The Hopf Algebra Corollary H A is a cocommutative Hopf algebra. Proof. Every graded connected bialgebra has an antipode. Now the Milnor-Moore theorem gives us Corollary H A is the universal enveloping algebra of its Lie algebra of primitive elements.
Definitions (Yeats [5]) Outline Graphs and Physical theories A graph consists of a set H of half edges, a set V of vertices where each half edge is adjacent to exactly one vertex and to either one half edge (internal edges) or no half edge (external edges). A combinatorial physical theory T consists of a set of half edge types, a set of edge types, and a set of vertex types. The edges and vertices have weights. (The edges are particles; the vertices are interactions.) A Feynman graph in a combinatorial physical theory T is a graph where the half edges have been given types from T and the resulting edges and vertices are also valid types in T.
(QED) Quantum electrodynamics QED describes photons and electrons interacting electromagnetically. Three half-edge types: half-photon, front half-electron, back half-electron. Two edge types: photon (wt 2), electron (wt 1) One vertex type: one of each half-edge type is incident (wt 0). A vertex and some Feynman graphs are shown below. e e γ
(QCD) Quantum chromodynamics QCD describes the interactions of quarks and gluons. Five half-edge types: half-gluon, front half-fermion, back half-fermion, front half-ghost, back half-ghost. Three edge types: gluon (wt 2), fermion (wt 1), ghost (wt 1) Four vertex types: shown below (wts from left to right 0, 0, 1, 0).
(scalar) These are two of the simplest renormalizable 1 theories, and are often used as examples in textbooks. φ 3 One half-edge type. One edge type (wt 2). One vertex type: has degree three (wt 0). φ 4 One half-edge type. One edge type (wt 2). One vertex type: has degree four (wt 0). 1 See page on Renormalizable theories in section.
Contraction G/γ Outline Suppose G is a Feynman graph in a theory T and γ is a subgraph of G. We ll contract G one connected component of γ at a time, so we may as well assume γ is connected. γ has type V external leg structure Replace all vertices and internal edges of γ with vertex v of type V. Connect all external edges of γ to v. γ has type E external leg structure Remove all vertices and half edges of γ. Connect the two half edges adjacent to the external edges of γ if they exist.
Contraction examples in QED Reading each display from left to right we have G, γ, G/γ. External leg structure V External leg structure E (from Yeats [5])
Insertion G v γ and G e γ Insertion is the opposite of contraction. Suppose G and γ are Feynman graphs in a theory T with γ connected. γ has type V external leg structure, v in G has same type Replace v with γ. Identify external edges of γ with half edges incident with v. γ has type E external leg structure, e in G has same type Cut e. Splice γ into cut.
Insertion examples in QED and φ 4 (from Yeats [5])
Divergence Outline We only consider Feynman graphs that are 2-connected. We associate a dimension D of space-time to a physical theory T. Definition Let L G be the loop number of the Feynman graph G, and let W G be the sum of the weights of all of the vertices and edges of G. The superficial degree of divergence of G is D L G W G We say G is divergent if this quantity is nonnegative. The superficial degree of divergence encodes how badly the integral associated to the graph diverges for large values of the integration variables. Yeats
Renormalizable theories Outline Definition The theory T in dimension D is renormalizable if graph insertion never changes the superficial degree of divergence of a graph. Space-time dimensions Our sample physical theories are renormalizable in the following space-time dimensions. QED D = 4 QCD D = 4 φ 3 D = 6 φ 4 D = 4
Computations of superficial degree of divergence
Category of labeled Feynman graphs F Objects Ob(F) = { labeled Feynman graphs } { } Morphisms Hom(G, G ) = {(γ, γ, f ) γ is a subgraph of G γ is a subgraph of G f : G/γ = γ } A little thought shows that this category satisfies all the properties listed on slides 3 and 4, so we have a Ringlel-Hall algebra H F that is a cocommutative Hopf algebra, and so the universal enveloping algebra of its Lie algebra of primitive elements ( P = 1 P + P 1).
The Ringel-Hall Hopf algebra H F Hall product G G = G a(g, G, G )G where { a(g, G, G ) = γ G γ = G and G /γ = G } Hall coproduct (G) = G G G G =G We want to look at the relationship of this algebra to the Connes-Kreimer Lie algebra.
Example of Hall product
Connes-Kreimer Lie algebra on Feynman Graphs is defined by the: Pre-Lie product { G G v = G v G if external leg structure of G is type V e G e G if external leg structure of G is type E This gives us the Lie bracket: Lie algebra [G, G ] = G G G G
A bracket computation (Kremnizer & Szczesny [3])
The isormorphism Outline Theorem The Lie algebras defined by the products and are isomorphic. Proof. The map is G Aut G G. See Connes-Kreimer [1] for details. And so, the Ringel-Hall algebra H F is the enveloping algebra of the Connes-Kreimer algebra on Feynman graphs.
Outline 1 A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann- Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys. 210 (2000), no. 1, 249273. 2 J.A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120, 361-377 (1995). 3 K. Kremnizer, M. Szczesny, Feynman graphs, rooted trees, and Ringel-Hall algebras, to appear in Comm. Math. Phys., arxiv:0806.1179. 4 O. Schiffmann, Lectures on Hall algebras. Preprint math.rt/0611617. 5 K. Yeats, Growth estimates for Dyson-Schwinger equations. Ph.D. thesis. Boston University, 2008.