On the structure and representations of the insertion-elimination Lie algebra
aa r X i v : . [ m a t h . QA ] N ov ON THE STRUCTURE AND REPRESENTATIONS OF THEINSERTION-ELIMINATION LIE ALGEBRA
MATTHEW SZCZESNY
Abstract.
We examine the structure of the insertion-elimination Lie alge-bra on rooted trees introduced in [CK]. It possesses a triangular structure g = n + ⊕ C .d ⊕ n − , like the Heisenberg, Virasoro, and affine algebras. Weshow in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight repre-sentations, and show that irreducible representations are uniquely determinedby a ”lowest weight” λ ∈ C . We show that each irreducible representation is aquotient of a Verma-type object, which is generically irreducible. Introduction
The insertion-elimination Lie algebra g was introduced in [CK] as a meansof encoding the combinatorics of inserting and collapsing subgraphs of Feynmangraphs, and the ways the two operations interact. A more abstract and universaldescription of these two operations is given in terms of rooted trees, which encodethe hierarchy of subdivergences within a given Feynman graph, and it is thisdescription that we adopt in this paper. More precisely, g is generated by twosets of operators { D + t } , and { D − t } , where t runs over the set of all rooted trees,together with a grading operator d . In [CK] g was defined in terms of its action ona natural representation C { T } , where the latter denotes the vector space spannedby rooted trees. For s ∈ C { T } , D + t .s is a linear combination of the trees obtainedby attaching t to s in all possible ways, whereas D − t .s is a linear combination ofall the trees obtained by pruning the tree t from branches of s . n + = { D + t } and n − = { D − t } form two isomorphic nilpotent Lie subalgebras, and g has a triangularstructure g = n + ⊕ C .d ⊕ n − as well as a natural Z –grading by the number of vertices of the tree t . The Hopfalgebra U ( n ± ) is dual to Kreimer’s Hopf algebra of rooted trees [K].This note aims to estblish a few basic facts regarding the structure and repre-sentation theory of g . We begin by showing that g is simple, which together withits infinite-dimensionality implies that it has no non-trivial finite-dimensional rep-resentations, and that any non-trivial representation is necessarily faithful. Wethen proceed to develop a highest-weight theory for g along the lines of [K1, K2]. Date : October 2007.
In particular, we show that every irreducible highest-weight representation of g is a quotient of a Verma-like module, and that these are generically irreducible.One can define a larger, ”two-parameter” version of the insertion-eliminationLie algebra e g , where operators are labelled by pairs of trees D t ,t (roughly speak-ing, in acting on C { T } , this operator replaces occurrences of t by t ). In the spe-cial case of ladder trees, e g was studied in [M, KM1, KM2]. The finite-dimensionalrepresentations of the nilpotent subalgebras n ± as well as many other aspects ofthe Hopf algebra U ( n ± ) were studied in [F]. Acknowledgements:
The author would like to thank Dirk Kreimer for manyilluminating conversations and explanations of renormalization as well as relatedtopics. This work was supported by NSF grant DMS-0401619.2.
The insertion-elimination Lie algebra on rooted trees
In this section, we review the construction of the insertion-elimination Liealgebra introduced in [CK], with some of the notational conventions introducedin [M].Let T denote the set of rooted trees. An element t ∈ T is a tree (finite, one-dimensional contractible simplicial complex), with a distinguished vertex r ( t ),called the root of t . Let V ( t ) and E ( t ) denote the set of vertices and edges of t ,and let | t | = V ( t )Let C { T } denote the vector space spanned by rooted trees. It is naturally graded,(2.1) C { T } = M n ∈ Z ≥ C { T } n where C { T } n = span { t ∈ T || t | = n } . C { T } is spanned by the empty tree, whichwe denote by . We have C { T } = < > C { T } = < • > C { T } = < •• > C { T } = < ••• , •• • > where <, > denotes span, and the root is the vertex at the top. If e ∈ E ( t ), bya cut along e we mean the operation of cutting e from t . This divides t into twocomponents - R c ( t ) containing the root, and P e ( t ), the remaining one. R e ( t ) and P e ( t ) are naturally rooted trees, with r ( R c ( t )) = r ( t ) and r ( P e ( t )) = (endpointof e). Note that V ( t ) = V ( R e ( t )) ∪ V ( P e ( t )). N THE STRUCTURE AND REPRESENTATIONS OF THE INSERTION-ELIMINATION LIE ALGEBRA3
Let g denote the Lie algebra with generators D + t , D − t , d , t ∈ T , and relations(2.2) [ D + t , D + t ] = X v ∈ V ( t ) D + t ∪ v t − X v ∈ V ( t ) D + t ∪ v t (2.3) [ D − t , D − t ] = X v ∈ V ( t ) D − t ∪ v t − X v ∈ V ( t ) D − t ∪ v t (2.4) [ D − t , D + t ] = X t ∈ T α ( t , t ; t ) D + t + X t ∈ T β ( t , t ; t ) D − t (2.5) [ D − t , D + t ] = d (2.6) [ d, D − t ] = −| t | D − t (2.7) [ d, D + t ] = | t | D + t where for s, t ∈ T , and v ∈ V ( s ) s ∪ v t denotes the rooted tree obtained byjoining the root of t to s at the vertex v via a single edge, and • α ( t t , t ; t ) = { e ∈ E ( t ) | R e ( t ) = t, P e ( t ) = t }• β ( t , t ; t ) = { e ∈ E ( t ) | R e ( t ) = t, P e ( t ) = t } Thus, for example[ D + • , D + •• • ] = D + •• • • + 2 D + ••• • − D + ••• • [ D −• , D − •• • ] = − D − •• • • − D − ••• • + D − ••• • [ D −• , D + •• • ] = 2 D + •• g acts naturally on C { T } as follows. If s ∈ T , viewed as an element of C { T } ,and t ∈ T , then D + t ( s ) = X v ∈ V ( s ) s ∪ v tD − t ( s ) = X e ∈ E ( s ) ,P e ( s )= t R e ( s ) d ( s ) = | s | s MATTHEW SZCZESNY Structure of g Let n + and n − be the Lie subalgebras s of g generated by D + t and D − t , t ∈ T .We have a triangular decomposition(3.1) g = n + ⊕ C .d ⊕ n − The relations 2.5, 2.6, and 2.7 imply that for every t ∈ T g t = < D + t , D − t , d > forms a Lie subalgebra isomorphic to sl . We have that g t ∩ g s = C .d if s = t .Assigning degree | t | to D + t , −| t | to D − t , and 0 to d equips g with a Z –grading. g = M n ∈ Z g n g possesses an involution ι , with ι ( D + t ) = D − t ι ( D − t ) = D + t ι ( d ) = − d Thus ι is a gradation-reversing Lie algebra automorphism exchanging n + and n − . Theorem 3.1. g is a simple Lie algebraProof. Suppose that
I ⊂ g is a proper Lie ideal. If x ∈ I , let x = P i x i , x i ∈ g i be its decomposition into homogenous components. We have[ d, x ] = X n nx n which implies that x n ∈ I for every n (because the Vandermonde determinant isinvertible) i.e. I = ⊕ n ∈ Z ( I ∩ g n ). Suppose now that x n ∈ g n , n >
0. We canwrite x n as a linear combination of n –vertex rooted trees(3.2) x n = X t ∈ T n α t · t We proceed to show that D + • ∈ I , where • is the rooted tree with one vertex.Let S ( x n ) ⊂ T n be the subset of n-vertex trees occurring with a non-zero α t in3.2. Given a rooted tree t , let St ( t ) denote the set of rooted trees obtained byremoving all the edges emanating from the root. Let St ( x n ) = [ s ∈ S ( x n ) St ( s )and let ξ ∈ St ( x n ) be of maximal degree. It is easy to see that [ D − ξ , x n ] is anon-zero element of g n −| ξ | . Starting with x n ∈ n + , x n = 0, and repeating thisprocess if necessary, we eventually obtain a non-zero element of g = < D + • > .Now, [ D −• , D + • ] = d , and since [ d, g ] = g ], this implies I = g . We have thus shownthat if I is proper, then I ∩ n + = 0 N THE STRUCTURE AND REPRESENTATIONS OF THE INSERTION-ELIMINATION LIE ALGEBRA5
Applying ι shows that I ∩ n − I ∩ C .d = 0. (cid:3) We can now use this result to deduce a couple of facts about the representationtheory of g . Corollary 3.1. If V is a non-trivial representation of g , then V is faithful. Corollary 3.2. g has no non-trivial finite-dimensional representations. The latter can also be easily deduced by analyzing the action of the sl subal-gebras g t as follows. Suppose that V is a finite-dimensional representation of g .To show that V is trivial, it suffices to show that it restricts to a trivial represen-tation of g t for every t ∈ T . This in turn, will follow if we can show that for a single tree t ∈ T , g t acts trivially, because this implies that d acts trivially, and C .d ⊂ g t plays the role of the Cartan subalgebra. Let V = M i =1 ··· k V δ i be a decomposition of V into d –eigenspaces - i.e. if v ∈ V δ i , then d.v i = δ i v . Since V is finite-dimensional, the set { δ i } is bounded, and so lies in a disc of radius R in C . If v ∈ V δ i then [ d, D + t ] = | t | D + t implies that D + t .v ∈ V δ i + | t | . Choosing a t ∈ T such that | t | > R shows that D + t .v = 0 for every v ∈ V .3.1. Lowest-weight representations of g . We begin by examining the ”defin-ing” representation C { T } of g introduced in section 2. Its decomposition into d –eigenspaces is given by 2.1. Given a representation V of g on which d is diag-onalizable, with finite-dimensional eigenspaces, and writing V = M δ V δ for this decomposition, we define the emphcharacter of V , char ( V, q ) to be theformal series char ( V, q ) = X δ dim ( V δ ) q δ The case V = C { T } , where dim ( V n ) is the number of rooted trees on n vertices,suggests that representations of g may contain interesting combinatorial infor-mation. The triangular structure 3.1 of g suggests that a theory of highest– orlowest–weight representations may be appropriate. Definition 3.1.
We say that a representation V of g is lowest–weight if thefollowing properties hold(1) V = ⊕ V δ is a direct sum of finite-dimensional eigenspaces for d .(2) The eigenvalues δ are bounded in the sense that there exists L ∈ R suchthat Re ( δ ) ≥ L . MATTHEW SZCZESNY
We call the δ the weights of the representation, and category of such represen-tations O . If V ∈ O , we say v ∈ V δ is a lowest-weight vector if n − v = 0. Since D − t decreases the weight of a vector by | t | , and the weights all lie in a half-plane,it is clear that every V ∈ O contains a lowest-weight vector.Recall that a representation V of g is indecomposable if it cannot be written as V = V ⊕ V for two non-zero representations. Let U ( h ) denote the universalenveloping algebra of a Lie algebra h . Lemma 3.1. If v ∈ V λ is a lowest-weight vector, then U ( n + ) .v is an indecom-posable representation of g Proof. U ( g ) .v is clearly the smallest sub-representation of V containing v . Thedecomposition 3.1 together with the PBW theorem implies that U ( g ) = U ( n + ) ⊗ C [ d ] ⊗ U ( n − )Because v is a lowest-weight vector, C [ d ] ⊗ U ( n − ) .v = C .v . It follows that U ( g ) .v = U ( n + ) .v . That the latter is indecomposable follows from the fact thatin U ( n + ) .v , the weight space corresponding to λ is one-dimensional, and so if U ( n + ) .v = V ⊕ V , then v ∈ V or v ∈ V . (cid:3) Observe that U ( n + ) .v = ⊕ ( U ( n + ) .v ) λ + k , k ∈ Z ≥ where ( U ( n + ) .v ) λ + k is spanned by monomials of the form(3.3) D + t D + t · · · D + t i .v with | t | + · · · | t i | = k .The category O contains Verma-like modules. For λ ∈ C , let C λ denote theone-dimensional representation of C .d ⊕ n − on which n − acts trivially, and d actsby multiplication by λ . Definition 3.2.
The g –module W ( λ ) = U ( g ) ⊗ C [ d ] ⊗ U ( n − ) C λ will be called the Verma module of lowest weight λ .Choosing an ordering on trees yields a PBW basis for n + , and thus also a basisof the form 3.3 for W ( λ ).Given a representation V ∈ O , and a lowest weight vector v ∈ V λ , we obtain amap of representations(3.4) W ( λ ) V v Lemma 3.2. If V ∈ O is an irreducible representation, then V is the quotient ofa Verma module. N THE STRUCTURE AND REPRESENTATIONS OF THE INSERTION-ELIMINATION LIE ALGEBRA7
Proof.
Since V ∈ O , V possesses a lowest-weight vector v ∈ V λ for some λ ∈ C .Since V is irreducible, V = U ( g ) .v = U ( n + ) .v . The latter is a quotient of W ( λ ). (cid:3) We have
Char ( W ( λ )) = q λ X n ∈ Z ≥ dim ( C { T } n ) q n = q λ Y n ∈ Z ≥ − q n ) P ( n ) where P ( n ) is the number of primitive elements of degree n in H K .3.2. Irreducibility of W ( λ ) . It is a natural question whether W ( λ ) is irre-ducible. In this section we prove the following result: Theorem 3.2.
For λ outside a countable subset of C containing , W ( λ ) isirreducible.Proof. Let v = 0 be a basis for W ( λ ) λ . W ( λ ) contains a proper sub-representationif and only if contains a lowest-weight vector w such that w / ∈ C .v . In W (0), D + • .v ∈ W (0) is a lowest-weight vector, since D −• D + • .v = D + • D −• .v + d.v = 0and D − t .v = 0 for all t ∈ T with | t | ≥ W (0) is not irreducible.If I = ( t , · · · , t k ) is a k –tuple of trees such that t (cid:22) t (cid:22) · · · (cid:22) t k in the chosen order, let D + I .v denote the vector(3.5) D + t k · · · D + t .v ∈ W ( λ ) w ∈ W ( λ ) λ + n is a lowest-weight vector if and only if(3.6) D − t .w = 0for all t such that | t | ≤ n . Writing w in the basis 3.5 w = X | I | = n α I D + I .v the conditions 3.6 translate into a system of equations for the coefficients α I . Forexample, if w ∈ W ( λ ) λ +2 , then w = α D + •• .v + α D + • D + • .v MATTHEW SZCZESNY and conditions D − •• .v = 0, D −• .w = 0 translate into λα + λα = 0 α + (2 λ + 1) α = 0The determinant of the corresponding matrix is 2 λ , and so for λ = 0, there is nolowest-weight vector w ∈ W ( λ ) λ +2 . For a general n , the system can be writtenin the form ( A + λB )[ α I ] = 0where A and B are matrices whose entries are non-negative integers. Let f n ( λ ) = dim ( Ker ( A + λB ))Then for every r ∈ N S n,r = { λ ∈ C | f n ( λ ) ≥ r } if proper, is a finite subset of C , since the condition is equivalent to the vanishinga finite collection of sub-determinants, each of which is a polynomial in λ . Theset of λ ∈ C for which W ( λ ) is irreducible is therefore [ n ∈ N { C \ S n, } The theorem will follow if S n, is proper for each n ∈ N . This follows from thefollowing Lemma. (cid:3) Lemma 3.3. Z (1) is irreducible.Proof. We begin by examining the representation C { T } . The degree 0 subspace C . g . Let M denote the quotient C { T } / C .
1. It iseasily seen that the exact sequence0 C C { T } 7→ M M has highest weight 1, and the subspace M can be identified withthe span of the tree on one vertex • . By the universal property of Verma modules,3.4 we have a map(3.7) W (1) M sending the lowest-weight vector of W (1) to • . Now, W (1) n is spanned by allvectors 3.5 such that | t | + · · · | t k | = n −
1, and so can be identified with the set offorests on n − M n can be identified with C { T } n . The operationof adding a root to a forest on n − n vertices yields an isomorphism W (1) n ∼ = M n . Thus, if the map 3.7 is a surjection,it is an isomorphism. This in turn, follows from the fact that M is irreducible. N THE STRUCTURE AND REPRESENTATIONS OF THE INSERTION-ELIMINATION LIE ALGEBRA9
It suffices to show that M n contains no lowest-weight vectors for n >
1. Thisfollows from an argument similar to the one used to prove 3.1. Let w ∈ M n , andwrite w = α t + · · · α k t k where | t i | = n and we may assume that α i = 0. In the notation of 3.1, let ξ ∈ St ( w ) be of maximal degree. Then D − ξ .w = 0Thus, M is irreducible, and hence isomorphic to W (1) by the map 3.7. (cid:3) References [CK] Connes, A.; Kreimer, D. Insertion and elimination: the doubly infinite Lie algebra ofFeynman graphs.
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Department of Mathematics Boston University, Boston, MA 02215
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