SLIDE 1
Lie Algebra of Differential Operators on Path Algebras
Fang Li (joint work with Li Guo) Zhejiang University 1
SLIDE 2 Differential Algebra
◮ A differential algebra is an associative algebra R together with a
linear operator d : R → R such that d(xy) = d(x)y + xd(y).
◮ Differential algebra originated from the algebraic study of differential
equations (Ritt and Kolchin) and is a natural yet profound extension
- f commutative algebra and the related algebraic geometry.
Differential algebra has also found important applications, such as to arithmetic geometry, logic and computational algebra, especially in the work of W. T. Wu on mechanical theorem proving in geometry.
◮ More recently, ideas of differential Galois theory has been applied in
the work of Connes and Marcolli on renormalization of QFT and motivic Galois groups. 2
SLIDE 3 Integral algebra and Rota-Baxter algebra
◮ In opposite to the differential operator, there is the integral operator
P : R → R such that P(x)P(y) = P(xP(y)) + P(P(x)y), ∀x, y ∈ R.
◮ In the Lie algebra context, this is the operator form of the classical
Yang-Baxter equation: [P(x), P(y)] = P[x, P(y)] + P[P(x), y], ∀x, y ∈ g.
◮ There is also the more general Rota-Baxter operator P : R → R:
P(x)P(y) = P(xP(y)) + P(P(x)y) + λP(xy), ∀x, y ∈ R, where λ is a fixed constant, and Baxter is an American mathematician.
◮ This concept has appeared in the Connes-Kreimer Hopf algebra
approach to renormalization of QFT (Connes and Kreimer, Comm.
- Math. Phy. (1999-2003), Ebrahimi-Fard, Guo and Kreimer, Integrable
renormalization I,II, J. Math. Phy. (2004), Ann. H. Poincare (2005),
- Comm. Math. Phy. (2006), Guo and Zhang, J. Algebra (2008)).
◮ Even more general is the concept of an O-operator, again applicable
to integrable systems (Bai, J. Phy. A (2007), Bai, Guo and Ni: Comm.
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SLIDE 4
Back to differential algebra
◮ Recently, differential algebra has found combinatorial connections.
For instance, differential structures were found on heap ordered trees (Grossman and Larson, Adv. Appl. Math, 2005) and on decorated rooted trees (Guo and Keigher, J. Pure and Appl. Algebra, 2008).
◮ In the current work, we consider differential algebra structures on
another combinatorially defined objects, namely the path algebras of quivers (Fang Li, J. Algebra (2009), · · · ).
◮ This gives a natural class of differential algebras of finite and infinite
dimensions.
◮ Most of the study on differential algebra up to date have been for
commutative algebras and fields. This paper can be regarded as a first step to extended the study to noncommutative algebras by studying their differential aspects.
◮ Through the realization of basic algebras and Artin algebras as
quotients of path algebras (Gabriel Theorem) and generalized path algebras, we hope this study will lead to the study of differential basic algebras and Artin algebras. 4
SLIDE 5 Quivers and their path algebras
◮ Recall that a quiver Γ is (algebraically) defined to be a quadruple
(Γ0, Γ1, h, t) where Γ0 is a set (of vertices), Γ1 (of edges) and map h, t : Γ1 → Γ0, giving the head h(p) and the tail t(p) of p. An arrow is a triple (t(p), p, h(p)), or more intuitively, •t(p)
p
− → •h(p).
◮ A path p in Γ is either a vertex p = v ∈ Γ0 or a sequence
(composition) of arrows p := •t(p1)
p1
− → •h(p1)=t(p1)
p2
− → •t(p2)=t(p3) · · · •h(pk−1)=t(pk )
pk
− → •h(pk ). Define t(p) = t(p1), h(p) = t(pk) and ℓ(p) = k.
◮ An oriented cycle is a path p with h(p) = t(p). ◮ Let P denote the set of paths of Γ. Define the product of two paths p
and q to be the composition pq if h(p) = t(q) and to be zero
- therwise. This defines an associative algebra structure on the linear
space kΓ := kP, called the path algebra of Γ. 5
SLIDE 6
Differential operators on a path algebra
◮ For an algebra A, we have two Lie algebras:
(i) The Lie algebra Lie(A) = (A, [, ]) with [x, y] = xy − yx ∀x, y ∈ A. (ii) The derivation Lie algebra Der(A) = {D ∈ End(A) | D(uv) = uD(v) + D(u)v ∀u, v ∈ A} with the Lie multiplication [D1, D2] := D1 ◦ D2 − D2 ◦ D1.
◮ For a ∈ A, we have an inner derivation
Da : A → A, Da(b) = (ada)(b) := ab − ba, b ∈ A. This gives a Lie algebra homomorphism D : Lie(A) → Der(A), D(a) = Da, a ∈ A.
◮ IDiff(A) := imD ⊆ Der(A) is a Lie ideal and ker D is precisely the
center C(A) of A. Denote EDiff(A) = Der(A)/IDiff(A).
◮ Our main interest is Diff(kΓ) = Der(kΓ) for a quiver Γ. ◮ It is well-known that C(kΓ) = k[x] if Γ is an oriented loop •v p
− → •v, C(kΓ) = k otherwise.
◮ Thus unless Γ is a loop, we have IDiff(kΓ) = kΓ/k ֒
→ Der(kΓ). 6
SLIDE 7 Questions to consider
◮ Existence of non-zero differential operators on kΓ; ◮ Structure of Der(kΓ); ◮ Relationship between the combinatorial structure of Γ and Der(kΓ)
and ODiff(kΓ).
◮ Structure on ODiff(kΓ); ◮ We will see that all the answers depend on an explicitly given basis
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SLIDE 8 Characterization of a derivation
◮ Recall that P is a k-basis of kΓ. So for a k-linear map D : kΓ → kΓ is
determined by the coefficients cp
q ∈ k in
D(p) =
cp
qq,
p ∈ P. Convention: the sum over an empty set is defined to be zero.
◮ Theorem 1. D : kΓ → kΓ is a differential operator if and only if D is
defined by D(v) =
ct(q)
q
q −
ct(q)
q
q for v ∈ Γ0, D(p) =
- q∈P\V,h(q)=t(p),t(q)=t(p)
ct(q)
q
qp +
- q∈P\V,t(q)=h(p),h(q)=h(p)
ct(q)
q
pq +
k
- i=1
- qi∈P\V,t(qi)=t(pi),h(qi)=h(pi)
cpi
qi p1 · · · pi−1qipi+1 · · · pk,
where P\Γ0 ∋ p = p1 · · · pk is the decomposition of p into arrows. 8
SLIDE 9
Standard basis of Der(kΓ)
◮ For r, s ∈ P\Γ0 with r ∈ Γ1 and h(s) = h(r), t(s) = t(r), define
Dr,s : kΓ → kΓ by the conditions (i). Dr,s(q) = δr,qs, q ∈ Γ0 ∪ Γ1. (ii). Dr,s(q1q2) = Dr,s(q1)q2 + q1Dr,s(q2), q1, q2 ∈ P.
◮ Then Dr,s defines a differential operator on kΓ. ◮ Theorem 2. Denote
B1 := {Ds | s ∈ P, h(s) = t(s)}. B2 := {Dr,s | r, s ∈ P, \Γ0 with r ∈ Γ1 and h(s) = h(r), t(s) = t(r)}. Then B := B1 ∪ B2 is a basis of Der(kΓ). 9
SLIDE 10 Existence of derivations
◮ Theorem 3. Der(kΓ) = 0 if and only if Γ contains an arrow. ◮ Proof. (⇐). If Γ contains an arrow p, then by Theorem 2, Dp,p is a
non-zero differential operator on kΓ. (⇒). If Γ does not contain any arrow, then a differential operator D on kΓ is determined by D(v), v ∈ Γ0. By Theorem 1, D(v) =
ct(q)
q
q −
ct(q)
q
q. Since Γ does contain any arrow, this sum is over an empty set. So D(v) = 0, ∀v ∈ Γ0 and hence D = 0. 10
SLIDE 11 Structure of the Lie algebra Der(kΓ)
◮ We list some sub-structures of Diff(kΓ) and their relations as follows: ◮ Indiff(kΓ) := ad(kΓ) = {Ds | s ∈ kΓ}. ◮ ODiff(kΓ) := Der(kΓ)/InDiff(kΓ). ◮ D1 := kB1 =the subspace of IndiffkΓ generated by
B1 := {Ds | s ∈ P, h(s) = t(s)}.
◮ D0 := kB0 =the subspace of D1 generated by
B0 := {Ds | s ∈ P, h(s) = h(t), but ℓ(s) ≥ 1}.
◮ DV := kBV =the subspace of D1 generated by BV := {Dv : v ∈ Γ0}. ◮ D2 := kB2 =the subspace of Diff(kΓ) generated by
B2 := {Dr,s | r, s ∈ P\Γ0, ℓ(r) = 1, h(s) = h(r), t(s) = t(r)}.
◮ Theorem 4. We have the commutative diagram of exact sequences
0 − − − − → Indiff(kΓ) − − − − → Diff(kΓ) − − − − → ODiff(kΓ) − − − − → 0
− − − → D0 + DV − − − − → D2 − − − − → D2/(D0 + DV) − − − − → 0 11
SLIDE 12
ODiff(kΓ) and the combinatorics of Γ
◮ In 0 −
− − − → IndiffkΓ − − − − → DiffkΓ − − − − → ODiff(kΓ) − − − − → 0 InDiff(kΓ) ∼ = kΓ/k (except when Γ is a oriented cycle), so it pretty much recovers the algebra structure of kΓ. One is wondering what extra information could ODiff(kΓ) provide.
◮ If Γ has an oriented cycle, then all the Lie algebras involved are
infinite dimensional. So we consider quivers without oriented cycles.
◮ Let Γ be a planar quiver with a fixed embedding into R2. A primitive
cycle is an unoriented cycle that contains no other unoriented cycles. Let Γp be the set of primitive cycles of Γ and let γp be its cardinality. It is one less than the number of connected components of R\Γ, so describes the topology of Γ.
◮ An almost oriented cycle is a pair (p, r) where p ∈ Γ1 and r ∈ P with
h(r) = h(p), t(r) = t(p) such that, for the inverse arrow p∗ of p, p∗ r is an oriented cycle. Let Γa be the set of almost oriented cycles of Γ and let γa be its cardinality.
◮ Theorem 5. Let Γ be a planar quiver with no oriented cycles. Then
dimk ODiff(kΓ) = γp + γa. 12
SLIDE 13
Structure of ODiff(kΓ)
◮ Theorem 5 suggests a canonical basis of
ODiff(kΓ) = Der(kΓ)/InDiff(kΓ).
◮ For each primitive cycle p consisting of a list of arrows (p1, · · · , pk),
define Dp = ±Dp1,p1 ± · · · ± Dpk,pk, where a ±pi is +pi if pi is in the clockwise direction and is −pi otherwise.
◮ Theorem 6. The set
{Dp | p ∈ Γp} ⊔ {Dp,r | (p, r) ∈ Γa} is a basis of ODiff(kΓ).
◮ The Lie algebra structure of ODiff(kΓ) can be given in terms of this
basis.
◮ THANK YOU!
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