On the $q$-analogue of the Maurer-Cartan equation

0.13 MB·English

by  Mauricio Angel

On the q-analogue of the Maurer-Cartan equation
6
Mauricio Angel and Rafael D´ıaz
0
0
2 February 2, 2008

n
a
J

8 Abstract
2
We consider deformations of the differential of a q-differential graded algebra.
]
A We prove that it is controlled by a generalized Maurer-Cartan equation. We find
Q explicit formulae for the coefficients ck involved in that equation.
.
h
t
a Introduction
m
[
Following the work of Kapranov [6] the interest on the theory of N-complexes has risen.

1
A delicate issue in the theory is the appropriate definition of a N-differential algebra. At
v
8 the moment there exists two options for such a definition, each describing very distinct
9
6 mathematical phenomena.
1
In [2] the notion of N-differential graded algebra A has been defined as follows: A
0
6 must be a graded associative algebra provided with an operator d : A → A of degree 1
0
/ such that d(ab) = d(a)b+(−1)a¯ad(b) and dN = 0.
h
t The study of the deformations of the differential of a N-dga has also been initiated
a
m in [2]. Applications to differential geometry are given in [3]. Applications to the study
:
v of A∞-algebras of depth N, are contained in [4].
i
X The other possible N-generalization of the notion of a differential algebra (see
r [7],[8],[9]) goes as follows: One fixed a primitive N-th root of unity q. One considers
a
q-differential graded algebras A defined by: A must be a graded associative algebra pro-
vided with an operator d : A → A of degree 1 such that d(ab) = d(a)b + qa¯ad(b) and
dN = 0.
In this paper we consider deformations of the differential of a q-differential graded
algebra. We prove that it is controlled by a generalized Maurer-Cartan equation. We
find explicit formulae for the coefficients c involved in that equation.
k
1
1 q-differential graded algebras and modules
Let k be a commutative ring with unity. For q ∈ k the q-numbers are given by [ ] :
q
N → k, k 7−→ [k] = 1 + q + ···qk−1 and [k] ! = [1] [2] ···[k] . We assume that on
q q q q q
k, for N ∈ N fixed, we can choose q ∈ k such that [N] = 0 and [k] is invertible for
q q
1 ≤ k ≤ N −1, in this case we refer to q as a primitive N-th root of unity.
Definition 1 A q-differential graded algebra or q-dga over k is a triple (A•,m,d)
where m : Ak ⊗Al → Ak+l and d : Ak → Ak+1 are k-modules morphisms satisfying
1) (A•,m) is a graded associative k-algebra.
2) d satisfies the q-Leibniz rule d(ab) = d(a)b+qa¯ad(b).
3) If q is a primitive N-th root of unity, then dN = 0, i.e., (A•,d) is a N-complex.
Definition 2 Let (A•,m ,d ) be a q-dga and M• be a k-module. A q-differential
A A
module (q-dgm) over (A•,m ,d ) is a triple (M•,m ,d ) where m : Ak ⊗ Ml →
A A M M M
Mk+l and d : Mk → Mk+1 are k-module morphisms satisfying the following properties
M
1. m (a,m (b,φ)) = m (m (a,b),φ), for all a,b ∈ A• and φ ∈ M•. We denote
M M M A
m (a,φ) by aφ, for all a ∈ A and φ ∈ M.
M
2. d (aφ) = d (a)φ+qa¯ad (φ) for all a ∈ A• and φ ∈ M•.
M A M
3. If q is a primitive N-th root of unity, then dN = 0.
M
2 N-curvature
Let (M•,m ,d ) be a q-dgm over a q-dga (A•,m ,d ). For a ∈ M1 we define a
M M A A
deformation of d as follows
M
Dφ = d φ+aφ, for all φ ∈ M•.
M
In order to obtain a q-dgm structure on (M•,m ,D) we require that DN = 0. In [KN]
M
the consecutive powers of D are calculated and the following formula is obtained
N−1
N
DNφ = dNφ+ (Dk−1a)dN−kφ+(DN−1a)φ,
M (cid:18)k(cid:19)
Xk=1 q
where N is the q-binomial coefficients given by
k q
(cid:0) (cid:1)
N [N] !
q
= .
(cid:18)k(cid:19) [N −k] ![k] !
q q q
2
For q a primitive N-th root of unity the formula above reduces to
DNφ = (DN−1a)φ, (1)
because in this case dN = 0 and N = 0 for 0 < k < N, since [N] = 1+q+···+qN−1 =
M k q q
1−qN = 0. (cid:0) (cid:1)
1−q
Formula (1) is the first step towards the construction of the q-analogue of the Maurer-
Cartan equation. A further step is required in order to write down DN−1a in terms of
d and a only. We shall work with a slightly more general problem: finding an explicit
M
expression for DNφ given in terms of d and a only. First we review some notations
M
from [AD].
For s = (s ,...,s ) ∈ Nn we set l(s) = n and |s| = s . For 1 ≤ i < n, s denotes
1 n i i >i
the vector given by s = (s ,...,s ), for 1 < i ≤ n,Ps stands for s = (s ,...,s ),
>i i+1 n <i <i 1 i−1
we also set s = s = ∅. N(∞) denotes the set ∞ Nn. By convention N0 = {∅}.
>n <1 n=0
For e ∈ End(M•) and s ∈ Nn we define e(sF) = e(s1)...e(sn), where e(l) = dl (e) if
End
l ≥ 1, e(0) = e and e∅ = 1. In the case that e ∈ End(M•) is given by
a
e (φ) = aφ, for a ∈ M1 fixed and all φ ∈ M•,
a
then e(l) = dl (e ) reduces to e(l) = e , thus
a End a a dl(a)
e(as) = ea(s1)···ea(sn) = eds1(a)···edsn(a).
For N ∈ N we define E = {s ∈ N(∞) : |s| + l(s) ≤ N} and for s ∈ E we define
N N
N(s) ∈ Z by N(s) = N −|s|−l(s).
We introduce a discrete quantum mechanical 1 by
1. V = N(∞).
1a discrete quantum mechanical system is given by the following data (1) A directed graph Γ (finite
or infinite). (2) A map L : E → R called the Lagrangian map of the system. An associated Hilbert
Γ
space H=CVΓ. Operators U :H→H, where n∈Z, given by
n
(U f)(y)= ω (y,x)f(x),
n n
xX∈VΓ
where the discretized kernel ω (y,x) admits the following representation
n
ω (y,x)= eiL(e).
n
γ∈PXn(Γ,x,y)eY∈γ
P (Γ,x,y) denotes the set of length n paths in Γ from x to y, i.e., sequences (e ,··· ,e ) of edges in Γ
n 1 n
such that s(e )=x, t(e )=s(e ), i=1,...,n−1 and t(e )=y.
1 i i+1 n
3
2. There is a unique directed edge e from vertex s to t if and only if t ∈ {(0,s),s,(s+
e )} where e = (0,.., 1 ,..,0) ∈ Nl(s), in this case we set source(e) = s and
i i
i−th
target(e) = t. |{z}
3. Edges are weighted according to the following table
source(e) target(e) v(e)
s (0,s) 1
s s q|s|+l(s)
s (s+e ) q|s<i|+i−1
i
The set P (∅,s) consists of all paths γ = (e ,...,e ), such that source(e ) = ∅,
N 1 N 1
target(e ) = s and source(e ) = target(e ). For γ ∈ P (∅,s) we define the weight
N l+1 l N
v(γ) of γ as
N
v(γ) = v(e ).
l
Yl=1
The following result is proven as Theorem 17 in [AD].
Theorem 3 Let (M•,m ,d ) be a q-dgm over a q-dga (A•,m ,d ). For a ∈ M1
M M A A
consider the map Dφ = d φ+aφ. Then we have
M
N−1
DN = c dk
k M
Xk=0
where
c = c (s,N)a(s) and c (s,N) = v(γ).
k q q
sX∈EN γ∈PXN(∅,s)
N(s)=k
si<N
Suppose that we have a q-dgm (M•,m ,d ), q a 3-rd primitive root of unity, and
M M
we want to deform it to D = d +a, a ∈ M1. So we required that D3 = 0. By Theorem
M
2
3 we must have c dk = 0. Let us compute the coefficients c . First, notice that
k M k
Xk=0
E = {∅,(0),(1),(2),(0,0),(1,0),(0,1),(0,0,0)}
3
We proceed to compute the coefficients c for 0 ≤ k ≤ 2 using Theorem 3.
k
k = 0
There are four vectors in E such that N(s) = 0, these are (2),(1,0),(0,1),(0,0,0).
3
4
For s = (2) the only path from ∅ to (2) of length 3 is ∅ → (0) → (1) → (2) with
weight 1 and e(2) = d2 (e ), thus c(s,3) = d2 (a).
a M a M
For s = (1,0) the only path from ∅ to (1,0) of length 3 is ∅ → (0) → (0,0) → (1,0)
(1,0)
with weight 1 and e = d (e )e , thus c(s,3) = d (a)e .
a M a a M a
For s = (0,1) the paths from ∅ to (0,1) of length 3 are ∅ → (0) → (0,0) → (0,1)
(0,1)
with weight q; and ∅ → (0) → (1) → (0,1) with weight 1. Since e = e d (e ), then
a a M a
c(s,3) = (1+q)e d (e ).
a M a
For s = (0,0,0) the only path from ∅ to (0,0,0) of length 3 is ∅ → (0) → (0,0) →
(0,0,0) whose weight is 1 and e(0,0,0) = a3, thus c(s,3) = a3.
a
k = 1
There are two vectors in E such that N(s) = 1, namely (1) and (0,0).
3
For s = (1), the paths from ∅ to (1) of length 3 are ∅ → ∅ → (0) → (1) with
weight 1; ∅ → (0) → (0) → (1) with weight q; ∅ → (0) → (1) → (1) with weight q2.
e(1) = d (e ) and thus c (s,3) = (1+q +q2) = 0.
a M a q
For s = (0,0) the paths from ∅ to (0,0) of length 3 are ∅ → (0) → (0) → (0,0) with
weight q. ∅ → ∅ → (0) → (0,0) with weight 1. ∅ → (0) → (0,0) → (0,0) with weight
q2. e(0,0) = a2 and c (s,3) = 0.
a q
k = 2
(0) is the only vector in E such that N(s) = 2.
3
The paths from ∅ to (0) of length 3 are ∅ → ∅ → ∅ → (0) with weight 1; ∅ →
∅ → (0) → (0) with weight q; ∅ → (0) → (0) → (0) with weight q2. e(0) = a and
a
c(s,3) = (1+q +q2) = 0.
So we have proven that the 3-curvature is given by
D3 = d2 (a)+d (a)a+(1+q)ad (a).
M M M
Notice that in case of q-commutativity αβ = qα¯β¯βα, the 3-curvature reduces to D3 =
d2 (a).
M
Suppose now that q is a 4-th primitive root of unity and we required that D4 = 0.
3
By Theorem 3 we must have c dk = 0. Notice that
k M
Xk=0
E = {∅,(0),(1),(2),(3),(0,0),(1,0),(0,1),(2,0),(0,2),(1,1),
4
(0,0,0),(1,0,0),(0,1,0),(0,0,1),(0,0,0,0)}
5
We proceed to compute the coefficients c for 0 ≤ k ≤ 3 using Theorem 3.
k
k = 3
∅ −→i ∅ −→ (0) −→j (0) the weight is qj and c = (1+q +q2 +q3)a = 0.
3
i+Xj=3
k = 2
∅ −→i ∅ −→ (0) −→j (0) −→ (0,0) −k→ (0,0) the weight is qjq2k and
i+Xj+k=2
c ((0,0),4)a(0,0) = (1+q2)a2.
q
∅ −→i ∅ −→ (0) −→j (0) −→ (1) −k→ (1) the weight is qjq2k and
i+Xj+k=2
c ((1),4)a(1) = (1+q2)d(a).
q
Finally, c = (1+q2)(a2 +d(a)).
2
k = 1
i j k l
∅ −→ ∅ −→ (0) −→ (0) −→ (0,0) −→ (0,0) −→ (0,0,0) −→ (0,0,0)
the weight is qjq2kq3l and c ((0,0,0),4)a(0,0,0) = (1+q +q2 +q3)a3 = 0.
q
i+j+Xk+l=1
i j k l
∅ −→ ∅ −→ (0) −→ (0) −→ (1) −→ (1) −→ (0,1) −→ (0,1)
the weight is qjq2kq3l.
i+j+Xk+l=1
i j k l
∅ −→ ∅ −→ (0) −→ (0) −→ (0,0) −→ (0,0) −→ (0,1) −→ (0,1)
the weight is qjq2kqq3l and c ((0,1),4)a(0,1) = ((1+q+q2+q3)+(1+q+q2+
q
i+j+Xk+l=1
q3))ad(a) = 0.
i j k l
∅ −→ ∅ −→ (0) −→ (0) −→ (0,0) −→ (0,0) −→ (1,0) −→ (1,0)
the weight is qjq2kq3l and c ((1,0),4)a(1,0) = (1+q +q2 +q3)d(a)a = 0.
q
i+j+Xk+l=1
i j k l
∅ −→ ∅ −→ (0) −→ (0) −→ (1) −→ (1) −→ (2) −→ (2)
the weight is qjq2kq3l and c ((2),4)a(2) = (1+q +q2 +q3)d2(a) = 0.
q
X
i+j+k+l=1
Finally, c = c ((0,0,0),4)a(0,0,0)+c ((0,1),4)a(0,1)+c ((1,0),4)a(1,0)+c ((2),4)a(2) =
1 q q q q
0.
k = 0
∅ −→ (0) −→ (0,0) −→ (0,0,0) −→ (0,0,0,0)
the weight is 1 and c ((0,0,0,0),4)a(0,0,0,0) = a4.
q
6
∅ −→ (0) −→ (0,0) −→ (0,0,0) −→ (0,0,1)
∅ −→ (0) −→ (1) −→ (0,1) −→ (0,0,1)
∅ −→ (0) −→ (0,0) −→ (0,1) −→ (0,0,1)
the weight is 1+q +q2 and c ((0,0,1),4)a(0,0,1) = (1+q +q2)a2d(a).
q
∅ −→ (0) −→ (0,0) −→ (1,0) −→ (0,1,0)
∅ −→ (0) −→ (0,0) −→ (0,0,0) −→ (0,1,0)
the weight is 1+q and c ((0,1,0),4)a(0,1,0) = (1+q)ad(a)a.
q
∅ −→ (0) −→ (0,0) −→ (0,0,0) −→ (1,0,0)
the weight is 1 and c ((1,0,0),4)a(1,0,0) = d(a)a2.
q
∅ −→ (0) −→ (0,0) −→ (1,0) −→ (2,0)
the weight is 1 and c ((2,0),4)a(2,0) = d2(a)a.
q
∅ −→ (0) −→ (0,0) −→ (0,1) −→ (0,2)
∅ −→ (0) −→ (1) −→ (0,1) −→ (0,2)
∅ −→ (0) −→ (1) −→ (2) −→ (0,2)
the weight is 1+q +q2 and c ((0,2),4)a(0,2) = (1+q +q2)ad2(a).
q
∅ −→ (0) −→ (0,0) −→ (1,0) −→ (1,1)
∅ −→ (0) −→ (1) −→ (0,1) −→ (1,1)
the weight is 1+q2 and c ((1,1),4)a(1,1) = (d(a))2.
q
∅ −→ (0) −→ (1) −→ (2) −→ (3)
the weight is 1 and c ((3),4)a(3) = d3(a).
q
Finally, c = a4 + (1 + q + q2)a2d(a) + (1 + q)ad(a)a + d(a)a2 + d2(a)a + (1 + q +
0
q2)ad2(a)+(d(a))2 +d3(a).
So we have that the 4-curvature is given by
D4 = (1+q2)(a2 +d(a))d2 +a4 +(1+q +q2)a2d(a)+(1+q)ad(a)a+
M
d(a)a2 +d2(a)a+(1+q +q2)ad2(a)+(d(a))2 +d3(a).
Intheremainderofthepaperweconsiderinfinitesimaldeformationsofaq-differential.
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Theorem 4 Let (M•,m ,d ) be a q-dgm such that dN = 0, consider the infinitesimal
M M M
deformation D = d +te, where e ∈ End1(M•) and t2 = 0, then
M
N−1
DN =  q|v|+w(v)dN−k−1,
M
Xk=0 v∈Par(XN,N−k−1)
 
where
Par(N,N −k −1) = {(v ,··· ,v )/ v = N},
0 N−k−1 i
X
|v| = v +···+v and w(k) = iv .
1 n i
P
Proof: By Theorem 3, DN = N−1c dk . Since t2 = 0, then
k=0 k M
P
(te)(s) = (te)(s1)···(te)(sl(s)) = tl(s)e(s) = 0 unless l(s) ≤ 1.
Thus E is given by
N
E = {(0),(1),··· ,(N −1)}.
N
For 0 ≤ k ≤ N−1, since N(s) = N−|s|−l(s) = k andl(s) = 1 we obtain |s| = N−k−1,
andtheonlyvectorsinE oflength1suchthat|s| = N−k−1ispreciselys = (N−k−1).
N
Thus
c = c (s,N)a(s) = c ((N −k −1)),N)a(s) = c ((N −k −1)),N)dN−k−1(a).
k q q q
sX∈EN
N(s)=k
si<N
Any path from ∅ to (N −k −1) of length N must be of the form
∅ → ··· → ∅ → (0) → ··· → (0) → (1) → ···(1) → ···(N −k −1) → ··· → (N −k −1)
p0 p1 p2 pN−k−1
| {z } | {z } | {z } | {z }
with p0 +p1 +···+pN−k−1 = N. The weight of such path is qp1qp2(1+1)...qpN−k−1(N−k) =
q|p|+w(p), where p = (p ,··· ,p ). Then
0 N−k−1
c (s,N) = v(γ) = q|p|+w(p).(cid:7)
q
X X
γ∈PN(∅,s) p∈Par(N,N−k−1)
Acknowledgement
We thank Sylvie Paycha and Nicol´as Andruskiewitsch.
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[3] M. Angel, R. D´ıaz, N-flat connections, Preprint math.DG/0511242.
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∞
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Mauricio Angel. Universidad Central de Venezuela (UCV). [email protected]
Rafael D´ıaz. Universidad Central de Venezuela (UCV). [email protected]
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