THE FROBENIUS THEOREM ON N -DIMENSIONAL QUANTUM HYPERPLANE
C. CIUPALĂ
Abstract. In this paper we introduce the universal ρ-differential calculus on a ρ-algebra and we prove the universality of
the construction. We also present submanifolds, distributions, linear connections and two different ρ-differential calculi
q
q
on SN
: the algebra of forms and the algebra of universal differential forms of SN
. Finally we prove the Frobenius
q
q
theorem on the N -dimensional quantum hyperplane SN in a general case, for any ρ-differential calculus on SN
.
1.
Introduction
The basic idea of noncommutative geometry is to replace an algebra of smooth functions defined on a smooth
manifold by an abstract associative algebra A which is not necessarily commutative. In the context of noncommutative geometry the basic role is the generalization of the notion of differential forms (on a manifold). A
(noncommutative) differential calculus over the associative algebra A is a Z-graded algebra Ω(A) = ⊕n≥0 Ωn (A)
(where Ωn (A) are A-bimodules and Ω0 (A) = A) together with a linear operator d : Ωn (A) → Ωn+1 (A) satisfying
d2 = 0 and d(ωω 0 ) = (dω)ω 0 + (−1)n ωdω 0 where ω ∈ Ωn (A).
There are studied some differential calculi associated to A, here we recall two of them: the algebra of forms
of A in [8] and the algebra of universal differential forms of A in [2]. These two differential calculi are studied
even in the case when A is a superalgebra in [10] and in [11], but when A is a ρ-algebra there is studied only the
algebra of forms of A in [1].
Received March 3, 2004.
2000 Mathematics Subject Classification. Primary 81R60, 16W99, 53C04.
Key words and phrases. noncommutative geometry, linear connections, submanifolds, distributions, quantum hyperplane.
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In this paper we introduce the notion of universal derivation of the order (α, β) , we construct the algebra of
universal differential forms of a ρ-algebra A and we prove the universality of the construction. We apply this to
q
the particular case of the N -dimensional quantum hyperplane SN
and thus we obtain a new differential calculus
q
q
q
Ωα (SN ) on SN , denoted the algebra of universal differential forms of SN
.
q
On the other hand we present submanifolds algebra on SN in the context of noncommutative geometry, we also
q
q
define linear connections, distributions on SN
and we prove the Frobenius theorem on SN
, for any ρ-differential
q
q
calculus ΩSN on SN .
2.
Differential calculus on ρ-algebras
In this section we define the differential calculus on a ρ-algebra A and we give two examples of such differential
calculus on A: the algebra of forms Ω (A) of a ρ-commutative algebra A from [1] and the algebra of universal
differential forms Ωα (A) of a ρ-algebra A. We briefly review the basic notions of ρ-algebras and ρ-modules on
ρ-algebras. (see [1], [3], [5] and [6] for details).
First we review the basic notions about ρ-algebras, ρ-bimodules over a ρ-algebra and the derivations on ρbimodules.
Let G be an abelian group, additively written, and let A be a G-graded algebra. A is a ρ-algebra if there is a
map ρ : G × G → k which satisfies the relations
(1)
ρ(a, b) = ρ(b, a)−1 and ρ(a + b, c) = ρ(a, c)ρ(b, c), for any a, b, c ∈ G.
The G-degree of a (nonzero) homogeneous element f of A is denoted by |f | . The ρ-algebra A is ρ-commutative
if f g = ρ (|f | , |g|) gf for any f ∈ A|f | and g ∈ A|g| .
The morphism f : M → N between the ρ-bimodules M and N over the ρ-algebra A is of degree β ∈ G if
f : Mα → Nα+β for any α ∈ Ġ, f (am) = ρ(α, |a|)af (m) and f (ma) = f (m)a for any a ∈ A|a| and m ∈ M.
Definition 1. Let α, β ∈ G and M a ρ-bimodule over the ρ-algebra A. A ρ-derivation of order (α, β) on M
(ρ-derivation of degree α and of G-degree β) is a linear map X : A → M , which fulfils the properties:
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1. X : A∗ → M∗+β ,
2. X(f g) = (Xf )g + ρ(α, |f |)f (Xg),for any f ∈ A|f | and g ∈ A.
The left product f X : A → M between the element f ∈ A and a ρ-derivation X of the order (α, β) is defined
in a natural way: by (f X)(g) = f X(g), for any g ∈ A.
Proposition 1. If is X a ρ-derivation of order (α, β) on M and f ∈ A|f | , then f X is a derivation of the
order (|f | + α, |f | + β) on M if and only if the algebra A is ρ-commutative.
Proof. We have to show that (f X)(gh) = ((f X)g)h + ρ(|f | + α, |g|)g(f X)h and f X : A∗ → M∗+|f |+|X| , for
any h ∈ A|h| and f ∈ A|f | .
(f X) (gh) = f (X(gh)) = f (X(g)h + ρ(α, |g|)gX(h))
= (f X)(g)h + ρ(α, |g|)f gX(h) = (f X)(g)h + ρ(α, |g|)ρ(|f | , |g| gf X(h)
= (f X)(g)h + ρ(α, |f | + |g|)g(f X)(h).
The second relation is obvious.
The linear map X : A → A is a ρ-derivation on A if it is a ρ-derivation of the order (|X| , |X|) . Next we denote
by ρ-Der(A) the ρ-bimodule over A of the all ρ-derivations on A.
Definition 2. Let A be a ρ-algebra. We say that Ω (A) =
⊕
(n,β)∈Z×G
0
Ωnβ (A) is a ρ-differential calculus on A if
Ω (A) is a Z × G-graded algebra, Ω (A) is a ρ-bimodule over A, Ω (A) = ⊕ Ω0β (A) = A and there is an element
β∈G
α ∈ G and ρ-derivation d : Ω (A) → Ω (A) of the order ((1, α) , (1, 0)) such that d2 = 0.
The first example of a ρ-differential calculus over the ρ-commutative algebra A is the algebra of forms (Ω(A), d)
of A from [1].
The second example of a ρ-differential calculus over a ρ-algebra is the universal differential calculus of A from
the next subsection.
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2.1.
The algebra of universal differential forms of a ρ-algebra
In this subsection we define the algebra of universal differential forms of the ρ-algebra A and we prove the
universality of the construction. First we introduce the universal derivation of the order (α, β) , α, β ∈ G.
Definition 3. Let M be a ρ-bimodule over A. The ρ-derivation D : A → M of the order (α, β) is universal
if for ρ-derivation D0 : A → N of the order (α, 0) , there is an unique morphism Φ : M → N of ρ-bimodules of
degree β such that D0 = Φ ◦ D.
Next we will construct an universal derivation over the ρ-algebra A.
Let α ∈ G, µ : A ⊗ A → A be the map µ (x ⊗ y) = ρ (α, |y|) xy and Ω1α A = ker (µ).
We define the map
(2)
d : A → Ω1α A
by
(3)
dx = 1 ⊗ x − ρ (α, |x|) x ⊗ 1,
for any x ∈ A.
Proposition 2. Ω1α A is a ρ-bimodule over A and the map d : A → Ω1α A is an universal derivation of the order
(α, 0) .
Proof. We have to show that a(b ⊗ c) ∈ Ω1α A, (b ⊗ c)a ∈ Ω1α A and (a(b ⊗ c))d = a((b ⊗ c)d), for any a, d ∈ A
and b ⊗ c ∈ Ω1α A. Indeed
µ(a(b ⊗ c)) = µ(ab ⊗ c) = ρ(α, |ab|)abc = ρ(α, |a|)a(ρ(α, |b|)bc) = 0.
The other relations are obvious.
It is easy to see that Ω1α A is the space {xdy : x, y ∈ A and d from (3)} .
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Now we show that d : A → Ω1α A is a ρ-derivation of the order (α, 0):
(da)b + ρ(α, |a|)adb = (1 ⊗ a − ρ(α, |a|)a ⊗ 1)b + ρ(α, |a|)a(1 ⊗ b − ρ(α, |b|)b ⊗ 1)
= 1 ⊗ ab − ρ(α, |a|)a ⊗ b+ρ(α, |a|)a ⊗ b − ρ(α, |a|+|b|)ab ⊗ 1
= 1 ⊗ (ab) − ρ(α, |ab|)(ab) ⊗ 1 = d(ab).
Let M be a ρ-bimodule over A and D:A → M a ρ-derivation of the order (α, β) . We define the map Φ : Ω1α A → M
in the following way:
Φ(a ⊗ b) = ρ(α, |a|)aD(b),
Ω1α A.
for any a ⊗ b ∈
Now we show that Φ is a morphism of ρ-bimodules over A:
Φ(a(b ⊗ c)) = φ((ab) ⊗ c) = ρ(α, |ab|)abD(c)
= ρ(α, |a|)a[ρ(α, |b|)bD(c)] = ρ(α, |a|)aΦ(b ⊗ c).
On the other hand
Φ((a ⊗ b)c) = Φ(a ⊗ (bc)) = ρ(α, |a|)aD(bc)
= ρ(α, |a|)a[D(b)c + ρ(α, |b|)bD(c)]
= ρ(α, |a|)aD(b)c + ρ(α, |b|)[ρ(α, |a|)ab]D(c)
= ρ(α, |a|)aD(b)c + ρ(α, |b|)µ(a ⊗ b)D(c)
a⊗b∈ker(µ)
=
ρ(α, |a|)aD(b)c = Φ(a ⊗ b)c.
Finally we get
(Φ ◦ d)(a) = Φ(1 ⊗ a − ρ(α, |a|)a ⊗ 1)
= ρ(α, |1|)D(a) − ρ(α, |a|)ρ(α, |a|)aD(1) = D(a)
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so we have proved that Φ ◦ d = D.
Let
Ωnα A = Ω1α A ⊗ . . . ⊗ Ω1α A,
A
A
|
{z
}
Ω0α A = A and Ωα A = ⊕ Ωnα A.
n≥0
n times
Naturally Ωα A is a ρ-bimodule over A and a algebra with the multiplication (ω, θ) 7→ ω ⊗ θ, for any ω, θ ∈ Ωα A.
A
Remark 1. The algebra Ωα A may be identified with the algebra generated by the algebra A and the derivations
da, a ∈ A which satisfies the following relations:
1. da is linear in a
2. the ρ-Leibniz rule: d(ab) = d(a)b + ρ(α, |a|)adb
3. d(1) = 0.
n
Ωα A is the space of n-forms a0 da1 ...dan , ai ∈ A for any 0≤ i ≤ n. Ωnα A is a ρ-bimodule over A with the left
multiplication
(4)
a(a0 da1 . . . dan ) = aa0 da1 . . . dan ,
and with the right multiplication given by:
n
n
X
X
(a0 da1 . . . dan )an+1 =
(−1)n−i ρ(α,
|aj |)(a0 da1 . . . d(ai ai+1 ) . . . dan+1 )
i=1
+ (−1)n ρ(α,
j=i+1
n
X
|aj |)a0 a1 da2 . . . dan+1 .
i=1
The multiplication in the algebra Ωα A is:
(a0 da1 . . . dan )(an+1 dan+2 . . . dam+n ) = ((a0 da1 . . . dan )an+1 )dan+2 . . . dam+n ),
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for any ai ∈ A, 0≤ i ≤ n + m, n, m ∈ N.
We define the G-degree of the n-form a0 da1 . . . dan in the following way
|a0 da1 . . . dan | =
n
X
|ai | .
i=0
It is obvious that |ωn · ωm | = |ωn | + |ωm | for any homogeneous forms ωn ∈ Ωnα A and ωm ∈ Ωm
α A.
Ωα A is a G0 = Z × G-graded algebra with the G0 degree of the n-form a0 da1 . . . dan defined by
!
n
X
0
|a0 da1 . . . dan | = n,
|ai | .
i=0
We may define the cocycle ρ0 : G0 × G0 → k on the algebra Ωα A thus:
(5)
0
0
0
nm
ρ (|a0 da1 . . . dan | , |b0 db1 . . . dbm | ) = (−1)
ρ
n
X
i=0
|ai | ,
m
X
!
|bi |
i=0
and thus we have that Ωα A is a ρ0 -algebra. It may be proved that the map d : Ωα A → Ωα A is a derivation of the
order ((1, α) , (1, 0)). Concluding we have the following result:
Theorem 1. (Ωα A, d) is a ρ-differential calculus over A.
Example 1. In the case when the group G is trivial then A is the usual associative algebra and Ωα A is the
algebra of universal differential forms of A.
ab
Example 2. If the group G is Z2 and the cocycle ρ is defined by ρ (a, b) = (−1) then A is a superalgebra.
In the case when α = 1 Ωα A is the superalgebra of universal differential forms of A from [11].
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3.
The Frobenius theorem on quantum hyperplane
q
In this section we give the Frobenius theorem on the N -dimensional quantum hyperplane SN
and we give the
equations of any globally integrable distributions and parallel with respect to a connection ∇. These results are
q
valid for any ρ-differential calculus on SN
. In the first subsection we present two different ρ-differential calculi on
q
q
q
SN : the first one is the algebra of forms Ω (SN
) on SN
form [1] and the second one is the algebra of universal
q
q
q
differential forms Ωα (SN ) on SN . Both of these ρ-differential calculi on SN
are different by differential calculus
q
q
on SN of Wess and Zumino [15]. In the second subsection we present submanifolds on SN
, in the third subsection
q
we review the basic notions about linear connections on SN and finally we will give the main results of this paper.
3.1.
q
Differential calculi on SN
q
First we review the basic notions about the N -dimensional quantum hyperplane SN
. For more details see [1] and
[3].
q
The N -dimensional quantum hyperplane SN
is the k-algebra generated by the unit element and N linearly
independent elements x1 , . . . , xN satisfying the relations: xi xj = qxj xi , i < j for some fixed q ∈ k, q 6= 0.
q
SN
is a ZN -graded algebra
∞
M
q
q
SN
=
(SN
)n1 ...,nN ,
n1 ,...,nN
q
(SN
)n1 ,...,nN
with
the one-dimensional subspace spanned by products xn1 ·. . .·xnN . The ZN -degree of xn1 ·. . .·xnN
is n = (n1 , . . . , nN ). The cocycle ρ : ZN × ZN → k is
N
P
0
ρ(n, n ) = q
nj n0k αjk
j,k=1
,
with αjk = 1 for j < k, 0 for j = k and −1 for j > k.
q
It may be proved that the N -dimensional quantum hyperplane SN
is a ρ-commutative algebra.
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q
q
Remark that the space of ρ-derivations ρ-Der(SN
) is a free SN
-module of rank N with ∂/∂x1 , . . . , ∂/∂xN as
the basis, where ∂/∂xi (xj ) = δij .
q
q
q
q
Remark 2. Let (ΩSN
, d) a ρ-differential calculus on SN
, with d : ΩSN
→ ΩSN
a ρ-derivation of the order
q
N
((1, α) , (1, 0)) , where α, .0 ∈ Z It is easy to see that ΩSN is generated by {x1 , . . . , xN } and there differentials
{y1 = dx1 , . . . , yN = dxN } with some relations between them.
q
Next we give some examples of ρ-differential calculi on SN
.
q
q
q
3.1.1. The algebra of forms Ω (SN
) of SN
. Ω (SN
) ([1], [3]) is the algebra determined by the elements
x1 , . . . , xN and y1 = dx1 , . . . , yN = dxN with the relations
(6)
xj xk = q αjk xk xj ,
yj yk = −q αjk yk yj ,
xj yk = q αjk yk xj .
q
3.1.2. The algebra of universal differential forms Ωα (A) of SN
. Next we will apply the construction of
q
the algebra of the universal differential forms of a ρ-algebra from the remark 1 to the ρ-algebra SN
and, thus, we
q
q
will give a new differential calculus on SN denoted by Ωα (SN ) .
q
q
Let α = (n1 , . . . , nN ) be an arbitrary element from ZN . Ωα (SN
) is the algebra generated by a ∈ SN
and the
symbols da ,which satisfies the following relations:
1. da is linear in a.
2. the ρ-Leibniz rule: d(ab) = (da)b + ρ(n, |a|)adb.
3. d(1) = 0.
q
Next we present the structure of the algebra Ωα (SN
).
We use the following notations yi = dxi , for any i ∈ {1, . . . , N }. By an easy computation we get the following
lemmas:
Lemma 1. yi xj = ρ(α + |xi | , |xj |)xj yi , for any i, j ∈ {1, . . . , N }.
Lemma 2. yj yi = ρ(α, |xi |)ρ(n + |xi | , |xj |)yi yj , for any i, j ∈ {1, . . . , N }.
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m−1
Lemma 3. d(xm
(α, |xi |)xm−1
yi , for any m ∈ N and i ∈ {1, . . . , N }.
i ) = mρ
i
Putting together the previous lemmas we obtain the following theorem which gives the structures of the algebra
q
Ωα (SN
):
q
Theorem 2. Ωα (SN
) is the algebra spanned by the elements xi and yi with i ∈ {1, . . . , N } which satisfies the
following relations:
1. xi xj = ρ(|xi | , |xj |)xj xi ,
2. yi xj = ρ(α + |xi | , |xj |)xj yi ,
3. yj yi = ρ(, |xi |)ρ(α + |xi | , |xj |)yi yj , for any i, j ∈ {1, . . . , N }.
3.2.
q
Submanifolds in SN
In this section we use the definition of submanifolds algebra in noncommutative geometry from [11] to introduce
submanifolds in the quantum hyperplane.
q
q
q
Let C be an ideal in SN
. We denote by Q = SN
/C the quotient algebra and by p : SN
→ Q the quotient map.
q
We consider the following two Lie ρ-subalgebras of ρ-DerSN :
q
GC = {X ∈ ρ-DerSN
/XC ⊂ C}
and
q
q
GA = {X ∈ ρ-DerSN
/X(SN
) ⊂ C}
q
q
We define the map π : GC → ρ-DerSN
by π(X)p(a) = p(Xa) for any a ∈ SN
and X ∈ GC .
q
q
Definition 4. The quotient algebra Q = SN
/C is a submanifold algebra of SN
if the map π is sujective.
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In this situation we obtain the following the short exact sequence of ρ-Lie algebras.
0 → GA → GC → ρ-DerQ → 0
(7)
q
SN
Let C be the ideal from
generated by the elements x1 , x2 , . . . , xp with p ∈ {1, . . . , N }. Then the algebra Q
generated by the elements xp+1 , . . . , xN and we have that A = C ⊕ Q.
Theorem 3. GC = (ρ- Der Q) ⊕ GA .
q
Proof. Let X be a colour derivation from ρ-DerSN
, then
X=
N
X
i=1
Xi
∂
∂xi
and
Xi = XiC + XiQ
XiQ
XiC
with
∈ C and
∈ Q for any i ∈ {1, . . . , N }.
If X ∈ GC then X(c) ∈ C for any c ∈ C so
X(c) =
N
X
N
XiC
i=1
=
N
X
i=1
|
X Q ∂c
∂c
+
X
∂xi i=1 i ∂xi
p
N
X
X
∂c
∂c
∂c
∈C
+
XiQ
+
XQ
∂xi i=1
∂xi i=p+1 i ∂xi
{z }
|
{z
}
XiC
∈C
∈C
XiQ
= 0 for any i ∈ {1, . . . , p}.
N
P
So any element X =
Xi ∂/∂xi from GC may be written is the following way
Results that
i=1
X = X GA + X Q
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where
X GA =
N
X
XiC
i=1
∂
∈ GA
∂xi
and
XQ =
N
X
XiQ
i=1
∂
∈ ρ-DerQ
∂xi
with XiQ = 0 for i ∈ {1, . . . , p}.
q
Corollary 1. Q is submanifolds algebra of SN
.
3.3.
q
Linear connections on SN
q
In this subsection we use the definition of linear connections on ρ-algebras from the paper [3]. Let (ΩSN
, d) be
N
P
q
∂
a ρ-differential calculus on SN
and n ∈ ZN . A linear connection along the field X= Xi ∂x
on the ρ-bimodule
i
i=1
q
q
Ω n SN
over SN
is a linear map
q
q
∇ : ρ-Der (SN
) → End (Ωn SN
)
of degree |X| such that
∇(X)(aω) = ∇X (aω) = ρ(|X| , |ω|)X(a)ω + a∇X ω
q
ρ-DerSN
,
q
for any X ∈
a∈
and ω ∈ Ωn SN
.
q
Using the structure of the free bimodule Ωn SN
we deduce that any such connection ∇ is well defined by the
j1 ,...,jn
q
connections coefficients Γi,i1 ,...,in ∈ SN defined by
(8)
q
SN
∇
∂
∂xi
n
(yi1 . . . yin ) = Γji,i1 ,...,j
y . . . yjn .
1 ,...,in j1
n
Remark 3. The connection coefficients Γji,i1 ,...,j
satisfy the some properties which depend on the choice of
1 ,...,in
q
the ρ-differential calculus (ΩSN , d) .
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q
q
q
Example 3. If (ΩSN
, d) is the algebra of forms Ω (SN
) of SN
we obtain that
n
n
Γji,i1 ,...,j
= −q αik ,ik+1 Γji,i1 ,...,j
1 ,...,ik ,ik+1 ,...,in
1 ,...,ik+1 ,ik ,...,in
(9)
for any i, i1 , . . . , in , j1 , . . . , jn ∈ {1, . . . , n}.
q
q
q
Example 4. If (ΩSN
, d) is the algebra of universal differential forms Ωα (SN
) of SN
using an easy computation
we obtain that
m
m
(10)
Γji,i1 ,...,j
= ρ(α, |yik |)ρ(α + |yik | , yik+1 )Γji,i1 ,...,j
.
1 ,...,ik ,ik+1 ,...,im
1 ,...,ik+1 ,ik ,...,im
3.4.
q
Distributions in SN
q
q
q
Next we introduce distributions on SN
. Let (ΩSN
, d) be a ρ-differential calculus on SN
.
q
q
q
Definition 5. A distribution D on SN
is a SN
-subbimodule of Ω1 SN
. The distribution D is globally integrable
q
q
q
if there is a subset B of SN such that D is the subspace generated by SN
d(B) and by d(B)SN
.
q
Definition 6. We say that the distribution D is parallel with respect to the connection ∇ : ρ-DerSN
→
1 q
End(Ω SN ) if
q
∇X (m) = 0, for any X ∈ ρ-DerSN
and for any m ∈ D.
q
Using the structure of ΩSN
we obtain the following structure theorem of globally integrable distributions.
q
q
Theorem 4. Any globally integrable distributions D determined by SN
d(B) and d(B)SN
where B is the subset
{x1 , . . . , xp }. In this situation we say that the distribution D has the dimension p.
3.4.1. The Frobenius theorem for quantum hyperplane. In this section we will give a Frobenius theorem
for N -dimensional quantum hyperplane which is obvious from the previous results.
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q
Theorem 5. The Frobenius theorem for quantum hyperplane. Any globally integrable distribution from SN
q
q
is given by a maximal submanifold algebra of SN and conversely any submanifold algebra of SN give a globally
integrable distribution with the same dimension.
q
q
Proof. Let D be a global integrable distribution from SN
. Then from the theorem 10, D is given by SN
dQ, where
q
Q is a subset with p elements from {x1 , . . . xN } . If we denote by C the ideal of SN generated by {x1 , . . . xN } \Q
q
q
and using the Corollary 9 we obtain that Q=SN
/C is a submanifold algebra of SN
of the dimension p.
q
Conversely, if Q is a submanifold algebra of the dimension p of SN results that there is an ideal C of the
q
q
dimension N − p of SN
such that Q = SN
/C. If we denote by {x1 , . . . , xN −p } is the subset of {x1 , . . . , xN } which
q
generates the ideal C then the set Q = {xN −p+1 , . . . , xN } generates a distribution of the dimension p of SN
. We may find the equations of an globally integrable distributions and parallel with respect to a connection ∇.
q
Theorem 6. Any globally integrable and parallel distribution D with respect to a connection ∇ : ρ-DerSN
→
1 q
End(Ω SN ) of dimension p is given by the following equations:
(11)
Γki,j = 0
for a subset I of {1, . . . , N } with p elements and for any i ∈ {1, . . . , N }, j ∈ I, k ∈ {1, . . . , N } \ I.
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e-mail: cciupala@yahoo.com
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