The algebra of one-sided inverses of a polynomial algebra

0.41 MB·English

by  V. V. Bavula

The algebra of one-sided inverses of a polynomial algebra
V. V. Bavula
0
1
0
2 Abstract

n We study in detail the algebra Sn in the title which is an algebra obtained from a poly-
Ja nomial algebra Pn in n variables by adding commuting, left (but not two-sided) inverses of
thecanonicalgeneratorsofPn. ThealgebraSn isnon-commutativeandneitherleft norright
4
Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved
2
that the classical Krull dimension of Sn is 2n; but the weak and the global dimensions of
] Sn are n. The prime and maximal spectra of Sn are found, and the simple Sn-modules are
A classified. It is proved that the algebra Sn is central, prime, and catenary. The set In of
R idempotent ideals of Sn is found explicitly. The set In is a finite distributive lattice and the
numberof elements in the set In is equal tothe Dedekind numberdn.
.
h Key Words: catenary algebra; the classical Krull, the weak, and the global dimensions;
t simple module, prime ideal.
a
m Mathematics subject classification 2000: 16E10, 16G99, 16D25, 16D60.
[ Contents

3 1. Introduction.
v
1 2. The ideals of Sn commute and satisfy thea.c.c..
4 3. Classification of simple Sn-modules.
6
0 4. The prime and maximal spectra of thealgebra Sn.
3. 5. Left or right Noetherian factor algebras of Sn.
0 6. The weak and global dimensions of the algebra Sn.
9
0 7. Idempotent ideals of thealgebra Sn.
:
v
i 1 Introduction
X
r
a Throughout, ring means an associative ring with 1; module means a left module; N := {0,1,...}
is the set of natural numbers; K is a field and K∗ is its group of units; P := K[x ,...,x ]
n 1 n
is a polynomial algebra over K; ∂ := ∂ ,...,∂ := ∂ are the partial derivatives (K-linear
1 ∂x1 n ∂xn
derivations) of P .
n
ThealgebrasS (seethedefinitionbelow)appearnaturallywhenonewantstodevelopatheory
n
of one-sided localizations. Letmegiveanexample. LetK[x]beapolynomialalgebrainavariable
x over the field K. When we invert the element x the resulting algebra K[x,x−1] has the same
properties as K[x]. This is not the case when we invert the element x on one side only, say, on
the left: yx = 1. Then the algebra S := Khx,y|yx = 1i has very different properties from the
1
polynomial algebra K[x]. It is non-commutative, not left and right Noetherian, not a domain, it
contains the ring of infinite dimensional matrices, etc. Moreover, the algebra S has properties
1
that are a mixture of the properties of the three algebras: K[x ], K[x ,x ], and the Weyl algebra
1 1 2
A :=Khx,∂|∂x−x∂ =1iif char(K)=0 (for example,as it is provedin the paper, the ideals of
1
thealgebraS commute,andeachproperidealofS butoneisauniqueproductofmaximalideals
1 1
(counted with multiplicity), and the lattice of ideals is distributive; the classical Krull dimension
of S is 2; the global homological dimension of S is 1; the Gelfand-Kirillov dimension of S is
1 1 1
2). The algebra S is a well-known primitive algebra [7], p. 35, Example 2. Over the field C of
1
1
complex numbers,the completionofthe algebraS is the Toeplitz algebra whichis the C∗-algebra
1
generated by a unilateral shift on the Hilbert space l2(N) (note that y = x∗). The Toeplitz
1 1
algebra is the universal C∗-algebra generated by a proper isometry.
Definition. The algebra S of one-sided inverses of P is an algebra generated over a field K
n n
by 2n elements x ,...,x ,y ,...,y that satisfy the defining relations:
1 n n n
y x =···=y x =1, [x ,y ]=[x ,x ]=[y ,y ]=0 foralli6=j,
1 1 n n i j i j i j
where [a,b]:=ab−ba, the commutator of elements a and b.
By the very definition, the algebra S is obtained from the polynomial algebra P by adding
n n
commuting,left (or right)inversesofits canonicalgenerators. Clearly,S =S (1)⊗···⊗S (n)≃
n 1 1
S⊗n where S (i):=Khx ,y |y x =1i≃S and
1 1 i i i i 1
S = Kxαyβ
n
α,β∈Nn
M
where xα := xα11···xαnn, α = (α1,...,αn), yβ := y1β1···ynβn, β = (β1,...,βn). In particular, the
algebra S contains two polynomial subalgebras P and Q := K[y ,...,y ] and is equal, as a
n n n 1 n
vector space, to their tensor product P ⊗Q . The canonical generators x , y (1 ≤ i,j ≤ n)
n n i j
determine the ascending filtration {Sn,≤i}i∈N on the algebra Sn in the obvious way (i.e. by
the total degree of the generators): S := Kxαyβ where |α| = α + ··· + α
n,≤i |α|+|β|≤i 1 n
(S S ⊆ S for all i,j ≥ 0). Then dim(S ) = i+2n for i ≥ 0, and so the Gelfand-
n,≤i n,≤j n,≤i+j L n,≤i 2n
Kirillov dimension GK(S ) of the algebra S is equal to 2n. It is not difficult to show (Lemma
n n
2.3) that the algebra S is neither left nor right Noetherian.(cid:0)Mor(cid:1)eover, it contains infinite direct
n
sums of left and right ideals.
Another (left and right) non-Noetherian algebras, so-called, the Jacobian algebras A (intro-
n
ducedin[2]),appearasalocalizationnotinthesenseofOreoftheWeylalgebrasA . Thegeneral
n
construction, proposed by the author, is as follows: given an algebra A, an A-module M, and a
set S = {a } of elements of the algebra A such that the maps a : M → M, m 7→ a m,
i i∈I i,M i
are invertible. The subalgebra S−1 ∗A of End (M) generated by the image of the algebra A
M K
in End (M) and the elements {a−1 } can be seen as a new way of localizing the algebra A.
K i,M i∈I
Clearly, S−1∗(S−1 ∗A) = S−1∗A, and the factor algebra A/ann (M) of A modulo the anni-
M M M A
hilator ann (M) of the A-module M is a subalgebra of S−1∗A. In general, as the example of
A M
the JacobianalgebrasA shows[2], the algebrasA and S−1∗Ahave differentproperties,and the
n M
localized algebra S−1∗A is not a left or right localization of the algebra A in the sense of Ore.
M
Definition, [2]. When char(K) = 0, the Jacobian algebra A is the subalgebra of End (P )
n K n
generated by the Weyl algebra A :=Khx ,...,x ,∂ ,...,∂ i and the elements H−1,...,H−1 ∈
n 1 n 1 n 1 n
End (P ) where H :=∂ x ,...,H :=∂ x .
K n 1 1 1 n n n
Clearly, A = A (1)⊗···⊗A (n) ≃ A⊗n where A (i) := Khx ,∂ ,H−1i ≃ A . The algebra
n 1 1 1 1 i i i 1
A contains all the integrations : P → P , p 7→ pdx , since = x H−1. In particular, the
n i n n i i i i
algebra A contains all (formal) integro-differential operators with polynomial coefficients. The
n
Jacobian algebra A appeared inR my study of the gRroup of polynRomial automorphisms and the
n
Jacobian Conjecture, which is a conjecture that makes sense only for polynomial algebras in the
classofallcommutativealgebras[3]. InordertosolvetheJacobianConjecture,itisreasonableto
believe that one should create a technique which makes sense only for polynomials; the Jacobian
algebrasareastepinthisdirection(theyexistforpolynomialsbutmakenosenseevenforLaurent
polynomials). The Jacobian algebras were studied in detail in [2]. Their relevance to S is the
n
fact that S is a subalgebra of A (Lemma 2.1), and this fact makes it possible to shorten proofs
n n
ofseveralresultsonS . Moreover,there aremanyparallelsbetweenthese twoclassesofalgebras.
n
2
Let us describe main results of this paper.
• (Proposition 4.1, Theorem 4.15) The algebra S is central, prime, and catenary. Every
n
nonzero ideal of S is an essential left and right submodule of S .
n n
• (Proposition4.3). Let A be a K-algebra. Then the algebra S ⊗A is a prime algebra iff the
n
algebra A is a prime algebra.
• (Theorems 2.7 and 2.8). The ideals of S commute (IJ = JI), and the set of ideals of S
n n
satisfy the a.c.c..
• (Theorem 4.11) The classical Krull dimension cl.Kdim(S ) of S is 2n.
n n
• (Theorem 4.12) For each ideal a of S , the set Min(a) of the minimal primes over a is a
n
finite, non-empty set.
• (Theorem4.13) ht(p)+cht(p)=cl.Kdim(S ), for all prime ideals p of S . Formulae for the
n n
height ht(p) and the co-height cht(p) are found explicitly (via combinatorial data).
• (Theorems 6.6, Corollary 6.8) The weak homological dimension and the left and right global
dimensions of S are equal to n.
n
• (Theorems 4.4 and 4.5) The prime and the maximal spectra of S are found.
n
• (Theorem 3.2) The simple S -modules are classified.
n
• (Corollary3.5)GK(M)≤n,forallsimpleS -modulesM. Moreover,GK(M)∈{0,1,...,n}.
n
• (Corollary 3.4) The annihilators separate the simple S -modules.
n
• (Corollary 3.3) The polynomial algebra P is the only faithful, simple S -module.
n n
• (Corollary 6.12) GK(M)+pd(M)=l.gldim(S ) for all simple S -modules M.
n n
• (Theorem 6.11) The projective dimension pd(M) of each the simple S -module M is found
n
explicitly.
• (Theorem 3.2.(4), Corollary3.6) For each simple S -module M, the endomorphism division
n
algebra EndS (M) is a finite field extension of K and its dimension over K is equal to the
n
multiplicity e(M) of the S -module M.
n
• (Lemma 4.2) There are precisely n height one prime ideals of the algebra S (they are given
n
explicitly).
• (Theorem 5.1) Let I be an ideal of S . Then the factor algebra S /I is left (or right)
n n
Noetherian iff the ideal I contains all the height one primes of S .
n
• (Corollary4.9) A prime ideal p6=S of S is an idempotent ideal (p2 =p) iff p is contained
n n
in all the maximal ideals of S iff p is a sum of height one prime ideals of S .
n n
• (Theorem7.1, Corollary7.5) The set I of idempotent ideals of S is found. The set I is a
n n n
finite distributive lattice and the number of elements in the set I is equal to the Dedekind
n
number d .
n
• (Theorem 7.2) Each idempotent ideal a of S distinct from S is a unique product and a
n n
unique intersection of the minimal (necessarily idempotent) prime ideals over a.
3
These results show that the algebra S has properties that are a mixture of properties of the
n
polynomial algebra P and the Weyl algebra A . This is not so surprising when we look at the
2n n
defining relations of the algebras S , A , and P .
n n 2n
The algebras S are fundamental non-Noetherian algebras;they are universalnon-Noetherian
n
algebras of their own kind in a similar way as the polynomial algebras are universal in the class
of all the commutative algebras and the Weyl algebras are universal in the class of algebras of
differential operators.
The algebra S often appears as a subalgebra or a factor algebra of many non-Noetherian
n
algebras. For example, S is a factor algebra of certain non-Noetherian down-up algebras as was
1
shown by Jordan [8] (see also [5], [10], [9]).
2 The ideals of S commute and satisfy the a.c.c.
n
In this section, it is proved that the algebra S is neither left nor right Noetherian (it contains
n
infinite direct sums of left and right nonzero ideals) but the set of all the ideals of the algebra S
n
satisfies the a.c.c. (Theorem 2.7), the ideals of the algebra S commute (Theorem 2.8).
n
The polynomial algebra P is a left End (P )-module, we denote the action of a linear map
n K n
a∈End (P )onanelementp∈P eitherbya(p)orbya∗p. Bytheverydefinition,theJacobian
K n n
algebra A is a subalgebra of End (P ).
n K n
Lemma 2.1 The algebra homomorphism S → A , x 7→ x , y 7→ H−1∂ , is a monomorphism
n n i i i i i
when char(K)=0.
Proof. In view of the natural isomorphisms S ≃ S⊗n and A ≃ A⊗n, it suffices to prove the
n 1 n 1
lemma when n = 1 (we drop the subscript ‘1’ in this case here and everywhere if this does not
lead to confusion). The homomorphism is correctly defined since H−1∂x=1. It remains to show
that its kernel is zero. Note that for each natural numbers i and j, we have
1
xiyj 7→xi(H−1∂)j =xi ∂j.
H(H +1)···(H +j−1)
If anelement a=a yj+a yj+1+···+a yj+k ∈S (where all a ∈K[x], and k ≥0) belongs
j j+1 j+k 1 s
to the kernel then
0=a∗xj =a yj ∗xj =a ,
j j
and so a=0, i.e. the kernel is zero. (cid:3)
WeidentifythealgebraS withitsisomorphiccopyinthealgebraA viathe abovemonomor-
n n
phism. Then, S ⊂A ⊂End (P ), and so P is a faithful S -module when char(K)=0.
n n K n n n
Corollary 2.2 The S -module P is simple and faithful.
n n
Proof. We have to prove that the S -module P is simple. For n = 1 and natural numbers i
n n
and j:
0 if j <i,
yi∗xj = (1)
(xj−i if j ≥i.
If p = λ xα ∈ P (where λ ∈ K) is a nonzero polynomial of degree, say d, then λ 6= 0 for
α n α β
some element β ∈Nn such that |β|=d. Then λ−1yβ∗p=1. This means that the S -module P
P β n n
is simple.
Suppose that a∗P = 0 for a nonzero element a = a yα ∈ A where a ∈ P , we seek a
n α α n
contradiction. Fix α such that a 6=0 and |α| is the least possible. Then 06=a = a∗xα =0, a
α α
contradiction. Therefore, P is a faithful S -module. (cid:3) P
n n
Later, we will see that P is the only simple and faithful S -module (Corollary 3.3).
n n
4
Example. Consider a vector space V = Ke and two shift operatorsonV, X :e 7→e
i∈N i i i+1
and Y : e 7→ e for all i ≥ 0 where e := 0. By Corollary 2.2 and (1), the subalgebra of
i i−1 −1
End (V) generated by the operators X anLd Y is isomorphic to the algebra S (X 7→x, Y 7→y).
K 1
Bytakingthen’thtensorpowerV⊗n = Ke ofV weseethatthealgebraS isisomorphic
α∈Nn α n
tothesubalgebraofEnd (V⊗n)generatedbythe2nshiftsX ,Y ,...,X ,Y thatactindifferent
K 1 1 n n
L
directions.
Whenn=1,by(1),foreachnaturalnumberi,theproductxiyi istheprojectionontotheideal
(xi) of the polynomial algebra K[x] in the decomposition K[x]=( Kxj) (xi). Therefore,
j<i
the elements of the algebra End (P ) (where i,j ∈N):
K n
L L
xi−j(xjyj −xj+1yj+1) if i≥j ,
E := (2)
ij (yj−i(xjyj −xj+1yj+1) if i<j ,
are the matrix units, i.e. E ∗xk =δ xi, k ≥0, where δ is the Kronecker delta. In particular,
ij jk jk
E E =δ E for all natural numbers i, j, k, and l. Therefore, the subring (without 1)
ij kl jk il
F := KE
ij
i,j∈N
M
of S is canonically isomorphic to the ring (without 1) of infinite dimensional matrices
1
M (K):=limM (K)= KE
∞ −→ d ij
i,j∈N
M
(via F → M (K), E 7→ E ) where E are the matrix units of M (K) and M (K) :=
∞ ij ij ij ∞ d
n
KE is the ring of d-dimensional matrices over K.
i,j=1 ij
We have another presentation of the matrix units:
L
(xiyi−xi+1yi+1)xi−j if i≥j ,
E := (3)
ij ((xiyi−xi+1yi+1)yj−i if i<j .
The formula (3) can be verified directly using the inclusion S ⊂End (P ). Now, combining (2)
1 K 1
and (3) we can write
E =xiyj −xi+1yj+1 =xiE yj, i,j ≥0. (4)
ij 00
The involution η on S . The algebra S admits the involution
n n
η :S →S , x 7→y , y 7→x , i=1,...,n,
n n i i i i
i.e. itisaK-algebraanti-isomorphism(η(ab)=η(b)η(a) foralla,b∈Sn)suchthatη2 =idSn,the
identity map on S . So, the algebra S is self-dual (i.e. it is isomorphic to its opposite algebra,
n n
η :S ≃Sop). This means that left and right algebraic properties of the algebra S are the same.
n n n
For n=1 and all i,j ∈N,
η(E )=E . (5)
ij ji
This follows from (4). The involution η acts on the ring F = M (K) as the transposition. In
∞
general case,
η(F )=F , (6)
n n
whereF :=F⊗n = n F(i)= KE ,F(i):= KE (i),E := n E (i)
n i=1 α,β∈Nn αβ s,t∈N st αβ i=1 αiβi
where E (i) := xαi(1−x y )yβi ∈ F(i) ⊂ S (i). Clearly, E E = δ E for all elements
αiβi iN i i iL 1 L αβ γρ β,γ αρ N
α,β,γ,ρ ∈ Nn where δ is the Kronecker delta function. The involution η acts on the ‘matrix’
β,γ
ring F as the transposition:
n
η(E )=E . (7)
αβ βα
5
The algebra S is neither left nor right Noetherian. By (2) and (3), for all i,j ≥0,
n
xE =E , yE =E (E :=0), (8)
ij i+1,j ij i−1,j −1,j
E x=E , E y =E (E :=0). (9)
ij i,j−1 ij i,j+1 i,−1
By(8)and(9),F isanidealofthealgebraS . NotethatF isanidealofthealgebraA (Corollary
1 1
2.6, [2]) (from this fact it also follows that F is an ideal of the algebra S since F ⊂S ⊂A ).
1 1 1
By (8) and (9), for all i,j ≥0,
xE =E x, E y =yE . (10)
ij i+1,j+1 ij i+1,j+1
By (8), for eachi≥0 andj ≥0, the left S -module S E = KE is isomorphic to the left
1 1 ij k≥0 kj
S -module K[x] via the isomorphism S E →K[x], E 7→1 (and so E 7→xk, k ≥0).
1 1 ij 0j L kj
Lemma 2.3 The algebra S is neither left nor right Noetherian. Moreover, it contains infinite
n
direct sums of left and right ideals.
Proof. The algebra is self-dual, so it suffices to prove, say, the first statement of the lemma.
Since Sn ≃S1⊗Sn−1, it suffices to prove the lemma when n=1. In this case, F =⊕i∈NS1Eii ≃
⊕i∈NP1 is the direct sum of infinitely many copies of the simple S1-module P1. (cid:3)
The elements x and y of S are not regular. Let r be an element of a ring R. The
i i n
element r is called regular if l.ann (r)=0 and r.ann (r)=0 where l.ann (r):={s∈R|sr =0}
R r R
is the left annihilator of r and r.ann (r):={s∈R|rs=0} is the right annihilator of r.
R
The next lemma shows that the elements x and y of the algebra S are not regular.
1
Lemma 2.4 1. l.annS1(x)=S1E00 = i≥0KEi,0 = i≥0Kxi(1−xy) and r.annS1(x)=0.
2. r.annS1(y)=E00S1 = i≥0KE0,i =L i≥0K(1−xLy)yi and l.annS1(y)=0.
Proof. 1. yx=1 impliesLr.annS1(x)=0.LSince E00x=(1−xy)x=x−x=0,
l.annS1(x)⊇S1E00 = KEi,0 = Kxi(1−xy).
i≥0 i≥0
M M
To prove the reverse inclusion note that the right K[x]-module S /K[x] is a direct sum of its
1
right submodules M := ∞ Kxiyj + K[x], i ∈ N. It follows from the equalities xiyjx =
i j=1
xiyj−1 that the kernel of the linear map in S /K[x] given by the multiplication by the ele-
L 1
ment x on the right is equal to i≥0Kxiy+K[x]. Clearly, l.annS1(x) ⊆ i≥0Kxiy+K[x] =
Ky i≥0KEi,0 K[x]. NowL, one can easily find that l.annS1(x)= i≥L0KEi,0, as required.
2. The second statement follows from the first by using the involution η:
LL L L
r.annS1(y) = η2(r.annS1(y))=η(l.annS1(η(y)))=η(l.annS1(x))=η(S1E00)
= η(E )S =E S = K(1−xy)yi,
00 1 00 1
i≥0
M
where we have used the fact that η(E )=E (see (5)). (cid:3)
00 00
It follows from (4) that
S =K xK[x] yK[y] F, (11)
1
the direct sum of vector spaces. Then M M M
S /F ≃K[x,x−1], x7→x, y 7→x−1, (12)
1
since yx=1, xy =1−E and E ∈F.
00 00
6
Example. Let V = Ke be a vector space. By taking the matrix of a linear map in V
i∈N i
with respect to the basis {ei}i∈N for the vector space V, we identify the algebra EndK(V) with
L
thealgebraofinfinitematrices{ a E isaninfinitesumwhereforeachj almostallscalars
i,j∈N ij ij
a = 0} where E are the matrix units. Let n := E and m := E . A matrix
ij ij P i≥0 i+1,i i≥1 i−1,i
of the form λ mi+λ E+ λ ni is called a multidiagonal matrix where λ ∈ K (and
i>0 −i 0 i>0 i P P i
the sums are finite). A matrix is called an almost multidiagonal if it becomes a multidiagonal
P P
matrix by adding a finite sum µ E , µ ∈K. The set of all almost multidiagonal matrices is
ij ij ij
a subalgebra of End (V) which is isomorphic to the algebra S , by (11) and (1) (n↔x, m↔y,
K 1
P
E ↔E ).
ij ij
For an element a of an algebra A, the subalgebra of A, Cen (a):={b∈A|ba=ab}, is called
A
the centralizer of the element a in the algebra A. By (8), (9), and (11),
CenS (x)=K[x] and CenS (y)=K[y]. (13)
1 1
We saythataK-algebraAis centralif its centreZ(A) is K. We denote by J(A) the setofall
the ideals of the algebra A. An ideal I of the algebra A is called a proper ideal if I 6= 0,A. The
classical KrulldimensionofthealgebraAisdenotedcl.Kdim(A). spec(A)andMax(A)denotethe
prime spectrum and the maximal spectrum of the algebra A respectively. A nonzero polynomial
a∈K[x] is a monic polynomial if its leading coefficient is 1. The socle soc(M) of a module M is
the sum of its semi-simple submodules, if they exist, and is zero otherwise.
Proposition 2.5 1. The algebra S is central.
1
2. Fa6=0 and aF 6=0 for all nonzero elements a∈S .
1
3. F is the smallest (with respect to inclusion) nonzero ideal of the algebra S (i.e. F is
1
contained in all thenonzero ideals of S ); F2 =F; F is an essential left and right submodule
1
of S ; F is the socle of the left and right S -module S .
1 1 1
4. ThesetJ(S )ofalltheidealsofthealgebraS is{0,F,S ,F+a(K[x]+K[y])wherea=a(x)
1 1 1
is a monic non-scalar polynomial of K[x] such that a(0)6=0}; and two such ideals are equal,
F +a(K[x]+K[y])=F +b(K[x]+K[y]), iff a=b.
5. IJ =JI for all ideals I and J of the algebra S .
1
6. spec(S )={0,F,m :=F+a(K[x]+K[y])where a∈K[x] is amonic irreducible polynomial
1 a
distinct from x}. In particular, S is a prime ring.
1
7. Max(S )={m |a is a monic irreducible polynomial distinct from x}.
1 a
8. Any proper ideal I of the algebra S such that I 6= F is a unique finite product of maximal
1
ideals, i.e. I = miaa where all but finitely many natural numbers ia are equal to zero; and
miaa = mjaa iff all ia =ja for all a.
Q
9. Qmiaa =Q miaa; miaa + mjaa = mamin(ia,ja) and miaa mjaa = mmaax(ia,ja). In
particular, the lattice J(S ) is distributive.
1
Q T Q Q Q Q TQ Q
10. The classical Krull dimension of the algebra S is 2.
1
Remark. For K = C, statement 6 above and the fact that F is the minimal nonzero ideal of
S were proved by Jordan (Corollary 7.6, [8]) using a different method.
1
Proof. 1. By (13), Z(S )=Cen(x)∩Cen(y)=K.
1
2. This follows at once from (11), (8), and (9).
3. F2 = F since F = M (K). The fact that F is the smallest nonzero ideal follows from
∞
statement 2. By statement 2, F is an essential left and right S -submodule of S . Then F is the
1 1
7
socleofthemoduleS1S1 sincetheS1-moduleF issemi-simple(seetheproofofLemma2.3). Using
the involution η we see that F is the socle of the right S -module S .
1 1
4. Let I be an ideal of the algebra S which is distinct from the ideals 0, F and S . Then
1 1
F ⊂ I, and, by (12), I = F +aK[x] for a monic non-scalar polynomial of K[x] with a(0) 6= 0
(since I 6=S ). The rest is obvious due to (12).
1
5. This follows from statements 3 and 4 (yx=1, xy =1−E , E ∈F).
00 00
6. Let I and J be nonzero ideals of the algebra S . Then both of them contains the ideal F,
1
by statement 3. Then IJ ⊇ F2 = F 6= 0. Therefore, 0 is a prime ideal, i.e. the algebra S is
1
prime. F is a prime ideal since S /F ≃K[x,x−1] is a Laurent polynomial algebra over a field (a
1
commutativedomain). Fromthis factit followsthat the ideal F +a(K[x]+K[y])fromstatement
4 is prime iff in addition the polynomial a is irreducible. This observation finishes the proof of
statement 6.
7. Statement 7 follows from statement 6.
8. Statement 8 follows from statement 7.
9. Statement 9 follows from statement 8.
10. It follows from the inclusion of prime ideals 0 ⊂ F ⊂ m that cl.Kdim(S ) = 2 since the
a 1
ideals m are maximal. (cid:3)
a
By Proposition 2.5.(7), the map
Max(K[x,x−1])→Max(S ), m7→F +S m=F +mS ,
1 1 1
is a bijection.
Let n≥2. By (11), each element a∈S =S ⊗S has a unique presentation
n 1 n−1
a=λ+ (yi⊗λ +xi⊗λ )+ E ⊗λ (14)
−i i ij ij
i≥1 i,j≥0
X X
for some elements λ,λ ,λ ∈ S . The next lemma is crucial in the proofs of Theorems 2.7
±i ij n−1
and 2.8).
Lemma 2.6 Let I and J be ideals of the algebra S =S ⊗S , n≥2. Then
n 1 n−1
1. I∩(F ⊗S )=F ⊗I for a unique ideal I of the algebra S .
n−1 n−1 n−1 n−1
2. The ideal I of S is the K-linear span in S of the coefficients λ,λ ,λ ∈S in
n−1 n−1 n−1 ±i ij n−1
(14) for all the elements a of the ideal I.
3. If I ⊆J then I ⊆J .
n−1 n−1
4. (IJ) =I J .
n−1 n−1 n−1
Proof. 1 and 2. The uniqueness in statement 1 is obvious (if F ⊗a = F ⊗b for two ideals a
and b of S then a = b, and vice versa). Let I′ be the K-linear span in statement 2. Then
n−1 n−1
I′ is an ideal of the algebra S and
n−1 n−1
I∩(F ⊗S )⊆F ⊗I′ .
n−1 n−1
To finish the proof of statements 1 and 2 it suffices to show that the reverse inclusion holds.
Let a ∈ I, and so we have the decomposition (14) for the element a. First, let us show that
E ⊗λ ∈ I for some natural numbers k and l. Fix sufficiently large natural numbers k and l.
kl
Then E ( E ⊗λ )E =0, and so
kl i,j≥0 ij ij ll
P
E aE =E λ+ (E E ⊗λ +E E ⊗λ )=E ⊗λ.
kl ll kl k,l+i ll −i k,l−i ll i kl
i≥1
X
Similarly, for sufficiently large natural numbers k and l, and for all natural numbers i ≥ 1,
E yiaE =E ⊗λ andE axiE =E ⊗λ . Forallnaturalnumbersi,E aE =E ⊗(λ+λ )
kl ll kl i kl ll kl −i ii ii ii ii
8
and so E ⊗ λ ∈ I since E ⊗ λ = E (E ⊗ λ)E ∈ I. For all natural numbers i > j,
ii ii ii ik kl li
E aE =E ⊗(λ +λ ) and E aE =E ⊗(λ +λ ), and so E ⊗λ ,E ⊗λ ∈I.
ii jj ij i−j ij jj ii ji −(i−j) ji ij ij ji ji
This finishes the proof of statements 1 and 2.
3. Statement 3 is obvious.
4. Statement 4 follows from statements 1 and 2. By statement 2, (IJ) ⊆ I J . By
n−1 n−1 n−1
statement 1, we have the inverse inclusion:
F ⊗(I J ) = (F ⊗I )(F ⊗J )=I∩(F ⊗S ) · J ∩(F ⊗S )
n−1 n−1 n−1 n−1 n−1 n−1
⊆ (IJ)∩(F ⊗S )=F ⊗(IJ) ,
n−1 n−1
and so I J ⊆(IJ) . (cid:3)
n−1 n−1 n−1
Theorem 2.7 The set J(S ) of ideals of the algebra S satisfies the ascending chain condition
n n
(the a.c.c., for short).
Proof. Recall that the set J(S ) satisfies the a.c.c. if each ascending chain of ideals in S
n n
stabilizes, i.e. has a largest element. We use induction on n. The case n = 1 follows from the
description of the set J(S ) (Proposition 2.5.(4)). Suppose that n≥2 and that the result is true
1
foralln′suchthatn′ <n. NotethatforanyalgebraA,thesetJ(A)ofitsidealssatisfiesthea.c.c.
iffthe A-bimodule AisNoetherian. Recallthe following(easy)generalizationofthe HilbertBasis
Theorem: An A-module M is Noetherian iff the K[x]⊗A-module K[x]⊗M is Noetherian. By
induction,the S -bimodule S is Noetherian,hence theK[x]⊗S -bimoduleK[x]⊗S is
n−1 n−1 n−1 n−1
Noetherian,henceK[x,x−1]⊗S -bimoduleK[x,x−1]⊗S isNoetherian. NotethatF⊗S
n−1 n−1 n−1
is an ideal of the algebra S = S ⊗ S such that S /(F ⊗ S ) ≃ K[x,x−1] ⊗ S is a
n 1 n−1 n n−1 n−1
NoetherianK[x,x−1]⊗S -bimodule, or,equivalently,a NoetherianS -bimodule. For anyideal
n−1 n
I of S , by Lemma 2.6.(1),
n
I∩(F ⊗S )=F ⊗I ,
n−1 n−1
for some ideal I of the algebra S . Therefore, the S -bimodule F ⊗S is Noetherian. It
n−1 n−1 n n−1
follows from the short exact sequence of S -bimodules:
n
0→F ⊗S →S =S ⊗S →K[x,x−1]⊗S →0 (15)
n−1 n 1 n−1 n−1
thatthe S -bimodule S is Noetheriansincethe S -bimodules F⊗S andK[x,x−1]⊗S are
n n n n−1 n−1
Noetherian. This proves that the set J(S ) satisfies the a.c.c.. (cid:3)
n
Definition. For an algebra A we say that its ideals commute if IJ =JI for all ideals I and J
of the algebra A.
Theorem 2.8 IJ =JI for all ideals I and J of the algebra S .
n
Proof. To prove the result we use induction on n. The case n = 1 is Proposition 2.5.(5). So,
let n>1 andwe assume that the resultholds for all n′ <n. By Lemma 2.6.(1), I∩(F ⊗S )=
n−1
F ⊗I for some ideal I of the algebra S . Using (15), we have the short exact sequence
n−1 n−1 n−1
of S -modules
n
0→F ⊗I →I →I →0
n−1
where I is an ideal of the algebra K[x,x−1]⊗S which is the image of the ideal I under the
n−1
epimorphism S →K[x,x−1]⊗S . It is obvious that two ideals I and I′ of the algebra S are
n n−1 n
equal iff I = I′ and I = I′. Note that (IJ) = I J = J I = (JI) (by
n−1 n−1 n−1 n−1 n−1 n−1 n−1 n−1
Lemma 2.6.(4)), and IJ =I ·J =J ·I =JI (by the same arguments and induction). Therefore,
IJ =JI. (cid:3)
The associated graded algebra gr(S ) and the algebra D .
n n
9
Definition. ThealgebraD isanalgebrageneratedoverafieldKby2nelementsx ,...,x ,y ,...,y
n 1 n n n
that satisfy the defining relations:
y x =···=y x =0, [x ,y ]=[x ,x ]=[y ,y ]=0 foralli6=j.
1 1 n n i j i j i j
Clearly, D ≃D⊗n and D = Kxαyβ. The canonical generators x , y (1 ≤i,j ≤n) of
n 1 n α,β∈Nn i j
the algebra Dn determine the ascending filtration {Dn,≤i}i∈N on the algebra Dn where Dn,≤i :=
Kxαyβ. Then dim(LD )= i+2n for i≥0, and so GK(D )=2n.
|α|+|β|≤i n,≤i 2n n
The associatedgraded algebragr(S ):= S /S is isomorphic to the algebraD .
L n(cid:0) (cid:1) i∈N n,≤i n,≤i−1 n
The nil-radical n = n(D ) of the algebra D (i.e. the sum of all the nilpotent ideals of D ) is
n n n
equal to Kxαyβ, and nn+1 = 0. LThe factor algebra D /n ≃ n K[x ,y ]/(y x ) is
∃i:αiβi6=0 n i=1 i i i i
the tensor product of the commutative algebras K[x ,y ]/(y x ).
i i i i
The inLvolution η of the algebra Sn respects the filtration {Sn,≤i}i∈N, iN.e. η(Sn,≤i)= Sn,≤i for
all i≥0; and so the associated graded algebra gr(S ) inherits the involution
n
η :D →D , x 7→y , y 7→x , i=1,...,n.
n n i i i i
In particular, the algebra D is self-dual. The algebra D is neither left nor right Noetherian
n n
as it contains the infinite direct sum D x ···x yβ of nonzero left ideals. The simple
β∈Nn n 1 n
D -modules can be easily described,
n
L \
D =D /n.
n n
In particular, all the simple D -modules are finite dimensional. The prime and maximal spectra
n
of the algebra D are easily found since Spcec(D )=Spec(D /n) and Max(D )=Max(D /n).
n n n n n
3 Classification of simple S -modules
n
In this section, we classify allthe simple S -modules (Theorem3.2.(1)). It is provedthat for each
n
simple Sn-module M, its endomorphism algebra EndSn(M) is a finite field extension of the field
K (Theorem3.2.(4)), andthe multiplicity e(M)ofM is equalto dim(EndS (M)) (Corollary3.6).
n
This is the second instance known to me when the multiplicity of a simple module is equal to the
dimensionofits endomorphismdivisionalgebra. In[4]this wasprovedforcertainsimple modules
over the ring D(P ) of differential operators on the polynomial algebra P over a perfect field of
n n
prime characteristic. Note that the algebra D(P ) is neither left nor right Noetherian and not
n
finitely generated either.
The algebra S is Zn-graded. The algebra S = S is a Zn-graded algebra where
S :=S ⊗···⊗nS , α=(α ,...,α ), n α∈Zn n,α
n,α 1,α1 1,αn 1 n L
xiS =S xi if i≥1 ,
1,0 1,0
S := S if i=0 ,
1,i  1,0
y|i|S1,0 =S1,0y|i| if i≤−1 ,
S1,0 := KhE00,E11,...i = K ⊕KE00 ⊕KE11 ⊕··· is a commutative non-Noetherian algebra
(KE ⊂KE ⊕KE ⊂··· is an ascending chain of ideals of the algebra S ).
00 00 11 1,0
The polynomialalgebraP = P is an Nn-gradedalgebra(and, automatically,a Zn-
n α∈Nn n,α
graded algebra) where P := Kxα. Moreover, S P ⊆ P for all α,β ∈ Zn. Therefore,
n,α n,α n,β n,α+β
the polynomial algebra P is a ZnL-graded S -module.
n n
Note that the involution η reverses the Zn-grading of the algebra S , i.e.
n
η(S )=S , α∈Zn,
n,α n,−α
and it acts as the identity map on the algebra S .
n,0
ForanalgebraA,letAdenotethesetofisoclassesofsimpleA-modules. ForasimpleA-module
M, [M] is its isoclass. We usually drop the brackets [,] if this does not lead to confusion.
b
The simple S -modules. The next Lemma gives all the simple S -modules.
1 1
10