Olav Arnfinn Laudal*
Institute of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Received date 7 October 2009; Accepted date 10 December 2010
Visit for more related articles at Journal of Generalized Lie Theory and Applications
We have previously introduced the notion of non-commutative phase space (algebra) associated to any associative algebra, defined over a field. The purpose of the present paper is to prove that this construction is useful in non-commutative deformation theory for the construction of the versal family of finite families of modules. In particular, we obtain a much better understanding of the obstruction calculus, that is, of the Massey products.
In , we sketched a physical “toy model,” where the space-time of classical physics became a section of a universal fiber space , defined on the moduli space of the physical systems we chose to consider (in this case, the systems composed of an observer and an observed, both sitting in a Euclidean 3-space). This moduli space was called the time-space. Time, in this mathematical model, was defined to be a metric ρ on the time-space, measuring all possible infinitesimal changes of the state of the objects in the family we are studying. This gave us a model of relativity theory, in which the set of all (relative) velocities turned out to be a projective space. Dynamics was introduced into this picture, via the general construction, for any associative algebra A, of a phase space Ph(A). This is a universal pair of a homomorphism of algebras, and a derivation, such that for any homomorphism of A into a k-algebra R, the derivations of A in R are induced by unique homomorphisms composed with d. Iterating this Ph(−)-construction, we obtained a limit morphism with image, and a universal derivationthe Dirac-derivation. A general dynamical structure of order n is now a two-sided δ-ideal σ in inducing a surjective homomorphism
In  and later in , we have shown that, associated to any such time space H with a fixed dynamical structure H(σ), there is a kind of “Quantum field theory”. In particular, we have stressed the point that, if H is the affine ring of a moduli space of the objects we want to study, the ring is the complete ring of observables, containing the parameters not only of the iso-classes of the objects in question, but also of all dynamical parameters. The choice made by fixing the dynamical structure σ, and reducing to the k-algebra H(σ), would classically correspond to the introduction of a parsimony principle (e.g. to the choice of some Lagrangian).
The purpose of this paper is to study this phase-space construction in greater detail. There is a natural descending filtration of two-sided ideals,.The corresponding quotients are finite dimensional vector spaces, and considered as affine varieties; these are our non-commutative Jet-spaces.
We will first see, in Section 2, that we may extend the usual prolongation-projection procedure of Elie Cartan to this non-commutative setting, and obtain a framework for the study of general systems of (non-commutative) PDEs; see also .
Then, in Section 4, the main part of the paper follows: the construction for finitely generated associative algebras A of the versal family of the non-commutative deformation functor of any finite family of finitely dimensional A-modules, based on the phase-space of a resolution of the k-algebra A.
Given a k-algebra A, denote by A/k − alg the category where the objects are homomorphisms of k-algebras κ : A → R, and the morphisms are commutative diagrams:
and consider the functor
defined by It is representable by a k-algebra-morphism, ith a universal family given by a universal derivation It is easy to construct Ph(A). In fact, let be a surjective homomorphism of algebras, with freely generated by the and put I = kerπ. Let,
where is a formal variable. Clearly there is a homomorphism and a derivation defined by putting the equivalence class ofSince both kill the ideal I, they define a homomorphism and a derivation To see that i0 and d have the universal property, be an object of the categoryAny derivation defines a derivation mapping be the homomorphism defined by
where sends I and dI to zero, and so defines a homomorphism such that the composition with is The unicity is a consequence of the fact that the images of i0 and d generate Ph(A) as k-algebra.
Clearly Ph(−) is a covariant functor on k − alg, and we have the identities,
with the last one associating d to the identity endomorphism of Ph(A). In particular, we see that i0 has a cosection, corresponding to the trivial (zero) derivation of A.
Let now V be a right A-module, with the structure morphism).We obtain a universal derivation:
defined by Using the long exact sequence
we obtain the non-commutative Kodaira-Spencer class
inducing the Kodaira-Spencer morphism
via the identity then the exact sequence above proves that there exist a such that This is just another way of proving that c(V ) is the obstruction for the existence of a connection,
It is well known, I think, that in the commutative case, the Kodaira-Spencer class gives rise to a Chern character by putting
and that if c(V ) = 0, the curvature of the connection ∇ induces a curvature class in a generalized Lie-algebra cohomology:
Any Ph(A)-module W, given by its structure map,
corresponds bijectively to an induced A-module structuretogether with a derivation defining an element Fixing this last element, we find that the set of Ph(A)-module structures on the A-module W is in one-to-one correspondence with Conversely, starting with an A-module V and an element we obtain a Ph(A)-module It is then easy to see that the kernel of the natural map
induced by the linear map
is the quotient
and the image is a subspace which is rather easy to compute; see examples below.
Remark 1. Defining time as a metric on the moduli space, Simp(A), of simple A-modules, in line with the philosophy of , noticing that is the tangent space of Simp(A) at the point corresponding to V , we see that the non-commutative space Ph(A) also parametrizes the set of generalized momenta, that is, the set of pairs of a point and a tangent vector at that point.
Example 2. (i) Let then obviously, and d is given by d(t) = dt, such that for we find with the notations of , that is, the non-commutative derivation of f with respect to t. One should also compare this with the non-commutative Taylor formula of loc.cit. If is an A-module, defined is an A-module, defined by the matrix and is defined in terms of the matrix then the Ph(A)-module is the -module defined by the action of the two matrices and we find
We have the following inequalities:
(ii) Let then we find
In particular, we have a surjective homomorphism
with the right-hand side algebra being theWeyl algebra. This homomorphism exists in all dimensions.We also have a surjective homomorphism,
that is, onto the affine algebra of the classical phase-space.
The phase-space construction may, of course, be iterated. Given the k-algebra A, we may form the sequence defined inductively by
Let be the canonical imbedding, and let be the corresponding derivation. Since the composition of and the derivation dn+1 is a derivation there exists by universality a homomorphism such that
Notice that we compose functions and functors from left to right. Clearly, we may continue this process constructing new homomorphisms
with the property
Notice also that we have the “bi-gone” and the “hexagone”
and, in general,
which is all easily proved by composing with Thus, the Ph* (A) is a semi-cosimplicial algebra with a cosection onto A. Therefore, for any object
the semi-cosimplicial algebra above induces a semi-simplicial k-vector space and one should be interested in its homology.
The system of k-algebras and homomorphisms of k-algebras has an inductive (direct) limit, together with homomorphisms satisfying
Moreover, the family of derivations define a unique derivation such that Put
The k-algebra has a descending filtration of two-sided ideals, with given inductively by
such that the derivation δ induces derivations Using the canonical homomorphismwe pull the filtration back tonot bothering to change the notation.
Definition 3. Let be the completion of in the topology given by the filtration The k-algebra will be referred to as the k-algebra of higher differentials, and D(A) will be called the k-algebra of formalized higher differentials. Put
Clearly, δ defines a derivation on D(A), and an isomorphism of k-algebras
and, in particular, an algebra homomorphism
inducing the algebra homomorphisms
which, by killing, in the right-hand side algebra, the image of the maximal ideal, of A corresponding to a point induces a homomorphism of k-algebras
and an injective homomorphism
see . More generally, let A be a finitely generated k-algebra and let be an n-dimensional representation (e.g. a point of Simpn(A)) corresponding to a two-sided ideal m = ker ρ of A. Then induces a homomorphism
and we will be interested in the image; see Section 4.
The k-algebras are our generalized jet spaces. In fact, any homomorphism of A-algebras
is a usual differential operator of order ≤ n on A. Notice also the commutative diagram
Here the upper vertical morphisms are injective, with the lower line being the sequence of symbols.
It is easy to see that the differential operators form an associative k-algebra, Diff(A). In fact, assume two differential operators
and consider the functorially defined diagram
then the product is defined by the composition
Let now V be, as above, a right A-module, with structure morphism Consider the linear map
Assume that the non-commutative Kodaira-Spencer class, defined above,
vanishes. Then, as we know, there exist a connection, that is, a linear map
such that It is also easy to see that this connection induces higher-order connections, that is, k-linear maps,
In fact, we just have to prove that ∇(n) is well defined, that is, we have to prove that
where we have put , we find
These higher-order connections will induce a diagram
where the lower line is the sequence of symbols. Notice that
as given above, by definition has the property that for all a ∈ A and all v ∈ V we have
Assume, in particular, that V and the A-moduleW are free of ranks p and q, respectively. Let be a family of A-homomorphisms defining a generalized differential operator
The solution space of is by definition There are natural generalizations of this set-up, which we will, hopefully, return to in a later paper, extending the classical prolongation-projection method of Elie Cartan to this non-commutative setting. See Example 4 for the commutative analogue.
In , we introduced the notion of a dynamical structure for a k-algebra A, as a two-sided δ-stable ideal σ ⊂ Ph∞ (A), or equivalently as the corresponding quotient A(σ) of the δ-algebra Any such A(σ)will be given in terms of a sequence of ideals, with the property that The solution space of such a system, should be considered as the non-commutative scheme parametrized by A(σ), that is, as the geometric system of all simple representations of A(σ); see .
This is, in a sense, dual to the classical theory of PDEs, as we will show by considering the following example, leaving the general situation to the hypothetical paper referred to above.
Example 4 (see ). (i) Let and consider the situation corresponding to a free particle (see ) that is, where we have obtained A(σ) by killing for every then the commutativization is a free A-module generated by the basis
Put The dual basis may be identified with a basis of the A-module of all (classical higher-order) differential operators of order less or equal to k. In fact, consider the composition
then, for f ∈ A we have
where we assume
and where is μpth-order derivation with respect to we let to be the identity operator on A..
Now, consider the commutativization of A(σ), as a k-linear space, and for every k ≥ 1,
as a family of affine spaces fibered over Simp1(A),
This family is defined by the homomorphism of k-algebras
Let then the system of equations
is a system of partial differential equations (an SPDE, for short). Suppose there is a solution, that is, an f ∈ A, such that
then, for every j, we must have
which amounts to extending the SPDE by, including together with the polynomials
where it should be clear how to interpret the indices. Let us denote by P the extended family of polynomials,
and let be the ideal, generated by the polynomials in P, contained in Denote by Sm := the corresponding subvariety. Clearly, the canonical map induced by the trivial derivation of has a canonical restriction Denote also by the restriction of the morphism defined above, to Sk. Classically, the system is called regular if all πk are fiber bundles, so smooth, for all Now, for any closed point of Spec(A), that is, for any point consider the sequence of fibers over and the corresponding sequence of maps An element corresponds exactly to an elementfor which
that is, to a formal solution of the SPDE. Thus, the projective limit of schemes is the space of formal solutions of the SPDE at
A fundamental problem in the classical theory of PDE is then the following.
Find necessary and sufficient conditions on the SPDE to be non-empty, and find, based on its structure. In particular, compute its dimension
We will not, here, venture into this vast theory, but just add one remark. The solution space is in fact a family, with parameter-space Simp1(A). Given any point the (formal) scheme, of formal solutions may have deformations.We might want to compute the formal moduli and relate the given family to the corresponding mini-versal family.
The tangent space of is given as
see . A tangent at the point of Simp1(A) is the value at t of a linear combination of the fundamental vector fields, the derivations of A. The map between the tangent space of the given family and the tangent space of H is then easily seen to be the following:
where is the class of the map, associating a to the class at The image of the tangent at of corresponding to Dj , in the tangent space of H, is zero if this map is a derivation. Now, this is exactly what we have arranged, together with any P ∈ p, and also including
in the ideal p. Thus, the map η is trivial, and the given pro-family is formally constant, as one probably should have suspected! Moreover, it is easy to see that if has a local section, then is formally constant at The basic problem is to find computable conditions under which the constancy of πk implies the surjectivity of pl1, and thereby the non-triviality of S(P)(t).
We will, hopefully, come back to these questions in a later paper.
and it is easy to see that and, of course, In particular, there is a homomorphism onto
(iii) Let now so that there are no natural surjective homomorphisms is, however, injective. The difference between examples (i) and (ii) is, of course, due to the fact that in the first case A is graded, and in the second it is not; see Section 4.
3 Non-commutative deformations of families of modules
In [5,6,7], we introduced non-commutative deformations of families of modules of non-commutative k-algebras, and the notion of swarm of right modules (or more generally of objects in a k-linear abelian category). Let denote the category of r-pointed not necessarily commutative k-algebras R. The objects are the diagrams of k-algebras
such that the composition of ι and π is the identity. Any such r-pointed k-algebra R is isomorphic to a k-algebra of The radical of R is the bilateral ideal Rad The dual k-vector space of Rad(R)/ Rad(R)2 is called the tangent space of R.
For r = 1, there is an obvious inclusion of categories where l, as usual, denotes the category of commutative local Artinian k-algebras with residue field k.
Fix a (not necessarily commutative) associative k-algebra A and consider a right A-module M. The ordinary deformation functor is then defined. Assuming has a finite k-dimension for i = 1, 2, it is well known (see  or ) that DefM has a pro-representing hull H, the formal moduli of M. Moreover, the tangent space of H is isomorphic to and H can be computed in terms of and their matric Massey products; see .
In the general case, consider a finite family of right A-modules. Assume that dimk ∞. Any such family of A-modules will be called a swarm. We will define a deformation functor generalizing the functor DefM above. Given an object consider the k-vector space and the left R-module It is easy to see that
Clearly, π defines a k-linear and left R-linear map
inducing a homomorphism of R-endomorphism rings,
The right A-module structure on the Vi s is defined by a homomorphism of k-algebras:
Notice that this homomorphism also provides each with an A-bimodule structure. Let Sets be the set of isoclasses of homomorphisms of k-algebras
such that where the equivalence relation is defined by inner automorphisms in the k-algebra inducing the identity on One easily proves that DefV has the same properties as the ordinary deformation functor and we may prove the following theorem (see ).
Theorem 5. The functorDefV has a pro-representable hull, that is, an object H of the category of pro-objects of together with a versal family
where m = Rad(H), such that the corresponding morphism of functors on
defined for is smooth and an isomorphism on the tangent level. Moreover, H is uniquely determined by a set of matric Massey products defined on subspaces
with values in
The right action of defines a homomorphism of k-algebras,
and the k-algebra O(V) acts on the family of A-modules extending the action of A. If for all i = 1, . . . , r, the operation of associating turns out to be a closure operation.
Moreover, we prove the crucial result.
Theorem 6 (a generalized Burnside theorem). Let A be a finite dimensional k-algebra, with k being an algebraically closed field. Consider the family of all simple A-modules, then
is an isomorphism.
We also prove that there exists, in the non-commutative deformation theory, an obvious analogy to the notion of pro-representing (modular) substratum H0 of the formal moduli H; see . The tangent space of H0 is determined by a family of subspaces
the elements of which should be called the almost split extensions (sequences) relative to the family V, and by a subspace
which is the tangent space of the deformation functor of the full subcategory of the category of A-modules generated by the family is the set of all indecomposables of some Artinian k-algebra A, we show that the above notion of almost split sequence coincides with that of Auslander; see .
Using this we consider, in [5,7], the general problem of classification of iterated extensions of a family of modules and the corresponding classification of filtered modules with graded components in the family V, and extension type given by a directed representation graph Γ. The main result is the following; see .
Proposition 7. Let A be any k-algebra and any swarm of A-modules, such that
(i) Consider an iterated extension E of V, with representation graph Γ. Then there exists a morphism of k-algebras
(ii) The set of equivalence classes of iterated extensions of V with representation graph Γ is a quotient of the set of closed points of the affine algebraic variety
(iii) There is a versal family of A-modules defined on containing as fibers all the isomorphism classes of iterated extensions of V with representation graph Γ.
and a sub-quotient together with natural k-algebra homomorphisms
and with the property that the A-module structure on c is extended to an O-module structure in an optimal way. We then defined an affine non-commutative scheme of right A-modules to be a swarm of right A-modules, such that is an isomorphism. In particular, we considered, for finitely generated k-algebras, the swarm consisting of the finite dimensional simple A-modules, and the generic point A, together with all morphisms between them. The fact that this is a swarm, that is for all objects we have is easily proved. We have in  proved the following result (see [7, Proposition 4.1] for the definition of the notion of geometric k-algebra)
Proposition 8. Let A be a geometric k-algebra, then the natural homomorphism
is an isomorphism, that is, is a scheme for A.
In particular, is a scheme for To analyze the local structure of we need the following lemma (see [7, Lemma 3.23]).
Lemma 9. Let be a finite subset of then the morphism of k-algebras,
is topologically surjective.
Proof. Since the simple modules are distinct, there is an obvious surjection
Put and consider for m ≥ 2 the finite dimensional k-algebra, Clearly, Simp(B) = V so that by the generalized Burnside theorem (see [5, Theorem 3.4]) we find
Consider the commutative diagram
where all morphisms are natural. In particular α exists since maps into and therefore induces the morphism α commuting with the rest of the morphisms. Consequently, α has to be surjective, and we have proved the contention.
Example 10. As an example of what may occur in rank infinity, we will consider the invariant problem Here we are talking about the algebra crossed product of C[x] with the group the product in A is given by There are two “points” (i.e. orbits) modeled by the obvious origin We may also choose the two points in line with the definitions of . Obviously, C[x] corresponds to the closure of the orbit This choice is the best if we want to make visible the adjacencies in the quotient, and we will therefore treat both cases.
We need to compute
and since x acts as zero on acts as identity on V1 and as a homogenous multiplication on V0, we find
Any is determined by its values Moreover, since in A we have
The left-hand side of the last equation is and the right-hand side is δ(x), and since this must hold for all we must have δ(x) = 0. Moreover, since it is clear that the continuity of δ implies that δ must be equal to α ln for some α ∈ C. (To simplify the writing, we will put log := ln(| · |).)Therefore,
The cup-product of this class, log ∪ log, sits in and is given by the 2-cocycle
This is seen to be a boundary, that is, there exists a map such that for all we have
Just put Therefore, the cup product is zero, and if we, in general, put
where n is the number of 1s in the first index, then computing the Massey products of the element log ∈ we find the nth Massey product
and this is easily seen to be the boundary of the 1-cochain
Therefore, all Massey products are zero. Of course, we have not yet proved that they could be different from zero, that is, we have not computed the obstruction group and found it non-trivial! Now this is unnecessary. Now, assume first
is determined by the values of Since we may find a trivial derivation such that subtracting from δ we may assume δ(x) = 0. But then the formula
from which it follows that
Now, since we find
which should hold for any pair of and any p. This obviously implies δ = 0.
This argument shows not only that
when but also when Finally, we find that the formula above,
shows that for
we have for all p. Therefore,
when However, when we find that δ with for p ≥ 1, survives. These will, as above, give rise to a logarithm of the real part of C ∗. Therefore, in this case The miniversal families look like
when V1 = C[x].
4 The infinite phase space construction and Massey products
Let, as above, be a family of A-modules. To compute the relevant cohomology for the deformation theory, that is, the we may use the Leray spectral sequence of , together with the formulas
where W is any A-bimodule. Choose a surjective morphism of a free k-algebra F onto A, and put I = ker μ, then we find that
where Der is the restriction of the derivations , Moreover, consider a commutative diagram of homomorphisms of algebras, in which is not yet included
and where J := ker π has square 0. The composition map induces an element independent upon the choice of ρ′ . If this (obstruction) element vanishes, then O is the restriction to I of a derivation Subtracting this from ρ′,we may assume that ρ′ (I) = 0, so there exists a lifting If there exists a lifting then we may obviously assume that O = 0.
Now, let represent a basis of and let denote the dual basis. Consider, the free matrix k-algebra (quiver) generated in slot (i, j) by the (formal) elements of Ei,j . There is a unique homomorphism
Denote by the same letter the completion of T1 with respect to the powers of the radical Rad(T1) := kerπ. Then Consider the k-algebra and the π-induced homomorphism
Clearly, π1 splits, and it is easy to see that.
is a derivation, therefore inducing a unique homomorphism, makes the following diagram commute
Now, we would have liked to extend this diagram, completing it with commuting homomorphisms,
However, as will be clear in the next construction, the obvious continuation of this procedure does not work. In fact, the formalized higher differentials D(A) is not really the natural phase-space to work with for all purposes. In an obvious sense it is too homogenous. We are therefore led to the construction of a kind of projective resolution of A. Consider as above a surjective homomorphism, with a free k-algebra, and Obviously are also free, and is a free algebra. Let exp(δ) : F →D(F) be defined as in Section 2 by
and denote by the induced homomorphism. Define
Clearly, there are only natural surjective homomorphisms, By functoriality, the diagram above induces another commutative diagram, which may be completed to the commutative diagram
where we, in expectation of later constructions, put
Now the map
is zero, and the resulting map is, as deformation of the family V, the universal family at the tangent level. Since is a free algebra over Ph(n)(F), there is lifting We want an induced ρ2. Consider the composition
lifting The restriction to I vanishes on I2 and induces a map
It is easily seen to be F-linear, both from left and right, and so it induces the obstruction
independent upon the choice of extension Now
may be identified with
which is a subspace of
Denote by T2 the free matrix algebra (quiver), in generated by just like the construction of T1 above, such that
We may now state and prove the main result of this paper.
Theorem 11. (i) For any finite family of (finite dimensional) A-modules, there is a homomorphism , making the following diagram commutative
such that the versal family
(ii) Moreover, may be constructed recursively, as a quotient of by annihilating a series of obstructions, on, defining a morphism in
Proof. We have above constructed an obstruction for lifting ρ1 to a ρ2. It is a unique element;
Obviously, the image
generates an ideal of T1, contained in and put
Then, there is a commutative diagram
In fact, since we have divided out with the obstruction, we know that the morphism
is the restriction of a derivation
Now change the morphism It is easily seen that for this new morphism, is zero, restricted to I, proving the existence of Recall that D1(A) = H1.
Now defines be the two-sided ideal in T1 generated by
and let us put
The diagram above induces a commutative diagram, constructed as above, but where is the problem,
Consider now the map ending up in
which clearly is killed by and therefore really is a matrix of vector spaces, as an As above, this map is easily seen to be a left and right linear map as F-modules, F acting on O(3) via Moreover, the induced element
is independent on the choice of Now, we define where σ3 is defined by the image of o3, and define as above. Since, by functoriality, the morphism
must induce the zero element in the corresponding
it must be the restriction of a derivation Now change by sending leaving the other values of the parameters unchanged. Then, a little calculation shows that the new maps each to zero, inducing a morphism We now have a new situation, given by a commutative diagram, not yet including
and it is clear how to proceed. This proves (i), and the rest is a consequence of the general theorem [3, Theorem 4.2.4].
We cannot replace H by D. This follows from the trivial Example 4(iii) above. However, if we are in a graded situation, things are nicer.
Corollary 12. Assume that A is a finitely generated, graded, in degree 1, k-algebra, and assume that V is a family of graded A-modules. Then there is a corresponding graded formal moduli and there is a commutative diagram,
such that the graded versal family