We give a short proof of the Buchstaber–Rees theorem concerning symmetric powers. The proof is based on the notion of a formal characteristic function of a linear map of algebras.
This paper is based on a talk given at the Robin K. Bullough Memorial Symposium in June 2009. Robin Bullough had been always interested in new algebraic and geometric structures arising from integrable systems theory (and mathematical physics in general) as well as in philosophical understanding of what ‘integrability’ is. We had numerous conversations on that with him in the last 10 years. The subject of this paper is related to both aspects. We use a method inspired by our studies in supermanifold geometry (namely, Berezinians and associated structures, ultimately rooted in quantum physics) to give a short direct proof of a theorem of Buchstaber and Rees concerning symmetric powers of algebras and spaces. Buchstaber and Rees’s notion of an ‘n-homomorphism’, motivated by the earlier studies of n-valued groups, cannot be separated from integrable systems in the broad sense. In recent years, an understanding of ‘integrability’ of various objects has spread, according to which an ‘integrable’ case of any notion (a system of ordinary differential equations, a function, a manifold, etc.) is a case somehow distinguished and discrete within the continuum of ‘generic’ (or ‘non-integrable’) cases. It is often related to some non-trivial algebraic identities. An approach to linear maps of algebras that we have put forward (see below) allows us to isolate a hierarchy of ‘good’ classes of such maps, which may be also regarded as ‘integrable’. The notions of algebra homomorphisms and ‘n-homomorphisms’ find their natural place in such a hierarchy based on the analysis of a formal characteristic function of a linear map of algebras that we introduced. We also see the next step of this hierarchy (the ‘p|q-homomorphisms’, which we hope to study further elsewhere).
In this paper, we give a brief exposition of our general method together with its very concrete application, which is in the title.
A theory of ‘n-homomorphisms’ (or ‘Frobenius n-homomorphisms’) of algebras was developed in a series of papers of V. M. Buchstaber and E. G. Rees (see particularly [1–3] and references therein). We recall the definition of an n-homomorphism in §3. This notion originated in the studies of an analogue for multi-valued groups of the Hopf algebra of functions on a group (namely, for an n-valued group, the co-product in such an analogue is an n-homomorphism) and was then identified with a structure discovered by Frobenius in his theory of higher group characters.
The main algebraic result of Buchstaber and Rees is the following fundamental theorem. For commutative associative algebras with unit A and B over real or complex numbers (the condition of the commutativity of A can be relaxed), under some technical assumptions on B, there is a one-to-one correspondence between the algebra homomorphisms SnA→B and the n-homomorphisms A→B. Here SnA⊂A⊗n is the symmetric power of A as a vector space with the algebra structure induced from A⊗n. In particular, when A=C(X) for a compact Hausdorff space X and , this gives the following extension of the classical theorem of Gelfand & Kolmogorov : for any n, the symmetric power SymnX=X×…×X/Sn is canonically embedded into the linear space C(X)* so that the image of the embedding is the set of all n-homomorphisms . The proof of this remarkable result in Buchstaber & Rees  is a tour de force of combinatorial ingenuity.
In Khudaverdian & Voronov , we suggested the following construction. For an arbitrary linear map f of a commutative algebra A into a commutative algebra B we introduce a formal ‘characteristic function’ (a formal power series in z) and consider classes of maps f such that Rf(a, z) is a genuine function. A formal analysis of the behaviour of Rf(a, z) ‘at the infinity’ leads to another crucial notion, of a ‘Berezinian’ on the algebra A associated with a linear map f : A→B—shortly, an ‘f-Berezinian’. The study of the characteristic function and the f-Berezinian allows us to single out certain classes of linear maps ‘with good properties’ among arbitrary linear maps of algebras, as follows.
Suppose the characteristic function Rf(a, z) is a linear function of the variable z. This precisely characterizes those linear maps f : A→B that are the ring homomorphisms.
Suppose the characteristic function Rf(a, z) is a polynomial function of the variable z. It can be shown that such a condition precisely characterizes the n-homomorphisms of algebras in the sense of Buchstaber and Rees (for some natural number n). Our construction gives a different approach to their theory. In particular, by using f-Berezinian (which is almost tautologically a multiplicative map), we obtain an effortless proof of the Buchstaber and Rees main theorem stated above. See corollary 4.3 from proposition 4.2 below, which is the crucial point of the proof; the way it is obtained is the main advantage of our approach. It also leads to other substantial simplifications of the theory.
The next class of ‘good’ linear maps of algebras f : A→B corresponds to the case when the characteristic function Rf(a, z) is rational. We call them the p|q-homomorphisms. Geometrically they correspond to some interesting generalization of symmetric powers. This is briefly discussed in §6. See also Khudaverdian & Voronov .
Our approach is inspired by the previous work on invariants of supermatrices in Khudaverdian & Voronov . There we investigated the rational function where A is an even operator on a superspace and Ber is its Berezinian (superdeterminant). It can be also written as where str stands for supertrace, owing to the relation BereX=estrX. Comparing the expansions of RA(z) at zero and at infinity allowed us to establish non-trivial relations for the exterior powers of operators and spaces (in the Grothendieck ring) and in particular obtain a new formula for the Berezinian as the ratio of certain polynomial invariants.
2. The formal characteristic function
Let A and B be two associative, commutative and unital algebras over or . We shall study linear maps f : A→B that are not assumed to be algebra homomorphisms. For such a fixed map f, we say that an arbitrary function ϕ : A→B is f-polynomial if its values ϕ(a), where a∈A, are given by a universal (i.e. independent of a) polynomial expression in f(a), f(a2),…. The ring of f-polynomial functions is naturally graded so that the degree of f(a) is 1, the degree of f(a2) is 2, etc.
The characteristic function of a linear map of algebras f  is defined as the formal power series with coefficients in B 2.1 It is a ‘function’ of both z and a∈A. The coefficients ψk(a) of the series (2.1) are f-polynomial functions of a of degree k, so that ψk(λa)=λkψk(a). Indeed, by differentiating equation (2.1) with respect to z we can see that ψk(a) can be obtained by the Newton type recurrent formulae: With respect to the linear map f, the characteristic function enjoys an obvious exponential property
For a given linear map f, its characteristic function obeys the relations 2.2 or 2.3 (making sense as formal power series). This directly follows from the definition. We shall use these relations later.
One can make the following formal transformation of the characteristic function of f aimed at obtaining its ‘expansion near infinity’. More precisely: Rf(a, z) is defined initially as a formal power series in z; it can be seen as the Taylor expansion at zero of some genuine function of the real or complex variable z if such a function exists. Assume that it exists and keeps the notation Rf(a, z) for it. We have, by a formal transformation, that near the infinity in z, where we have denoted χ=f(1). Here, we assume whatever we may need for the calculation, e.g. that a−1 exists, and so on. Initially f(1)∈B; an assumption that there is a Laurent expansion at infinity forces one to conclude that χ must be a number in . We also observe that the formal expression arises as the coefficient of the leading term at infinity.
We are not using this heuristic argument in the next sections; however, it may be helpful for understanding our approach. Instead of discussing how this formal calculation can be made rigorous, we shall go around it and apply arguments more specific for a particular case.
3. From characteristic function to n-homomorphisms
Suppose that the formal power series (2.1) terminates, i.e. Rf(a, z) is a polynomial function in z for all a∈A, and that the degree of Rf(a, z) is bounded by some independent of a. We claim that in this case f(1)=n where n is a natural number and that Rf(a, z) is a polynomial of degree n, i.e. the degree of Rf(a, z) is at most n for all a and is exactly n for some a (provided some technical assumption for the target algebra B).
Indeed, let . Consider . We show first that the element χ∈B is a natural number. We have , where (1+z)χ is considered as a formal power series: But Rf(a, z) is a polynomial of degree at most N. Hence χ(χ − 1)⋯(χ − k + 1) = 0 for all k>N. If in an algebra B, the equation b(b−1)(b−2)⋯(b−k)=0 implies that b=j for some j=0,1,…,k, the algebra B is called ‘connected’ . This is satisfied, for example, if B does not have divisors of zero. Provided such a condition for B holds, we conclude that χ=n for some integer n between 1 and N.
Now we show that the value gives the degree of the polynomial function Rf(a, z). For an arbitrary a, we apply the above identity (2.2) to obtain 3.1
(note that ). More explicitly, the right-hand side of equation (3.1) has the form 3.2 Since the functions ψk(a) are f-polynomial of degrees k, all terms in the bracket are inhomogeneous polynomials in z−1 of degrees 0,1,…,N, respectively. We are given that Rf(a, z) is a polynomial in z; it follows that . By multiplying in equation (3.2) through and comparing with the expansion in equation (2.1), we conclude that the degree of Rf(a, z) in z is at most n, for any a. In particular, for a=1, we have Rf(1,z)=(1+z)n where the degree is exactly n. This completes the proof of the claim above.
Let us consider a linear map f : A→B such that its characteristic function Rf(a, z) is a polynomial of degree n in z, i.e. it is at most n for all a and it is exactly n for some a. As we have found, the integer n is necessarily the value of f at 1∈A.
We shall show now that the class of such maps coincides with the class of n-homomorphisms introduced by Buchstaber and Rees. Let us recall their definition. For a given linear map f : A→B, Buchstaber and Rees defined maps for all k=1,2,…, by a ‘Frobenius recursion formula’: Φ1=f, and
The class of the linear maps f : A→B such that their characteristic functions Rf(a, z) are polynomials of degree n in z coincides with the class of the Buchstaber–Rees Frobenius n-homomorphisms.
From the recursion formula, it is easy to show by induction that the multi-linear maps Φk are symmetric. Therefore, they are defined by the restrictions to the diagonal. Using induction again, one deduces that the functions φk(a)= Φk(a,…,a) obey Newton-type recurrence relations similar to those satisfied by our functions ψk(a) defined as the coefficients of the expansion (2.1). From here one can establish that Φk(a,…,a)=k!ψk(a), so Φk(a1,…,ak) can be recovered from ψk(a) by polarization (see remark below). Therefore, the identical vanishing of the Frobenius maps Φk for all is equivalent to the characteristic function Rf(a, z) being a polynomial of degree . Suppose the characteristic function Rf(a, z) is a polynomial of degree n, i.e. of degree for all a and exactly n for some a. Then f(1)=χ is an integer between 1 and n; if it is less than n, then the degree of Rf(a, z) is less than n for all a as shown above. Hence f(1)=n. Conversely, if f(1)=n, then Rf(1,z)=(1+z)n and Rf(a, z) is a polynomial of degree n as claimed.
Here is a formula for the polarization of a homogeneous polynomial of degree k (the restriction of a symmetric k-linear function to the diagonal): 3.3 Here Φk(a,…,a)=k!ψk(a). For example, and The relation between homogeneous polynomial functions and symmetric multi-linear functions is standard, but explicit formulas are not easy to find in the literature.
It may be noted that the ‘ghost’ of our characteristic function Rf(a, z) (to which we came motivated by the study of Berezinians in ) did appear in relation to n-homomorphisms but was never recognized. We can see these instances in hindsight. The initial definition of an ‘n-ring homomorphism’ in Buchstaber & Rees  (with a slightly different normalization from that adopted later) used a certain monic polynomial p(a, t)=tn−… of degree n. That definition was abandoned starting from Buchstaber & Rees  in favour of the Frobenius recursion. In hindsight one can relate the Buchstaber & Rees polynomial p(a, t) with our characteristic function when it is a polynomial of a given degree n, by the formula p(a, t)=tnRf(a,−1/t). In Buchstaber & Rees , corollary 2.12, the maps Φk(a,…,a) for an n-homomorphism were assembled into a certain generating function, which was then re-written in the exponential form. The resulting power series can be identified with our characteristic function. That series was used only in the proof of their theorem 2.9 about the sum of n- and m-homomorphisms, while the central theorem 2.8 concerning the relation of the Frobenius n-homomorphisms with the algebra homomorphisms of the symmetric powers was obtained in Buchstaber & Rees  by a long combinatorial argument.
4. Berezinian and a proof of the key statement
Let f : A→B be an arbitrary linear map of algebras. We define formally the f-Berezinian on A as a map berf : A→B by the formula when it makes sense. (Compare the Liouville formulas for matrices, and , with the ordinary and super determinants, respectively.) Note that the f-Berezinian appeared in the heuristic calculation in §2 above as the leading coefficient of the characteristic function Rf(a, z) ‘at infinity’.
The function berf is multiplicative: (whenever both sides make sense).
By the definition
This holds even if the algebra B is non-commutative, but f is a ‘trace’, i.e. f(a1a2)=f(a2a1).
Let f be an n-homomorphism. Then is an f-polynomial function of a with values in B: and is well defined for all a.
For an n-homomorphism,
Consider the equality (compare formulas (3.1) and (3.2); we have legitimately set N=n). Collecting all the terms of degree n in z, we arrive at the identity where the right-hand side is by the definition berf(a).
This proposition is the crucial step in our proof of the Buchstaber–Rees theorem.
For an n-homomorphism f, the function ψn(a) is multiplicative in a.
The multiplicativity of the function ψn(a) for n-homomorphisms is the central fact in the Buchstaber–Rees theorem. Establishing it was the main difficulty of the proof in Buchstaber & Rees , where it was deduced by complicated combinatorial arguments. In our approach this fact comes about almost without effort.
The apparatus of characteristic functions allows us to obtain easily many other facts. For example, if f and g are n- and m-homomorphisms A→B, respectively, then the exponential property of characteristic functions immediately implies that f+g is an (n+m)-homomorphism, since its characteristic function is the product of polynomials of degrees and . If g is an m-homomorphism A→B and f is an n-homomorphism B→C, then . Since we know that Rg(a, z) is a polynomial in z of degree at most m, and the f-Berezinian berfb is a polynomial in b∈B of degree n, we conclude that Rf∘g(a, z) has the degree at most nm in z, therefore f∘g is an nm-homomorphism. (The first statement was established in Buchstaber & Rees  by a similar argument, see remark 3.3, while the second statement was obtained in Buchstaber & Rees  in a much harder way as a corollary of the main theorem.)
5. A completion of the proof
Let us formulate the main theorem of Buchstaber and Rees.
Theorem 5.1 Buchstaber and Rees 
There is a one-to-one correspondence between the n-homomorphisms A→B and the algebra homomorphisms SnA→B. The algebra homomorphism F : SnA→B corresponding to an n-homomorphism f : A→B is given by the formula 5.1 where on the left-hand side a linear map from SnA is written as a symmetric multi-linear function.
Here Φn(a1,…,an) is the top non-vanishing term of the Frobenius recursion for f.
Basing on the key result established in the previous section (corollary 4.3), we can complete the proof of this theorem as follows.
Let be the set of all n-homomorphisms from an algebra A to an algebra B. We shall construct two mutually inverse maps between the spaces and , thus establishing their one-to-one correspondence: To every algebra homomorphism we shall assign an n-homomorphism , and to every n-homomorphism we shall assign an algebra homomorphism , in the following way.
It is convenient to introduce an n×n matrix with entries in the algebra A⊗n=A⊗…⊗A, where The map is a matrix representation A→Mat(n, A⊗n). Consider an equation 5.2 The coefficients of the determinant on the left-hand side take values a priori in the algebra A⊗n, but they actually belong to the subalgebra SnA. Let equation (5.2) hold identically in z. We shall show that, for a given F , equation (5.2) uniquely defines f so that we can set α(F):=f, and conversely, for a given f, equation (5.2) uniquely defines F so that we can set β(f):=F. (Then the maps α and β will automatically be mutually inverse.)
Suppose is given. Then, by comparing the linear terms in equation (5.2), we see that f should be given by the formula 5.3 We have .
The element a⊗1⊗⋯⊗1+…+1⊗1⊗⋯⊗a appears in Buchstaber & Rees  where it is denoted Δ(a).
Take equation (5.3) as the definition of f. Evidently, it is a linear map A→B. Calculate its characteristic function. We have (note that both and F can be swapped with functions, as we did in the calculation above, because they are algebra homomorphisms). So equation (5.2) is indeed satisfied. In particular, the characteristic function of f is a polynomial of degree n. Hence f is an n-homomorphism A→B. We have constructed the desired map .
Suppose now is given. We wish to define F from equation (5.2). We need to show the existence of a linear map F : SnA→B and that it is an algebra homomorphism. Assume that a linear map F satisfying equation (5.2) exists. We shall deduce its uniqueness. By developing the left-hand side of equation (5.2), we see that this equation specifies F on all the elements of SnA of the form , including . In particular, we should have 5.4 Take equation (5.4) as the definition of F on such elements. Note that the elements of the form linearly span SnA, so if a linear map F : SnA→B satisfying equation (5.4) exists, this formula determines it uniquely. Moreover, by replacing a by 1+az in equation (5.4), we see that equation (5.2) will be automatically satisfied. For the existence, observe that berf(a)=ψn(a) and ψn is the restriction on the diagonal of the symmetric multi-linear map (1/n!)Φ : A×…×A→B, which corresponds to a linear map SnA→B. It is the map F we are looking for. (We recover the Buchstaber–Rees formula (5.1).) It remains to check that so obtained F is indeed an algebra homomorphism. This immediately follows from corollary 4.3, which tells that F is multiplicative on the elements a⊗…⊗a, where it is ψn(a), because the elements a⊗…⊗a span the whole algebra SnA. So we have arrived at the desired map , and the maps α and β are mutually inverse by the construction.
This concludes the proof.
6. Rational characteristic functions and p|q-homomorphisms
Our notion of characteristic function readily suggests the next step in the investigation of linear maps of algebras with good properties (after the n-homomorphisms). We shall consider it now briefly.
Suppose the characteristic function Rf(a, z) of a linear map f : A→B is not a polynomial in z, but a rational function that can be written as the ratio of polynomials of degrees p and q. We call such a linear map a p|q-homomorphism. One can deduce that for a p|q-homomorphism the value f(1) must be an integer: χ=f(1)=p−q.
The negative −f of a ring homomorphism f is a 0|1-homomorphism. The difference f(p)−f(q) of a p-homomorphism f(p) and a q-homomorphism f(q) is a p|q-homomorphism. In particular, a linear combination of algebra homomorphisms of the form where is a p|q-homomorphism with , and . (This follows from the exponential property of the characteristic function.)
The geometric meaning of p|q-homomorphisms is related to a certain generalization of the notion of symmetric powers.
Consider a topological space X. We define its p|q-th symmetric power Symp|q(X) as the identification space of Xp+q=Xp×Xq with respect to the action of the group Sp×Sq together with the relations The algebraic analogue of Symp|q(X) is the p|q-th symmetric power of a commutative associative algebra with unit A, which we denote Sp|qA. We define the algebra Sp|qA as the subalgebra μ−1(Sp−1A⊗Sq−1A) in SpA⊗SqA, where μ : SpA⊗SqA→Sp−1A⊗Sq−1A⊗A is the multiplication of the last arguments.
For the algebra of polynomials in one variable , it can be shown that the algebra Sp|qA is the algebra of all polynomial invariants of p|q by p|q matrices. In more detail, consider even matrices p|q by p|q where the entries are regarded as indeterminates of appropriate parity (over complex numbers). The general linear supergroup GL(p|q) acts on such matrices by conjugation. ‘Polynomial invariants’ of p|q by p|q matrices are the polynomial functions of the matrix entries invariant under such action. Denote their algebra by Ip|q. The problem is to describe the algebra Ip|q in terms of functions of the p+q eigenvalues λi,μα similar to the classical case q=0. The restriction of a polynomial invariant of p|q by p|q matrices to the diagonal matrices is clearly a Sp×Sq-invariant polynomial (or an element of SpA⊗SqA where ). Such a polynomial f(λ1…,λp,μ1,…,μq) separately symmetric in λi and μα extends to an element of Ip|q if and only if it satisfies an extra condition that is equivalent to f∈Sp|qA. (Here, there is a great difference with the rational invariants, for which no extra condition arises.) This is a non-trivial theorem that can be traced to Berezin  (see , p. 294, see also ). See discussion in Khudaverdian & Voronov .
(The algebra described in the example above and its deformations have recently become important in integrable systems, e.g. .)
There is a relation between algebra homomorphisms Sp|qA→B and p|q-homomorphisms A→B. To each homomorphism Sp|qA→B canonically corresponds a p|q-homomorphism A→B.
For the algebra of functions on a topological space X, the element x=[x1,…,xp+q]∈Symp|q(X) defines a p|q-homomorphism by the formula This gives a natural map from Symp|q(X) to the dual space A* of the algebra A=C(X), which generalizes the Gelfand–Kolmogorov map X→A* and the Buchstaber–Rees map Symn(X)→A*.
By using formulas from Khudaverdian & Voronov , the condition that f : A→B is a p|q-homomorphism can be expressed by the algebraic equations 6.1 for all and all a∈A, where ψk(f,a)=ψk(a) are the coefficients in the expansion of the characteristic function (2.1). The determinants arising in equation (6.1) are the well-known Hankel determinants. The condition that they identically vanish replaces the condition ψk(a)=0 for all for an n-homomorphism.
We see that the image of Symp|q(X) in A*, where A=C(X), under the map x↦f=evx satisfies equations (6.1). The system (6.1) can be regarded as an infinite system of polynomial equations for the ‘coordinates’ f(a) of a point f in the infinite-dimensional vector space A*.
A conjectured statement is that the solutions of equations (6.1) give precisely the image of Symp|q(X) in A*. This would be an exact analogue of the Gelfand–Kolmogorov and Buchstaber–Rees theorems. The corresponding algebraic statement should be a one-to-one correspondence between the p|q-homomorphisms A→B and the algebra homomorphisms Sp|qA→B extending the correspondence given by our formula (5.2).
We thank V. M. Buchstaber for discussions.
One contribution of 13 to a Theme Issue ‘Nonlinear phenomena, optical and quantum solitons’.
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