Fractional residues

Aleksandrs Mihailovs Department of Mathematics University of Pennsylvania Philadelphia, PA 19104-6395 mihailov@math.upenn.edu http://www.math.upenn.edu/∼mihailov/

March 5, 1998

Abstract

Invariants of generalized tensor fields on a line are classified using special polynomials P_{mk}^{(−1/λ)} introduced here for this purpose. For the case of positive characteristic, a new invariant of formal power series, a width, is defined. Some applications to the geometric quantization of a line and conformal quantum field theory are discussed as well.

1 Introduction

Differential forms ϕ(t)dt on a line have a well-known invariant,

Res_{0} ϕ(t)dt = a_{−1},  (1)

where

ϕ(t) = ∑_{i=ord ϕ(t)}^{∞} a_{i}t^{i}.  (2)

For quadratic differential forms ψ(t)(dt)^{2}, one can construct an invariant by a composition of an invariant mapping

ψ(t)(dt)^{2} ↦ \sqrt{ψ(t)}dt  (3)

and a residue (1).

Why do we need the invariants of quadratic differential forms? One of the reasons is that the pairing

(ψ(t)(dt)^{2}, α(t)\frac{d}{dt}) = Res_{0} ψ(t)α(t)dt  (4)

identifies the space of the quadratic differentials with a dual space to the Lie algebra of vector fields on a line. Kirillov’s orbit method [2, 4, 5] associates the orbits of the group of authomorphisms of a line in that space with irreducible unitary representations of this group. Thus, for geometric quantization of a line, we need to describe the orbits and the invariants of quadratic differentials.

The first calculations for that case were done by the founder of the orbit method, Alexandre Kirillov, in [3]. I was his student at that time, and I found the explicit formulas for the invariants, some of them were announced in [3] with an indication of my priority. These results are presented in section 2. Most of them are based on the studying of special polynomials P_{mk} parametrizing the orbits of the co-adjoint representation of the group of authomorphisms of a line.

More generally, the composition of the invariant mapping

ψ(t)(dt)^{−λ} ↦ (ψ(t))^{−1/λ}dt  (5)

and a residue (1) defines an invariant of generalized differential forms ψ(t)(dt)^{−λ}. This nontrivial invariant allows us to describe the orbits of the group of authomorphisms in the space of generalized differential forms, utilizing special polynomials P_{mk}^{(−1/λ)} which for λ = −2 coinside with polynomials P_{mk} introduced in section 2. The orbits and invariants of generalized differential forms are described in section 3.

Sections 2 and 3 deal with an arbitrary field of characteristic 0. Almost without changes, the results can be transfered to the restricted case of a positive characteristic p > 0. The results related to the geometric quantization of a line in the restricted case for p > 0, corresponding to section 2, are described briefly in section 4.

For the general, not restricted, case of a field f of a positive characteristic p > 0, the situation is much more complicated. In that case even functions have a lot of additional invariants. If f is not a perfect field, there are formal power series which don’t have a polynomial normal form, see (54). However, all the orbits are closed, and the space of orbits can be metrized by a complete metrics. Section 5 presents these results. Also, at the end of section 5 I define a new invariant (67) of formal vector fields for p > 0, I called it a width.

Special polynomials P_{mk}^{(−1/λ)} describing the orbits and invariants of formal tensor fields on a line, naturally appear in some other fields of mathematics as well. An application of them to a particular problem from a quantum field theory is discussed in section 6.

2 Polynomials P_{mk}

Let f be an arbitrary field of characteristic 0. Denote W_{1} the Lie algebra of f-derivations of f[[t]], the (associative) algebra of the formal power series in one variable. In other words, elements of W_{1} are formal vector fields on a line 𝔸^{1}(f) = ft, i. e. expressions

a = ∑_{i=−1}^{∞} a_{i}l_{i},  (6)

with a_{i} ∈ f and l_{i} = t^{i+1} \frac{∂}{∂t}, with generators l_{i} satisfying

[l_{i}, l_{j}] = (j − i)l_{i+j}.  (7)

W_{1} has a natural decreasing filtration

W_{1} = L_{−1} ⊃ L_{0} ⊃ L_{1} ⊃ L_{2} ⊃ …  (8)

where L_{n} is the Lie subalgebra of W_{1}, consisting of elements (6) with a_{i} = 0 if i < n. Since [L_{m}, L_{n}] ⊆ L_{m+n}, Lie algebra L_{n} is an ideal of L_{m} for all m such that 0 ≤ m < n; and we can define Lie algebras L_{mn} = L_{m}/L_{n} for 0 ≤ m < n.

It follows directly from the definition, that L_{mn} is a Lie f-algebra of dimension n − m with a basis (l_{i} + L_{n})_{m≤i<n} satisfying

[l_{i} + L_{n}, l_{j} + L_{n}] = \begin{cases}(j − i)(l_{i+j} + L_{n}) & for i + j < n, \\ 0 & otherwise. \end{cases}  (9)

Below we’ll write l_{i} instead of l_{i} + L_{n}, where it won’t cause an ambiguity, meaning that the brackets [,] in L_{mn} satisfy (7) with l_{i+j} = 0 for i + j ≥ n.

Filtration (8) defines a filtration

L_{mn} ⊃ L_{m+1,n} ⊃ ⋯ ⊃ L_{nn} = 0  (10)

with

[L_{in}, L_{jn}] = \begin{cases}L_{i+j,n} & for i ≠ j, i + j < n, \\ L_{2i+1,n} & for i = j, 2i < n, \\ 0 & otherwise. \end{cases}  (11)

according to (9). Thus L_{mn} are solvable Lie algebras, nilpotent for m > 0 and commutative for 2m + 1 ≥ n.

Denote U_{mn} the universal enveloping algebra of L_{mn}. (10) implies

U_{mn} ⊃ U_{m+1,n} ⊃ ⋯ ⊃ U_{nn} = f  (12)

with U_{in} = f[l_{i}, l_{i+1}, …, l_{n−1}] for (n − 1)/2 ≤ i ≤ n − 1 since L_{in} is commutative in these cases.

Theorem 1. Let m ≥ 0. If n ≤ 2m + 2, U_{m+1,n+1} is commutative. If n ≥ 2m + 2, the center of U_{m+1,n+1} is f[l_{n}, l_{n−1}, …, l_{n−m}] for odd n and f[l_{n}, l_{n−1}, …, l_{n−m}, P_{mk}(l_{n}, l_{n−1}, …, l_{n/2})] for even n, where k = (n/2) − m and polynomials P_{mk} can be defined as the coefficients of a generating function

∑_{k=1}^{∞} P_{mk}(x_{0}, x_{1}, …, x_{m+k})z^{k} = \frac{\sqrt{∑_{k=0}^{∞} x_{k}x_{0}^{k−1}z^{k}} − \sqrt{∑_{k=0}^{m} x_{k}x_{0}^{k−1}z^{k}}}{x_{0}^{m}z^{m}}  (13)

A canonical projection L_{m} → L_{mn} induces a canonical inclusion of the spaces of f-linear forms, L_{mn}^{∗} → L_{m}^{∗}, and one has an infinite flag

0 = L_{mm}^{∗} ⊂ L_{m,m+1}^{∗} ⊂ L_{m,m+2}^{∗} ⊂ ⋯ ⊂ L_{m}^{∗}  (14)

with L_{m}^{∗} = ⋃_{k} L_{m,m+k}^{∗}.

Denote G_{0} = Gal(f((t))/f), the group of the automorphisms of f((t)), the field of power series, leaving the constants stable. Elements g ∈ G_{0} can be uniquely determined by their values g(t) = ct + o(t) with c ∈ f^{∗} i. e. c ≠ 0.

For a positive integer n, denote G_{n} a subgroup of G_{0} consisting of elements g satisfying g(t) = t + o(t^{n}). For m ≤ n, G_{n} is a normal subgroup of G_{m}. Denote G_{mn} = G_{m}/G_{n}.

For m > 0, there is a standard isomorphism between G_{mn} and an (algebraic) adjoint group of a nilpotent Lie algebra L_{mn}. The standard action of G_{mn} on L_{mn}^{∗} coincides with the co-adjoint representation. The group G_{m} also may be considered as an adjoint group of a pronilpotent Lie algebra L_{m} for m > 0, and the standard action of G_{m} on L_{m}^{∗} may be called a co-adjoint representation.

Theorem 2. Let m ≥ 0. If n ≤ 2m + 2, all the orbits of a co-adjoint representation of G_{m+1,n+1} in L_{m+1,n+1} are points, and every point is an orbit; all orbits are in a general position. If n > 2m + 2, the orbits in a general position of a co-adjoint representation G_{m+1,n+1} in L_{m+1,n+1}, can be parametrized by m + 1 numbers c_{0} ∈ f^{∗}, c_{1}, …, c_{m} ∈ f for odd n: they are affine planes of dimension n − 2m − 1 defined by equations l_{n} = c_{0}, l_{n−1} = c_{1}, …, l_{n−m} = c_{m}; or by m + 2 numbers c_{0} ∈ f^{∗}, c_{1}, …, c_{m+1} ∈ f for even n, in which case they are affine varieties of dimension n − 2m − 2 defined by equations l_{n} = c_{0}, l_{n−1} = c_{1}, …, l_{n−m} = c_{m}, P_{mk}(l_{n}, l_{n−1}, …, l_{n/2}) = c_{m+1} with k = (n/2) − m and P_{mk} defined by (13). Each orbit in a general position of a co-adjoint representation of G_{m+1,n+1} in L_{m+1,n+1}^{∗}, is an orbit of a co-adjoint representation of G_{m+1,i+1} in L_{m+1,i+1}^{∗} for all i ≥ n, as well as of G_{m+1} in L_{m+1}^{∗}. Each orbit of a co-adjoint representation of G_{m+1,i+1} in L_{m+1,i+1}^{∗}, is an orbit in a general position of a co-adjoint representation of G_{m+1,n+1} in L_{m+1,n+1}^{∗} for some n such that min(2m + 2, i) ≤ n ≤ i. Each orbit of a co-adjoint representation of G_{m+1} in L_{m+1}^{∗} is an orbit in a general position of a co-adjoint representation of G_{m+1,n+1} in L_{m+1,n+1}^{∗} for some n ≥ 2m + 2.

Let us study polynomials P_{mk} mentioned in Theorems 1 and 2 in more details.

Theorem 3. A polynomial P_{mk} is homogeneous of degree k and equalized of weight m + k.

P_{mk}(x_{0}, …, x_{m+k}) = ∑_{\begin{matrix}π⊢m+k \\ π_{1}>m \end{matrix}} \binom{1/2}{p_{1}, …, p_{m+k}}x_{0}^{p_{0}}x_{1}^{p_{1}} … x_{m+k}^{p_{m+k}},  (15)

where π = (1^{p_{1}}2^{p_{2}} …) is supposed to be a partition of m+k with the largest part π_{1} > m; p_{0} = k−ℓ(π) where ℓ(π) denotes the length of a partition π. The least common multiple of the denominators of the coefficients of P_{mk} equals 2^{2k−s(k)} for m = 0 or 2^{2k−s(k−1)−1} for m > 0, where s(k) is the sum of digits of the binary expression of k. Also,

P_{mk}(x_{0}, …, x_{m+k}) = \frac{(−1)^{k−1}}{(2k − 2)!! · 2} ∫_{0}^{x} det \binom{dx}{A},  (16)

where x = (x_{m+1}, …, x_{m+k}), dx = (dx_{m+1}, …, dx_{m+k}) and

A = (\begin{matrix}(2k − 2)x_{0} & & (2k − 3)x_{1} & & … & & kx_{k−2} & & (k − 1)x_{k−1} \\ 0 & & (2k − 4)x_{0} & & … & & (k − 1)x_{k−3} & & (k − 2)x_{k−2} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & 0 & & … & & 2x_{0} & & x_{1} \end{matrix})  (17)

is a (k − 1) × k matrix; \binom{dx}{A} denotes k × k matrix obtained from A by adding a first row dx. Also,

P_{0k}(x_{0}, …, x_{k}) = \frac{(−1)^{k−1}}{(2k)!!} det \binom{x′}{A},  (18)

where x′ = (x_{1}, 2x_{2}, …, kx_{k}) and A as in (17). If m ≥ k − 1, then

P_{mk}(x_{0}, …, x_{m+k}) = \frac{(−1)^{k−1}}{(2k − 2)!! · 2} det \binom{x}{A}  (19)

with x = (x_{m+1}, …, x_{m+k}) and matrix A defined above in (17). One has P_{m1} = \frac{1}{2}x_{m+1}; P_{02} = \frac{1}{8}(4x_{0}x_{2} − x_{1}^{2}) and P_{m2} = \frac{1}{4}(2x_{0}x_{m+2} − x_{1}x_{m+1}) for m ≥ 1; P_{03} = \frac{1}{16}(8x_{0}^{2}x_{3} − 4x_{0}x_{1}x_{2} + x_{1}^{3}), P_{13} = \frac{1}{16}(8x_{0}^{2}x_{4} − 2x_{0}(2x_{1}x_{3} + x_{2}^{2}) + 3x_{1}^{2}x_{2}), P_{m3} = \frac{1}{16}(8x_{0}^{2}x_{m+3} − 4x_{0}(x_{1}x_{m+2} + x_{2}x_{m+1}) + 3x_{1}^{2}x_{m+1}) for m ≥ 2; P_{04} = \frac{1}{128}(64x_{0}^{3}x_{4} − 16x_{0}^{2}(2x_{1}x_{3} + x_{2}^{2}) + 24x_{0}x_{1}^{2}x_{2} − 5x_{1}^{4}), P_{14} = \frac{1}{32}(16x_{0}^{3}x_{5} − 8x_{0}^{2}(x_{1}x_{4} + x_{2}x_{3}) + 6x_{0}(x_{1}^{2}x_{3} + x_{1}x_{2}^{2}) − 5x_{1}^{3}x_{2}), P_{24} = \frac{1}{32}(16x_{0}^{3}x_{6} − 4x_{0}^{2}(2x_{1}x_{5} + 2x_{2}x_{4} + x_{3}^{2}) + 6x_{0}(x_{1}^{2}x_{4} + 2x_{1}x_{2}x_{3}) − 5x_{1}^{3}x_{3}), P_{m4} = \frac{1}{32}(16x_{0}^{3}x_{m+4} − 8x_{0}^{2}(x_{1}x_{m+3} + x_{2}x_{m+2} + x_{3}x_{m+1}) + 6x_{0}(x_{1}^{2}x_{m+2} + 2x_{1}x_{2}x_{m+1}) − 5x_{1}^{3}x_{m+1}) for m ≥ 3. Also,

\begin{matrix}P_{0k}(1, 1, …, 1) = & \frac{(2k − 1)!!}{(2k)!!} \\ P_{mk}(1, 1, …, 1) = & \frac{(2k − 3)!!}{(2k − 2)!! · 2} for m ≥ k − 1. \end{matrix}  (20)

If m = 0, the sum (15) has p(k), the number of partitions of k, nonzero items. If m ≥ k − 1, the sum (15) has p(0) + p(1) + ⋯ + p(k − 1) nonzero items.

Referring to x_{0}, x_{1}, …, x_{m} as constants, one obtains from (16),

dP_{mk} = \frac{∂P_{mk}}{∂x_{m+1}} dx_{m+1} + ⋯ + \frac{∂P_{mk}}{∂x_{m+k}} dx_{m+k} = det \binom{dx}{A}.  (21)

Expanding the determinant along the first row, we get determinant formulas for partial derivatives:

Corollary 1. For an integer i so that 1 ≤ i ≤ m,

\frac{∂P_{mk}}{∂x_{m+i}} = (−1)^{i+1} det A_{i},  (22)

where A_{i} is a matrix obtained from A by deleting i-th column.

Lemma 1. Let r be an arbitrary commutative ring, \frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{k}} some derivations of r, P ∈ r and

dP =^{def} \frac{∂P}{∂x_{1}} dx_{1} + ⋯ + \frac{∂P}{∂x_{k}} dx_{k} = det (\begin{matrix}dx_{1} & & & … & & & dx_{k} \\ a_{21} & & & … & & & a_{2k} \\ . & . & . & . & . & . & . & . & . & . & . & . \\ a_{k1} & & & … & & & a_{kk} \end{matrix})  (23)

with a_{ij} ∈ r. Then P satisfies the following system of partial differential equations:

\{ \begin{matrix}a_{21} \frac{∂P}{∂x_{1}} + ⋯ + a_{2k} \frac{∂P}{∂x_{k}} = 0, \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ a_{k1} \frac{∂P}{∂x_{1}} + ⋯ + a_{kk} \frac{∂P}{∂x_{k}} = 0. \end{matrix}  (24)

Proof. Expanding the determinant

det (\begin{matrix}a_{i1} & & … & & & a_{ik} \\ a_{21} & & … & & & a_{2k} \\ . & . & . & . & . & . & . & . & . & . & . & . \\ a_{i1} & & … & & & a_{ik} \\ . & . & . & . & . & . & . & . & . & . & . & . \\ a_{k1} & & … & & & a_{kk} \end{matrix}) = 0  (25)

along the first row, we get the corresponding equation of the system (24).

Proof of Theorem 1. By Gelfand’s Lemma [1], for the standard representation of a Lie algebra having a basis (x_{0}, …, x_{N}), on the polynomial algebra f[x_{0}, …, x_{N}], one has

T (x_{i})P = ∑_{j=0}^{N}[x_{i}, x_{j}]\frac{∂P}{∂x_{j}}  (26)

for all i from 0 to N and P ∈ f[x_{0}, …, x_{N}]. Thus the algebra of invariants of the given Lie algebra in f[x_{0}, …, x_{N}], can be described as the algebra of solutions of a system of partial differential equations

\{ \begin{matrix}∑_{j=0}^{N}[x_{0}, x_{j}]\frac{∂P}{∂x_{j}} = 0, \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{j=0}^{N}[x_{N}, x_{j}]\frac{∂P}{∂x_{j}} = 0. \end{matrix}  (27)

For a Lie algebra L_{m+1,n+1} with odd n, the system (27) where x_{i} = l_{n−i} and N = n − m − 1, has an upper triangular matrix with a non-zero main diagonal. Thus, by induction, invariants don’t depend on l_{i} with m < i < n − m. If n is even, the matrix of the coefficients of system (27) for L_{m+1,n+1} has a form

(\begin{matrix}A & B \\ 0 & C \end{matrix})  (28)

where A is defined in (17), and C is an upper triangular square matrix with a non-zero main diagonal. By induction, the same as for the case of an odd n, one can deduct that invariants don’t depend on l_{i} for m < i < n/2. Further, the invariants satisfy the system of partial differential equations with a matrix A. Now, utilizing Lemma 1 and formulas (16), (21), we get an additional invariant P_{mk}(l_{n}, …, l_{n/2}), with k = (n/2) − m.

It follows from the general theory of invariants of nilpotent Lie algebras [7], that the algebra of invariants of L_{m+1,n+1} discussed above, for even n, is f[l_{n}, l_{n−1}, …, l_{n−m}, P] with unknown polynomial P. Notice that our polynomial P_{mk}(l_{n}, …, l_{n/2}) with k = (n/2) − m, is not included in any algebras f[l_{n}, l_{n−1}, …, l_{n−m}, P] ⊆ f[l_{n}, …, l_{n/2}] such that P ∉ f[l_{n}, …, l_{n−m}, P_{mk}(l_{n}, …, l_{n/2})], since

P_{mk}(l_{n}, …, l_{n/2}) = \frac{1}{2}l_{n}^{k−1}l_{n/2} + l_{n}Q_{mk}(l_{n}, …, l_{(n/2)+1}) + c_{mk}l_{n−1}^{k−1}l_{n−m−1}  (29)

for some polynomial Q_{mk} and nonzero constant c_{mk}, meaning that P_{mk} is a linear polynomial of l_{n/2} with coprime coefficients.

l_{n}, …, l_{n−m} are central elements of L_{m+1,n+1}. Connections between invariants and the center of U_{m+1,n+1}, the universal enveloping algebra, are well known now, and can be found in [1].

Another proof of Theorem 6, based on the studying of generating functions (13), will be given in the next section. Theorem 3 (except the determinant formula (18)) follows from a comparison between these two proofs of Theorem 1. The determinant formula can be obtained by differentiation of the corresponding generating function (13) and observing the conditions on coefficients; similar calculations can be found in [6] and [8]. Theorem 2 follows from Theorem 1 and the results of [7].

Some of results of this section were announced in [3].

3 Fractional residues

The same as in the previous section, let f be a field of characteristic 0. For λ, μ ∈ f denote F_{λμ} = f[[t]]t^{μ}(dt)^{−λ}, a linear topological f-space with the topology induced from the standard topology of f[[t]], the algebra of formal power series, assuming a discrete topology of f. Lie algebras L_{m} and groups G_{m}, with m > 0, naturally act on these spaces. The purpose of this section is to study the algebras I_{λμ}^{m} of polynomial invariants of these actions.

Elements e_{k} = t^{k+μ}(dt)^{−λ}, where k = 0, 1, 2, …, form a topological basis of F_{λ,μ}. Denote (x_{k})_{0≤k∈ℤ} the dual basis of the topological f-space F_{λ,μ}^{∗} of linear forms on F_{λ,μ}.

Theorem 4. Let m be a non-negative integer. If μ ≠ (m + k + 1)λ for any positive integers k, then I_{λμ}^{m+1} = f[x_{0}, x_{1}, …, x_{m}]. If λ = μ = 0, then I_{λμ}^{m+1} = f[x_{0}, x_{1}, …, x_{m} + 1]. If λ ≠ 0, μ = (m + k + 1)λ for a positive integer k, and −1/λ ≠ n for any positive integer n < k, then I_{λμ}^{m+1} = f[x_{0}, x_{1}, …, x_{m}, P_{mk}^{(−1/λ)}(x_{0}, x_{1}, …, x_{m+k})] where polynomial P_{mk}^{(−1/λ)} is defined by a generating function

∑_{k=1}^{∞} P_{mk}^{(−1/λ)}(x_{0}, x_{1}, …, x_{m+k})z^{k} = \frac{(∑_{i=0}^{∞} x_{i}x_{0}^{i−1}z^{i})^{−1/λ} − (∑_{i=0}^{m} x_{i}x_{0}^{i−1}z^{i})^{−1/λ}}{x_{0}^{m}z^{m}} .  (30)

If −1/λ = n for a positive integer n and μ = (m + k + 1)λ for a positive integer k > n, then I_{λμ}^{m+1} = f[x_{0}, x_{1}, …, x_{m}, P_{mk}^{(n)}(x_{0}, x_{1}, …, x_{m+k})/x_{0}^{k−n}] where polynomial P_{mk}^{(n)} is defined above.

Theorem 5. Let m be a non-negative integer. If μ ≠ (m + k + 1)λ for any positive integers k, then the orbits in a general position of the standard representation of G_{m+1} in F_{λμ} can be parametrized by (m + 1) numbers c_{0} ∈ f^{∗}, c_{1}, …, c_{m} ∈ f: they are affine planes of codimension m + 1 given by equations x_{0} = c_{0}, x_{1} = c_{1}, …, x_{m} = c_{m}. If λ ≠ 0, μ = (m+k +1)λ for a positive integer k, then the orbits in a general position of the standard representation of G_{m+1} in F_{λμ} can be parametrized by (m + 2) numbers c_{0} ∈ f^{∗}, c_{1}, …, c_{m}, c_{m+1} ∈ f: they are affine varieties of codimension m + 2 given by equations x_{0} = c_{0}, x_{1} = c_{1}, …, x_{m} = c_{m}, P_{mk}^{(−1/λ)}(x_{0}, x_{1}, …, x_{m+k}) = c_{m+1} if −1/λ ≠ n for any positive integer n < k, or x_{0} = c_{0}, x_{1} = c_{1}, …, x_{m} = c_{m}, P_{mk}^{(−1/λ)}(x_{0}, x_{1}, …, x_{m+k})/x_{0}^{k−n} = c_{m+1} if −1/λ = n for a positive integer n, where polynomial P_{mk}^{(−1/λ)} is defined by (30). Each orbit in a general position of the standard representation of G_{m+1} in F_{λμ} is an orbit of the standard representation of G_{m+1} in F_{λ,μ−i} for each nonnegative integer i. Each orbit of the standard representation of G_{m+1} in F_{λμ} is an orbit in a general position of the standard representation of G_{m+1} in F_{λ,μ+i} for a nonnegative integer i, with the only exception when λ = 0 and μ is a non-positive integer: then sets c + O are also orbits for any c ∈ f^{∗} and O, an orbit in general position of the standard representation of G_{m+1} in F_{0i} for a positive integer i.

Let us study polynomials P_{mk}^{(−1/λ)} mentioned in Theorems 4 and 5 in more details.

Theorem 6. A polynomial P_{mk}^{(−1/λ)} is homogeneous of degree k and equalized of weight m + k.

P_{mk}^{(−1/λ)}(x_{0}, …, x_{m+k}) = ∑_{\begin{matrix}π⊢m+k \\ π_{1}>m \end{matrix}} \binom{1/λ}{p_{1}, …, p_{m+k}}x_{0}^{p_{0}}x_{1}^{p_{1}} … x_{m+k}^{p_{m+k}},  (31)

where π = (1^{p_{1}}2^{p_{2}} …) is supposed to be a partition of m+k with the largest part π_{1} > m; p_{0} = k−ℓ(π) where ℓ(π) denotes the length of a partition π. Also,

P_{mk}^{(−1/λ)}(x_{0}, …, x_{m+k}) = \frac{1}{(k − 1)!(−λ)^{k}} ∫_{0}^{x} det \binom{dx}{A},  (32)

where x = (x_{m+1}, …, x_{m+k}), dx = (dx_{m+1}, …, dx_{m+k}) and

A = (\begin{matrix}(k − 1)λx_{0} & & ((k − 1)λ + 1)x_{1} & & … & & ((k − 1)λ + (k − 2))x_{k−2} & & (k − 1)(λ + 1)x_{k−1} \\ 0 & & (k − 2)λx_{0} & & … & & ((k − 2)λ + (k − 3))x_{k−3} & & (k − 2)(λ + 1)x_{k−2} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & 0 & & … & & λx_{0} & & (λ + 1)x_{1} \end{matrix}) (33)

is a (k − 1) × k matrix; \binom{dx}{A} denotes k × k matrix obtained from A by adding a first row dx. Also,

P_{0k}^{(−1/λ)}(x_{0}, …, x_{k}) = \frac{1}{k!(−λ)^{k}} det \binom{x′}{A},  (34)

where x′ = (x_{1}, 2x_{2}, …, kx_{k}) and A as in (33). If m ≥ k − 1, then

P_{mk}^{(−1/λ)}(x_{0}, …, x_{m+k}) = \frac{1}{(k − 1)!(−λ)^{k}} det \binom{x}{A}  (35)

with x = (x_{m+1}, …, x_{m+k}) and matrix A defined above in (33). One has P_{m1}^{(−1/λ)} = −\frac{1}{λ}x_{m+1}; P_{02}^{(−1/λ)} = −\frac{1}{λ}x_{0}x_{2} + \frac{λ+1}{2λ^{2}}x_{1}^{2} and P_{m2}(−1/λ) = −\frac{1}{λ}x_{0}x_{m+2} + \frac{λ+1}{λ^{2}}x_{1}x_{m+1} for m ≥ 1; P_{03}^{(−1/λ)} = −\frac{1}{λ}x_{0}^{2}x_{3} + \frac{λ+1}{λ^{2}}x_{0}x_{1}x_{2} − \frac{(λ+1)(2λ+1)}{6λ^{3}}x_{1}^{3}, P_{13}^{(−1/λ)} = −\frac{1}{λ}x_{0}^{2}x_{4} + \frac{λ+1}{2λ^{2}}x_{0}(2x_{1}x_{3} + x_{2}^{2}) − \frac{(λ+1)(2λ+1)}{2λ^{3}}x_{1}^{2}x_{2}, P_{m3}^{(−1/λ)} = −\frac{1}{λ}x_{0}^{2}x_{m+3} + \frac{λ+1}{λ^{2}}x_{0}(x_{1}x_{m+2} + x_{2}x_{m+1}) − \frac{(λ+1)(2λ+1)}{2λ^{3}}x_{1}^{2}x_{m+1} for m ≥ 2; P_{04}^{(−1/λ)} = −\frac{1}{λ}x_{0}^{3}x_{4} + \frac{λ+1}{2λ^{2}}x_{0}^{2}(2x_{1}x_{3} + x_{2}^{2}) − \frac{(λ+1)(2λ+1)}{2λ^{3}}x_{0}x_{1}^{2}x_{2} + \frac{(λ+1)(2λ+1)(3λ+1)}{24λ^{4}}x_{1}^{4}, P_{14}^{(−1/λ)} = −\frac{1}{λ}x_{0}^{3}x_{5} + \frac{λ+1}{λ^{2}}x_{0}^{2}(x_{1}x_{4} + x_{2}x_{3}) − \frac{(λ+1)(2λ+1)}{2λ^{3}}x_{0}(x_{1}^{2}x_{3} + x_{1}x_{2}^{2}) + \frac{(λ+1)(2λ+1)(3λ+1)}{6λ^{4}}x_{1}^{3}x_{2}, P_{24}^{(−1/λ)} = −\frac{1}{λ}x_{0}^{3}x_{6}+\frac{λ+1}{2λ^{2}}x_{0}^{2}(2x_{1}x_{5}+2x_{2}x_{4}+x_{3}^{2})−\frac{(λ+1)(2λ+1)}{2λ^{3}}x_{0}(x_{1}^{2}x_{4}+2x_{1}x_{2}x_{3})+\frac{(λ+1)(2λ+1)(3λ+1)}{6λ^{4}}x_{1}^{3}x_{3}, P_{m4}^{(−1/λ)} = −\frac{1}{λ}x_{0}^{3}x_{m+4}+ \frac{λ+1}{λ^{2}}x_{0}^{2}(x_{1}x_{m+3}+x_{2}x_{m+2}+x_{3}x_{m+1})− \frac{(λ+1)(2λ+1)}{2λ^{3}}x_{0}(x_{1}^{2}x_{m+2}+2x_{1}x_{2}x_{m+1})+ \frac{(λ+1)(2λ+1)(3λ+1)}{6λ^{4}}x_{1}^{3}x_{m+1} for m ≥ 3. Also,

\begin{matrix}P_{0k}^{(−1/λ)}(1, 1, …, 1) = & (−1)^{k}\binom{1/λ}{k}, \\ P_{mk}^{(−1/λ)}(1, 1, …, 1) = & \frac{(−1)^{k}}{λ} \binom{1/λ}{k − 1} for m ≥ k − 1. \end{matrix}  (36)

If −1/λ ≠ n for any positive integer n < k, the sum (31) has p(k) nonzero items for m = 0, or p(0) + p(1) + ⋯ + p(k − 1) nonzero items for m ≥ k − 1. If −1/λ = n for a positive integer n < k, then the sum (31) has p_{n}(k), the number of partitions of k with length ≤ n, nonzero items for m = 0, or p_{n}(0) + p_{n}(1) + ⋯ + p_{n}(k − 1) nonzero items for m ≥ k − 1.

Proof of Theorem 4. Analogously to the proof of Theorem 6, one can notice that for a representation T of a Lie f-algebra with a basis (l_{i})_{i∈J}, in the algebra f[x_{0}, x_{1}, …], one has the series of equalities

T (l_{i})P = ∑_{j∈J}(l_{i}x_{j})\frac{∂P}{∂x_{j}} .  (37)

Continuing as in the proof of Theorem 1, we get a proof of Theorem 4.

There is also another proof. Consider a residue Res_{0} = x_{k} ∈ I_{−1,−k−1} where k is a nonnegative integer. For such a non-negative k, if μ = (k + 1)λ and λ ≠ 0, then one has an invariant polynomial mapping

\begin{matrix}P^{(−1/λ)} : F_{λμ}|_{x_{0}=1} & ⟶ F_{−1,−k−1} \\ ht^{μ}(dt)^{−λ} & ↦ (ht^{μ}(dt)^{−λ})^{−1/λ} = h^{−1/λ}t^{−k−1}dt. \end{matrix}  (38)

Composing this invariant mapping with a standard residue Res_{0}, we obtain a polynomial invariant for every positive integer k,

Res_{0} ∘P^{(−1/λ)} : F_{λμ}|_{x_{0}=1} ⟶ f.  (39)

Noticing that x_{0} is also a G_{1}-invariant, we can extend (39) first to a rational G_{1}-invariant

Res_{0} ∘P^{(−1/λ)} ∘ (\frac{·}{x_{0}} ) : F_{λμ}|_{x_{0}≠0} ⟶ f  (40)

and then to a polynomial G_{1}-invariant

P_{0k}^{(−1/λ)} = (·x_{0}^{k}) ∘ Res_{0} ∘P^{(−1/λ)} ∘ (\frac{·}{x_{0}} ) : F_{λμ} ⟶ f.  (41)

The rest of the proof can be done by utilizing the standard techniques from [7].

Noticing that bilinear transformations

P : F_{λμ} × F_{λ′μ′} ⟶ F_{λ+λ′,μ+μ′}, (e_{i}, e_{j}) ↦ e_{i+j}  (42)

are invariant, one obtains a bilinear invariant for λ + λ′ = −1 when μ + μ′ is a negative integer:

Res_{0} ∘P : F_{λμ} × F_{λ′μ′} ⟶ f.  (43)

Thus,

F_{λμ}^{∗} ≃ (⋃_{i} F_{−1−λ,i−μ}) /F_{−1−λ,−μ} =^{def} F_{−1−λ,−μ}^{−}.  (44)

In particular,

L_{m}^{∗} = F_{1,m+1} ≃ F_{−2,−1−m}^{−}  (45)

and

L_{mn}^{∗} ≃ F_{−2,−1−n}/F_{−2,−1−m}.  (46)

That explains the identity

P_{mk} = P_{mk}^{(1/2)}  (47)

following from (13) and (30). Caution. ‘An orbit in a general position’, here and in the previous chapter, doesn’t mean the ‘orbit of maximal dimension’ or the ‘orbit of minimal codimension’. For instance, the following two series of the orbits of the co-adjoint representation of G_{1,5}: given by equations

l_{4} = c_{0}, \frac{1}{2}l_{4}l_{2} − \frac{1}{8}l_{3}^{2} = c_{1}  (48)

with c_{0} ∈ f^{∗}, c_{1} ∈ f, and by equations

l_{4} = 0, l_{3} = c_{0}  (49)

with c_{0} ∈ f^{∗}, both have the maximal dimension 2, but only (48) are orbits in a general position of the co-adjoint representation of G_{1,5} in L_{1,5}^{∗}.

4 Positive characteristic, a restricted case

Let f be an arbitrary field of a positive characteristic p. Denote W_{1} the restricted Lie p-algebra of f-derivations of f[t]/(t^{p}). Elements of W_{1} can be written in a form

a = ∑_{i=−1}^{p−2} a_{i}l_{i}  (50)

with a_{i} ∈ f and l_{i} = t^{i+1} \frac{d}{dt}. The basic elements l_{i} satisfy (7) meaning l_{i+j} = 0 for i + j ≥ p − 1. Also l_{i}^{p} = 0 for i ≠ 0, and l_{0}^{p} = l_{0}.

The same as in section 2, consider filtration

W_{1} = L_{−1} ⊃ L_{0} ⊃ ⋯ ⊃ L_{p−2}  (51)

assuming that L_{n} is a Lie p-algebra consisting of expressions (50) with a_{i} < 0 for i < n. Define L_{mn} = L_{m}/L_{n} for 0 ≤ m ≤ n ≤ p − 1, supposing that L_{m,p−1} = L_{m}. Denote U_{mn} the restricted universal enveloping algebra of L_{mn}.

Theorem 7. Let m ≥ 0. If n ≤ 2m + 2, then U_{m+1,n+1} is commutative. If p − 2 ≥ n ≥ 2m + 2, the center of U_{m+1,n+1} is f[l_{n}, l_{n−1}, …, l_{n−m}]/(l_{n}^{p}, l_{n−1}^{p}, …, l_{n−m}^{p}) for odd n, or f[l_{n}, l_{n−1}, …, l_{n−m}, P_{mk}(l_{n}, l_{n−1}, …, l_{n/2})]/((l_{n}^{p}, …, l_{n/2}^{p}) ⋂ f[l_{n}, l_{n−1}, …, l_{n−m}, P_{mk}(l_{n}, l_{n−1}, …, l_{n/2})]) for even n, where k = (n/2) − m, and P_{mk} defined by (13).

Denote G_{0} the group of automorphisms of f[t]/(t^{p}), leaving the constants stable, and G_{n} with 1 ≤ n ≤ p − 1, the subgroup of G_{0} of automprphisms g(t) = t + o(t^{n}). Also denote G_{mn} = G_{m}/G_{n} for 0 ≤ m ≤ n ≤ p − 1.

Theorem 8. Theorem 2 is true mutatis mutandis.

The proofs of Theorems 7 and 8 can be obtained the same way as the proofs of Theorems 1 and 2, mutatis mutandis.

5 Formal singularities

Let f be an arbitrary field of characteristic p ≥ 0. Denote F = f((t)) and G_{0} = Gal(F/f). We suppose that F has a valuation, a filtration and a topology, as usual, assuming a discrete topology of f. One can check that all of the elements of G_{0} are automatically continous automorphisms preserving the valuation, and are defined uniquely by their values g(t) = ct + o(t) with c ∈ f^{∗}, i. e. c ≠ 0.

For a positive integer n, denote G_{n} the subgroup of G_{0} of automorphisms g such that g(t) = t + o(t^{n}). The filtration

G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ …  (52)

defines a topology (‘given by a filtration’) in every G_{n}.

Theorem 9. Let n be a nonnegative integer. G_{n}-orbit of a formal power series h ∈ F, is open iff h ∈ F \f((t^{p})). The statement, ‘G_{n}-orbit of h ∈ f((t^{p^{m}}))\f((t^{p^{m+1}})) is open in the relative topology of a field f((t^{p^{m}}))’ is true for all h ∈ F \ f, iff f is a perfect field. All the series h ∈ F \ f((t^{p})) have a polynomial normal form, i. e. G_{n}(h) ⋂ f[t^{−1}, t] ≠ ∅. All the series h ∈ F have a polynomial normal form, iff f is a perfect field. Every G_{n}-orbit in F is closed. The canonical projection

β : F ⟶ F/G_{n}, h ↦ G_{n}(h)  (53)

is closed iff either f is a finite field, or p = 0. The orbit space F/G_{n} can be metrized by a complete metrics.

Proof. Start with a counterexample to the existence of a polynomial normal form for the case of imperfect field. Let f be an imperfect field, c ∈ f \ f^{p} and

h = ∑_{i=1}^{∞} c^{p_{i}}t^{p^{i}}  (54)

where

p_{i} = 1 + p + p^{2} + ⋯ + p^{i} = \frac{p^{i−1} − 1}{p − 1}.  (55)

Since h satisfies

h − ch^{p} = t^{p},  (56)

one has

g(h) − cg(h)^{p} = g(t)^{p}  (57)

for any g ∈ G_{n}. Suppose that g(h) is a polynomial of degree d. Then d ≥ p, because as we noticed, g saves the valuation, and ord h = p. Calculating coefficients at t^{d^{p}} in (57), we obtain

c = − (\frac{g(t)_{d}}{g(h)_{d}} )^{p} ∈ f^{p},  (58)

a contradiction. Thus, the series (54) doesn’t have a polynomial normal form for any imperfect field f.

The next interesting fact, that all the orbits are closed, follows from a minimality principle, one of the formulations of Hilbert’s theorem about bases of polynomial rings, and the following

Lemma 2. Let K be an algebraically closed field containing f; 𝔸^{m} an affine K-space of a finite dimension m, and 𝔸^{m}(f) the set of f-rational points of 𝔸^{m}. For each polynomial function Φ : 𝔸^{m} → f and for each f-linear subspace L of the f-linear space K, one can find an f-closed affine algebraic variety S ⊆ 𝔸^{m} such that S(f) = Φ^{−1}(L) ⋂ 𝔸^{m}(f).

In calculations involving the field F, the following lemma is extremely useful.

Lemma 3. Let p > 0 , k =^{p} \overline{… k_{i} … k_{1}k_{0}} ∈ ℤ_{p}, a p-adic integer, q_{1} =^{p} \overline{… q_{1i} … q_{11}q_{10}}, q_{2} = p\overline{… q_{2i} … q_{21}q_{20}}, ⋯ ∈ ℤ_{>0}, a finite sequence of nonnegative integers, q = q_{1} + q_{2} + . . .. Then

\binom{k}{q_{1}, q_{2}, …} ∈ ℤ_{p},  (59)
\binom{k}{q_{1}, q_{2}, …} ≡ ∏_{i=0}^{∞} \binom{k_{i}}{q_{1i}, q_{2i}, …} mod p,  (60)
\binom{k}{q_{1}, q_{2}, …} ≢ 0 mod p iff ∀i, k_{i} ≥ q_{1i} + q_{2i} + ….  (61)

For (61), one has ∀i, ν_{p}(q_{i}) ≥ ν_{p}(k) where ν_{p} denotes the p-adic valuation.

Proof. If k ∈ ℤ_{>0}, a positive integer, then (59) is clear, (60) follows from the particular case of binomial coefficients, which is well-known, and (61) follows from (60). Since ℤ is dense in ℤ_{p}, we can extend (59), (60) and (61) to k ∈ ℤ_{p} by continuity.

By the way, we obtained all the formulas (59), (60) and (61) for negative integers k as well, just from the case of positive k and density of ℤ in ℤ_{p}. For me, that is a very interesting p-adic trick.

The last sentence of Theorem 9 follows from

Lemma 4. Suppose that a group Γ acts on a metric space (X, d) by isometries such that all the orbits are closed. Then

(i) Function D : (X/Γ)^{2} → ℝ, (A, B) ↦ inf_{(a,b)∈A×B} d(a, b) is a metric on X/Γ defining the quotient topology.

(ii) D(A, B) = d(a, B) for any a ∈ A, (A, B) ∈ (X/Γ)^{2}.

(iii) A canonical injection in : X/Γ → H(X) where (H(X), d_{H}) is the space of non-empty closed subsets of X with Hausdorf’s metrics

d_{H}(A, B) = max{sup_{a∈A} d(a, B), sup_{b∈B} d(A, b)},  (62)

is an isometry and in(X/Γ) is a closed subset of H(X).

(iv) If (X, d) is a complete metric space, then (X/Γ, D) is a complete metric space as well.

Theorem 9 shows that the situation for p > 0 is much more complicated than it is for p = 0 which was studied in section 3. In addition to invariants described previously, there are a lot of new invariants for p > 0. For instance, the following functions F → ℤ ⋃{∞} are invariant:

ord_{0}(h) = ord(h − x_{0}(h)),  (63)
md(h) = max{m ∈ ℤ|h ∈ f((t^{p^{m}}))},  (64)
ord_{md}(h) = max ord{h − a|a ∈ f((t^{p^{md(h)+1}}))},  (65)

where

h = ∑_{i=ord h}^{∞} x_{i}(h)t^{i}  (66)

with x_{i}(h) ∈ f and x_{ord h}(h) ≠ 0.

The next example is more interesting. Determine w : F → ℤ,

w(h) = max \{[\frac{ord_{md}(h) − m}{|m|_{p}^{−1} − | ord_{md}(h)|_{p}^{−1}} ] | x_{m}(h) ≠ 0, 0 ≠ m < ord_{md}(h)\}  (67)

for ord_{md}(h) > ord_{0}(h), or w(h) = 0 otherwise. Here

|m|_{p}^{−1} =^{def} max \{p^{d} | \frac{m}{p^{d}} ∈ ℤ, d ∈ ℤ\}  (68)

Letter w in (67) is the first letter of the word width.

Proposition 1. w is G_{0}-invariant.

6 An application to QFT

Polynomials P_{mk}^{(−1/λ)} describing the orbits and invariants of formal tensor fields on a line, naturally appear in some other fields of mathematics as well. Here is just an example.

According to [9], denote

P_{2} = u_{2} − u_{1}^{2},  (69)
P_{k+1} = \frac{1}{k + 2} (∑_{i=1}^{k}((i + 2)u_{i+1} − 2u_{1}u_{i})\frac{∂P_{k}}{∂u_{i}} − 2ku_{1}P_{k} − ∑_{i=2}^{k−1} P_{i}P_{k+1−i})  (70)

for k > 2.

Theorem 10. For k > 1,

P_{k}(u_{1}, u_{2}, …, u_{k}) = \frac{1}{1 − k} P_{0k}^{(1−k)}(1, u_{1}, u_{2}, …, u_{k}).  (71)

Proof. As usual in analogous cases, after guessing the answer, the proof can be done by induction on k. Limited by the space and time, I omit superfluous details.

Acnowledgements. I would like to thank Alexandre Kirillov and Fan Chung Graham, as well as my mother and my Beautiful and Wonderful wife, Bette.

References

[1] Israel Gelfand, The center of an infinitesimal group ring, Matem. Sb. 26 (1950), p. 103–112.

[2] Alexandre Kirillov, Elements of the representation theory, Nauka, Moscow (1972).

[3] Alexandre Kirillov, Representations of some infinite dimensional Lie groups, Vestnik Mosc. Univ. Mat, Mech (1974), no. 1, p. 75–83.

[4] Alexandre Kirillov, The orbit method. I. Geometric quantization, Representation theory of groups and algebras, Contemp. Math., 145, Amer. Math. Soc., Providence, RI (1993), p. 1–32.

[5] Alexandre Kirillov, The orbit method. II. Infinite-dimensional Lie groups and Lie algebras, Representation theory of groups and algebras, Contemp. Math., 145, Amer. Math. Soc., Providence, RI (1993), p. 33–63.

[6] Aleksandrs Mihailovs, The Petrovsky numbers and multiplicities of representations, Master’s thesis, University of Latvia (1995).

[7] Aleksandrs Mihailovs, Diagrams of representations (1998).

[8] Herbert Wilf, Generationfunctionology, Academic Press, 3rd. ed. (1997).

[9] Denis Yuriev, Quantum conformal field theory as an infinite-dimensional noncommutative geometry, Uspekhi Mat. Nauk 46 (1991), no. 4(280), 115–138, 192; translation in Russian Math. Surveys 46 (1991), no. 4, 135–163.