GENERIC ALGEBRAS: RATIONAL PARAMETRIZATION AND NORMAL FORMS

VLADIMIR L. POPOV^{∗}

Abstract. For every algebraically closed field k of characteristic different from 2, we prove the following:

(1) Generic finite dimensional (not necessarily associative) kalgebras of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent over k rational functions in the structure constants.

(2) There exists an “algebraic normal form”, to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis, namely: there are two finite systems of nonconstant polynomials on the space of structure constants, {f_{i}}_{i∈I} and {b_{j}}_{j∈J}, such that the ideal generated by the set {f_{i}}_{i∈I} is prime and, for every tuple c of structure constants satisfying the property b_{j}(c) ≠ 0 for all j ∈ J, there exists a unique new basis of this algebra in which the tuple c′ of its structure constants satisfies the property f_{i}(c′) = 0 for all i ∈ I.

1. Introduction

Hereinafter k denotes an algebraically closed field k of arbitrary characteristic different from 2. Let V be a finite dimensional vector space over the field k. We put

n := dim_{k}V, M := V^{∗} ⊗ V^{∗} ⊗ V.

Putting in correspondence to an element ∑ ℓ ⊗ ℓ′ ⊗ v ∈ M the bilinear map

V × V → V, (a, b) ↦ ∑ ℓ(a)ℓ′(b)v,

we obtain a well-defined bijection between M and the set of all bilinear maps V × V → V. This bijections commutes with the natural action of the group

G := GL(V)

∗ Supported by grants ���� �����������, ��������������, and the granting program Contemporary Problems of Theoretical Mathematics of the Mathematics Branch of the Russian Academy of Sciences.

on M and its action on the the set of all bilinear maps V × V → V given by the formula (g · φ)(a, b) = g · (φ(g^{−1} · a, g^{−1} · b)).

The problem of classification of G-orbits in M is linked with the applications in the theory of continuous and discrete dynamical systems; see [3], [6], [9]. In these papers, the case n = 2 is considered (under the restriction char k ≠ 3 supplementary to char k ≠ 2): in [3],

[6] symmetric, and in [9] arbitrary generic maps are considered. The first main result of paper [9] is the proof of the existence of

(i) a nonempty open subset U in M and (ii) four G-invariant rational functions f_{1}, f_{2}, f_{3}, f_{4} on M defined

everywhere on U

such that every two tensors a, b ∈ U lie in the same G-orbit if and only if f_{i}(a) = f_{i}(b) for all i. Thus generic G-orbits in M are bijectively parametrized by points of the image of rational map

U → A^{4}, m ↦ (f_{1}(m), f_{2}(m), f_{3}(m), f_{4}(m)).  (1)

The second main result of paper [9] is the proof of the existence of a “normal form” to which every tensor in U can be uniquely transformed by means of an element of G; the set of all normal forms is algebraic, i.e., has the appearance U ∩ {m ∈ M | h_{i}(m) = 0 ∀ i ∈ I} for some finite set {h_{i}}_{i∈I} of polynomial functions on M. In [9], the set U, the functions f_{i}, the normal forms, and the elements of G transforming tensors into normal forms are explicitly given by bulky formulas, and the proofs are largely based on cumbersome explicit calculations.

According to the classical Rosenlicht theorem [15, Thm. 2], for any action of an algebraic group on an irreducible variety, orbits of points in general position are separated by a finite tuple of invariant rational functions. Therefore, leaving aside the explicit formulas, only the claim about the number of separating invariants (4, what is equal to dim M− dim G) does not follow from Rosenlicht’s theorem. This claim has the following important complements (a), (b), and (c), which are not noted in [9]:

(a) The field k(M)^{G} of all G-invariant rational functions on M is a finite purely inseparable extension of the field k(f_{1}, …, f_{4}), and for char k = 0 these fields coincide (this follows from [2, Sect. 18.2, Thm.] because k(f_{1}, …, f_{4}) separates G-orbits of points of an open subset of M).

(b) By [9, Prop. 4.1], in M there are points with finite G-stabilizers. Therefore, max_{m∈M} dim G·m=dim G=4. Since the transcendence degree of k(M)^{G} over k is dim M − max_{m∈M} dim G · m (see [12, Sect. 2.3, Cor.]), this means that its is equal to 4.

(c) The functions f_{1}, …, f_{4} are algebraically independent over k, and therefore the image of map (1) contains a nonempty open subset of A^{4} (this follows from (a) and (b)).

Property (c) is specific for the considered action: given the negative solution to the Noether problem [16], for a linear action of an algebraic group, in general, it is impossible to separate orbits of points in general position by an algebraically independent system of rational invariants.

Here we show that both main results of paper [9], together with complements (a) (in a strengthened form) and (c), hold in arbitrary dimension n. Namely, we prove the following statements (part of them is intentionally formulated in the form close to the applications oriented papers [3], [6], [9]).

Theorem 1 (Rationality of the field of G-invariants of M). The field k(M)^{G} is a rational extension of the field k of transcendental degree n^{3} − n^{2}.

Theorem 1 and Rosenlicht’s theorem [15, Thm. 2] imply the following statement (its part concerning the case n = 1 is clear):

Corollary 1 (G-equivalence of points in general position in M). Let n ⩾ 2. There exist a set {f_{i}}_{i∈I} of n^{3} − n^{2} algebraically independent over k rational G-invariant functions on M, and a finite set {h_{j}}_{j∈J} of nonconstant polynomial functions on M such that:

(i) for any points a and b of a dense open subset {m∈M | h_{j}(m)≠

0 ∀j ∈J} of M the conditions (a) G · a = G · b,

(b) f_{i}(a) = f_{i}(b) for all i ∈ I are equivalent;

(ii) the set {f_{i}}_{i∈I} generates the field k(M)^{G} over k. If n = 1, then k(M)^{G} = k, and M \ {0} is a single G-orbit.

Theorem 2 (The existence of normal forms for M). There exist two finite sets {s_{p}}_{p∈P} and {t_{q}}_{q∈Q} of nonconstant polynomial functions on M, such that:

(i) the closed subset S:={m∈M | s_{p}(m)=0 ∀p∈P } is irreducible, and the map k(M)^{G} → k(S), f ↦ f|_{S} is a well-defined kisomorphism of fields;

(ii) for every point a of the open dense subset {m ∈ M | t_{q}(m) ≠ 0 ∀q ∈Q} of M, there exists a unique element g ∈ G, for which g · a ∈ S.

Actually, we use another interpretation of elements of M, considering them as k-algebra structures (not necessarily associative) on V in the sense of Bourbaki, i.e., ring structures, for which k is the domain of external operators; see [4], [18]. Fixing such a structure on V is equivalent to fixing a bilinear map φ: V × V → V that defines the product of elements a, b ∈ V by formula ab := φ(a, b). This gives a bijection between M and the set of all such structures, which assigns to an element m = ∑ ℓ ⊗ ℓ′ ⊗ v ∈ M the k-algebra structure on V such that the product of elements a, b ∈ V is defined by the formula

ab := ∑ ℓ(a)ℓ′(b)v.  (2)

We denote this k-algebra by {V, m} and call elements of the space M multiplications. For every element g ∈ G and every k-algebra {V, m}, the map {V, m} → {V, g · m}, a ↦ g · a is an isomorphism of kalgebras and every isomorphism between k-algebras is obtained in this way. Thus, k-algebras {V, a} and {V, b} are isomorphic if and only if G · a = G · b, and the automorphism group of the k-algebra {V, m} is the G-stabilizer of the multiplication m ∈ M.

Let θ ∈ GL(M) be the involution induced by permuting the first two factors in V^{∗} ⊗ V^{∗} ⊗ V:

θ: M → M, ℓ ⊗ ℓ′ ⊗ v ↦ ℓ′ ⊗ ℓ ⊗ v.

Since char k ≠ 2, every multiplication m ∈ M can be written in the form m = \frac{1}{2}(m + θ(m)) + \frac{1}{2}(m − θ(m)), which implies that

\begin{matrix}M & = C ⊕ A, \\ C : & = \{m ∈ M | θ(m) = m\}, \\ A : & = \{m ∈ M | θ(m) = −m\}. \end{matrix} \}  (3)

The algebra {V, m} is commutative, i.e., ab = ba in (2) (respectively, anticommutative, i.e., ab = −ba (2)) if and only if m ∈ C (respectively, m ∈ A). The subspaces C and A in M are G-invariant.

The proofs of Theorems 1 and 2 are based on the triviality claims in the following two theorems:

Theorem 3. For points m in general position in C, the algebras {V, m} are simple commutative algebras with trivial automorphism group.

The analogous statement holds for M:

Theorem 4. For points m in general position in M, the algebras {V, m} are simple and have trivial automorphism group.

Theorems 1, 2, and 4 imply Corollary 2. There exists a system of simple pairwise nonisomorphic n-dimensional k-algebras rationally parametrized by n^{3}−n^{2} parameters algebraically independent over k.

We do not know whether the field k(C)^{G} is rational over k or not, however, Theorem 3 implies the following

Corollary 3. The field k(C)^{G} is a stably rational extension of the field k of the transcendence degree (n − 1)n^{2}/2.

Using Theorem 3, one proves the analogue of Theorem 2 for C:

Theorem 5 (The existence of normal forms for C). The formulation of Theorem 2 holds true if M in it is replaced by C.

The following analogue of Corollary 2 follows from Theorems 3, 5 and Corollary 3:

Corollary 4. There exists a system of simple pairwise nonisomorphic commutative n-dimensional k-algebras rationally parametrized by points of an (n − 1)n^{2}/2-dimensional stably rational variety.

In the proof of Theorem 1, we use the following general statement:

Lemma 1. Let L_{1}, L_{2} L_{3} be finite dimensional vector spaces over k, each of which is endowed with a linear action of an affine algebraic group H. Assume that

(i) the actions of H on L_{1} and L_{2} are locally free; (ii) k(L_{1})^{H} is rational over k; (iii) dim L_{3} ⩾ dim L_{1}.

Then k(L_{2} ⊕ L_{3})^{H} is also rational over k.

Theorems 3 and 4 are proved in Section 2. Lemma 1, Theorems 1, 2, 5, and Corollary 3 are proved in Section 3.

Terminology and notation

In what follows, variety means algebraic variety over k in the sense of Serre.

The topological terms are related to the Zariski topology.

If K/L is a field extension, then K is called rational^{1} over L if either K is a finitely generated purely transcendental extension of L or K = L. If there exists a field extension S/K such that S is rational over K and over L, then K is called stably rational over L.

For every nonempty set T of transformations of a set M, we denote by M^{T} the set of all joint fixed points of all transformation from T.

1pure in the terminology of [5].

We freely use the standard notation and conventions of the theory of algebraic groups and invariant theory [2], [12], [19]. In particular, we say that a certain property holds for points in general position in variety X if it holds for all points of an open dense subset of X (depending on the property under consideration). Action of an algebraic group G on a variety X means regular action, i.e., such that the map G × X → X, (g, x) ↦ g · x defining this action is a morphism. If X is irreducible, then π_{G,X}: X −−→ X---G denotes a rational quotient for this action, i.e., X---G is a variety (defined up to birational isomorphism) and π_{G,X} is a dominant rational map such that π_{G,X}^{∗}(k(X---G) = k(X). An action of G on X is called locally free if the G-stabilizers of points in general position in X are trivial.

2. Proofs of Theorems 3 and 4

It will be convenient to fix a basis e_{1}, …, e_{n} in V and assume that G = GL_{n}(k) (indentifying elements of G with their matrices in this basis).

Let ℓ^{1}, …, ℓ^{n} be the dual basis in V^{∗}. Then the elements

c_{r}^{pq} := (ℓ^{p}⊗ℓ^{q}+ℓ^{q}⊗ℓ^{p})⊗e_{r} ∈ C, where 1 ⩽ p ⩽ q ⩽ n, 1 ⩽ r ⩽ n, (4)

(somehow ordered) constitute a basis in C, and so

dim C = n^{2}(n + 1)/2.  (5)

¿From (5), (3) and dim M = n^{3} we get

dim A = (n − 1)n^{2}/2.  (6)

Below we use the following known (see, e.g., [1, Lemma 3]) statement:

Lemma 2. Let an algebraic group H act on a variety X and let Y be a closed subset of X. Then N_{H}(Y) := {h ∈ H | h · Y ⊆ Y } is a closed subgroup of H and dimension of the closure in X of the set dim H · Y is at most dim H + dim Y − dim N_{H}(Y).

Proof. Regarding the closedness see [2, Prop. I, 1,7]. The inequality is proved by applying the fiber dimension theorem to the morphism H × Y → X, (h, y) ↦ h · y. □

Proof of the claim of simplicity in Theorem 3.

Here, mutatis mutandis, the same arguments are applicable as in the proof of simplicity of {V, m} for m in general position in M, given in [10, p. 129]. For n = 1 the claim is clear, therefore let n ⩾ 2.

For every integer d ∈ Z, 0 < d ⩽ n, put

I_{d} := {m ∈ C | {V, m} has a d-dimensional two-sided ideal}.

If a commutative k-algebra {V, m} has a two-sided d-dimensional ideal I, then for every element g ∈ G the subspace g · I is a two-sided ddimensional ideal of {V, g · m}. Since G acts transitively on the set of all d-dimensional linear subspaces of V, there exists g such that g · I is the k-linear span V_{d} of the vectors e_{1}, …, e_{d}. Hence

I_{d} = G · L_{d}, where  (7)
L_{d} := {m ∈ C | V_{d} is the two-sided ideal of {V, m}}.  (8)

Consider in G the subgroup

P_{d} := {g ∈ G | g · V_{d} = V_{d}}.  (9)

It follows from (8), (9) that P_{d} · L_{d} = L_{d}, which, in view of (7) and Lemma 2, implies that

dim I_{d} ⩽ dim G + dim L_{d} − dim P_{d}  (10)

and, since P_{d} is a parabolic subgroup of G, that the set I_{d} is closed in C (see, e.g., [19, Sect. 2.13, Lemma 2]).

It follows from (4) that V_{d} is the two-sided ideal of a commutative k-algebra {V, m}, where

m = ∑_{{1⩽p⩽q⩽n \atop 1⩽r⩽n}} α_{pq}^{r}c_{r}^{pq}, α_{pq}^{r} ∈ k,  (11)

if and only if α_{pq}^{r} = 0 for all triples (p, q, r) from the range of summation in (11) satisfying the conditions:

(i) p or q is not greater than d; (ii) r > d.

This and (8) imply that L_{d} is a linear subspace in C and, since in total there are (n−d)(\binom{n+1}{2}−\binom{n−d+1}{2}) = (n−d)d(2n−d+1)/2 these triples, we obtain

dim L_{d} = dim C − (n − d)d(2n − d + 1)/2.  (12)

¿From dim G = n^{2}, dim P_{d} = n^{2} − d(n − d), and (12) we deduce that the right-hand side of inequality (10) is equal to dim C − d(n − d)(2n − d + 1)/2 + d(n − d), which, for d < n, is smaller than dim C. Hence C \ I_{d} is, for every d < n, a nonempty open subset of C. Therefore, the intersection of all these open subsets is a nonempty open subset of C such that for every its point m the k-algebra {V, m} is simple. This proves the claim of simplicity of algebra in Theorem 3. □

Proof of the claim of triviality in Theorem 3.

1. As above, since for n = 1 the claim is clear, we assume further that n ⩾ 2.

Since the G-stabilizer G_{x} of every point x ∈ C is an algebraic subgroup of G, the claim will be proved if shall prove the existence in C the nonempty open subsets C_{(s)} and C_{(u)} such that G_{x} for every x ∈ C_{(s)} (respectively, for every x ∈ C_{(u)}) does not contain nonidentity semisimple (respectively, unipotent) elements.

2. First we prove the existence of C_{(s)}.

The plan is the following. We shall use that every semisimple element of the group G is conjugate to an element of its fixed maximal torus, see [2, Thms. 11.10, 10.6, Prop. 11.19]. Namely, let X(T) be the group (algebraic) characters T → k^{∗}, and let Δ be the weight system of the T -module C, i.e., the set of all characters μ ∈ X(T) such that the dimension of its weight subspace

C_{μ} := {m ∈ C | t · m = μ(t)m for every element t ∈ T }  (13)

is positive. We have the decomposition

C = ⨁_{μ∈Δ} C_{μ}.  (14)

In view of (13), (14), for every element t ∈ T, the equality

C^{t} := {m ∈ C | t · m = m} = ⨁_{{μ∈Δ | μ(t)=1}} C_{μ}.  (15)

holds. Since the set Δ is finite, it follows from (15) that, when t runs through T, the space C^{t}, being the sum of some of the weight subspaces C_{μ}, runs through only finitely many T -invariant linear subspaces in C. We shall find an integer h such that for every nonidentity t ∈ T the inequalities

dim C^{t} ⩽ h,  (16)
h + dim G − dim T = h + n^{2} − n < dim C = n^{2}(n + 1)/2.  (17)

hold. In view of Lemma 2, it then follows from (16), (17) that the closure of the set G · C^{t} in C is a proper subvariety of C. Since, as is explained above, the set of these closures is finite, the complement to their union is a nonempty open subset of C. ¿From the above it follows that it can be taken as C_{(s)}.

3. We now proceed to implement this plan. As T we take the maximal torus consisting of diagonal matrices. Put

ε_{i}: T → k^{∗}, diag(t_{1}, …, t_{n}) ↦ t_{i}.  (18)

Then ε_{1}, …, ε_{n} is a basis of the free abelain group X(T) and

t · e_{i} = ε_{i}(t)e_{i}, t · ℓ_{i} = ε_{i}^{−1}(t)ℓ_{i} for every element t ∈ T . (19)

Formulas (19) and (4) imply the inclusion c_{r}^{pq} ∈ C_{ε_{p}^{−1}ε_{q}^{−1}ε_{r}}, from which we obtain

\begin{matrix}Δ & = Δ′ ∪ Δ′′, where \\ Δ′ & := \{ε_{p}^{−1}ε_{q}^{−1}ε_{r} | 1 ⩽p⩽q ⩽ n, 1⩽r ⩽n, r ≠p, r ≠q\}, \\ Δ′′ & := \{ε_{s}^{−1} | 1⩽s⩽n\}, and \\ dim C_{μ} & = \begin{cases}1 & if μ ∈ Δ′, \\ n & if μ ∈ Δ′′. \end{cases} \end{matrix}\} (20)

Every character α ∈ X(T) defines a partition of the set Δ as the union of mutually disjoint sequences of weights μ_{1}, …, μ_{d} ∈ Δ having the properties

(i) μ_{i+1} = αμ_{i} for every i = 1, …, d − 1, (ii) α^{−1}μ_{1} ∉ Δ and αμ_{d} ∉ Δ.

We call such sequences α-series.

The kernel of action of G on C is trivial: being a normal subgroup, it lies in the center of G, hence also in T, and it follows from (18), (19), and the inclusion Δ′′ ⊂ Δ that the kernel of action of T on C is trivial.

Fix in T a nonidentity element t = diag(t_{1}, …, t_{n}).

If t_{1} = … = t_{n} ≠ 1, then it follows from (20) that C^{t} = {0}, so C \ C^{t} is a nonempty open subset of C.

Now let t_{s} ≠ t_{s+1} for some s, or, in other words,

α_{s}(t) ≠ 1, where α_{s} := ε_{s}ε_{s+1}^{−1}.  (21)

It follows from (20) and the definition of α-series that all sequences of forms (22)–(27) listed below are the α_{s}-series for every p, q, and r satisfying the specified restrictions:

ε_{s}^{2}ε_{s+1}, ε_{s}^{−1}, ε_{s+1}^{−1}, ε_{s+1}^{−2}ε_{s};  (22)
ε_{p}^{−1}ε_{s}^{−1}ε_{s+1}, ε_{p}^{−1}, ε_{j}^{−1}ε_{s+1}^{−1}ε_{s}, p ≠ s, s + 1;  (23)
ε_{p}^{−1}ε_{q}^{−1}ε_{r}, r ≠ p, q and among p, q, r there is no s and s + 1; (24)
ε_{p}^{−1}ε_{q}^{−1}ε_{s+1}, ε_{p}^{−1}ε_{q}^{−1}ε_{s}, p ≠ s, s + 1, q ≠ s, s + 1;  (25)
ε_{s}^{−1}ε_{q}^{−1}ε_{r}, ε_{s+1}^{−1}ε_{q}^{−1}ε_{r}, r ≠ s, s + 1, q ≠ s, s + 1, r ≠ q;  (26)
ε_{s}^{−2}ε_{r}, ε_{s}^{−1}ε_{s+1}^{−1}ε_{r}, ε_{s+1}^{−2}ε_{r}, r ≠ s, s + 1.  (27)

Since for every weight from Δ there exists only one of the α_{s}-series of the forms (22)–(27) in which it lies, there are no other α_{s}-series. The number of different α_{s}-series of every form (22)–(27) is given in the following table where in the top raw are listed the forms of α_{s}-series and in the bottom the specified numbers:

\begin{matrix}(22) & (23) & (24) & (25) & (26) & (27) \\ 1 & n − 2 & (n − 2)^{2}(n − 3)/2 & (n − 1)(n − 2)/2 & (n − 2)(n − 3) & n − 2 \end{matrix}

It follows from (21) and property (i) in the definition of α-series that t can not lie in the intersection of kernels of any two neighboring weights of every α_{s}-series. Therefore, the set of weights of any α_{s}-series of form (22) that contains t in its kernel is either empty, or contains only one weight, or exactly two non-neighboring weights. In view of (20) this shows that the sum of dimensions of the weight spaces of this set is not greater than n + 1.

Similarly, we obtain that if μ runs through all the weights of some α_{s}-series of form (23), (24), (25), (26), or (27) that contains t in its kernel, then the number ∑ dim C_{μ} is not greater than, respectively, n, 1, 1, 1, or 2 (for form (23), we use here that n ⩾ 2).

It follows from this and the above table that if μ runs through all α_{s}-series of form (22), (23), (24), (25), (26), or (27) that contains t in its kernel, then the number ∑ dim C_{μ} is not bigger than, respectively, n + 1, n(n − 2), (n − 2)^{2}(n − 3)/2, (n − 1)(n − 2)/2, (n − 2)(n − 3), or 2(n − 2).

Since Δ is the union of all α_{s}-series, this shows that the inequality (16) holds for

\begin{aligned}h = n & + 1 + n(n − 2) + (n − 2)^{2}(n − 3)/2 \\ & + (n − 1)(n − 2)/2 + (n − 2)(n − 3) + 2(n − 2). \end{aligned}  (28)

A simple direct calculation then shows that inequality (17) follows from (28). This completes the proof of the existence of C_{(s)}.

4. We now turn to the proof of the existence of C_{(u)}.

First, we prove that the G-stabilizers of points m in general position in C are finite, or, equivalently, that the automorphism groups of the k-algebras {V, m} are finite. Since the set of points of C, whose Gstabilizer has minimal dimension, is open in C, in order to prove the finiteness it suffices to find a single multiplication m_{0} ∈ C such that the automorphism group of the k-algebra {V, m_{0}} is finite. Note, however, that from the triviality of the automorphism group of the k-algebra {V, m_{0}} we can not conclude that the automorphism group of the kalgebra {V, m} is trivial for points m in general position in C, see [12, Sect. 6.1, Example 1].

Define m_{0} ∈ C by the following multiplication table:

e_{i}e_{j} := \begin{cases}e_{i} & if j = i, \\ 0 & if j ≠ i. \end{cases} for all 1 ⩽ i, j, ⩽ n.  (29)

Then A := {V, m_{0}} is the direct sum of one-dimensional subalgebras: A = A_{1} ⊕ ⋯ ⊕ A_{n}, where e_{i} is the basis of A_{i}, and the multiplication is given by the formula e_{i}^{2} = e_{i} showing that e_{i} is the identity in the k-algebra A_{i}.

All one-dimensional two-sided ideals I of A are exhausted by the ideals A_{1}, …, A_{n}. Indeed, let a vector v = α_{1}e_{1} + ⋯ + α_{n}e_{n}, where α_{1}, …, α_{n} ∈ k, be a basis of I, and let α_{p} ≠ 0. Then e_{p}v = α_{p}e_{p} in view of (29). On the other hand, since I is a one-dimensional two-sided ideal, e_{p}v = λv for some λ ∈ k. It then follows from α_{p} ≠ 0 that λ ≠ 0, hence v = α_{p}λ^{−1}e_{p}. Therefore, I = A_{p}.

Now let σ be an automorphism of the k-algebra A. Then σ(A_{i}) for every i is a one-dimensional two-sided ideal of A, hence, as is proved, it coincides with A_{j} for some j and the restriction of σ to A_{i} is an isomorphism of k-algebras A_{i} → A_{j}. Since isomorphism maps identity to identity, σ(e_{i}) = e_{j}. Thus the automorphism group of the k-algebra A leaves invariant the finite set {e_{1}, …, e_{n}}. Since e_{1}, …, e_{n} is a basis in V, its action on this set has no kernel. Therefore, this group is finite^{2}.

5. If char k = 0, then the order of every nonidentity unipotent element is infinite, therefore, the existence of C_{(u)} follows from the proved finiteness of the G-stabilizers of points in general position in C. In the case of char k > 0, the order of every unipotent element is finite, therefore, this argument is not applicable. Consider this case.

Let u ∈ G be a nonidentity unipotent element, let C be the finite cyclic group that it generates, and let kC be its group algebra over k. The order of C is a positive power char k. It follows from the Jordan normal form theory (see also [8]), that for every positive integer d not bigger than C, there is a unique up to isomorphism indecomposable d-dimensional kC-module M_{d}, and there are no other indecomposable kC-modules. In particular, I := M_{1} is a one-dimensional trivial kCmodule. The said implies that M_{d} is isomorphic to its dual module M_{d}^{∗}. In a certain basis, the matrix of the linear operator M_{d} → M_{d}, m ↦ u · m is the Jordan block with eigenvalue 1, and therefore,

dim_{k}M_{d}^{C} = 1.  (30)

If the kC-modules P and Q are isomorphic, we write P ≃ Q. If N is a finite dimensional kC-module, we denote by N^{⊕r} and N^{⊗r} respectively the direct sum and the tensor product of r > 0 copies of the kC-module N; we put N^{⊗0} := I^{⊕ dim_{k} N}. By |N| we denote the number of indecomposable kC-submodules, whose direct sum is N (by the Krull–Schmidt theorem this number is well-defined). According to [13, Lemma 2.1],

2Its order is n! since the element of G inducing a permutation of e_{1}, …, e_{n} is an automorphism of the k-algebra A.

|M_{p} ⊗ M_{q}| = min{p, q} for every p and q.  (31)

Lemma 3. For every finite dimensional kC-modules P and Q and every positive integer d, the following inequality holds:

|P ⊗ Q| ⩽ |P ⊗ I^{⊕ dim_{k} Q}|.  (32)

Proof of Lemma 3. Decompose P and Q as the direct sums of indecomposable kC-submodules:

P ≃ ⨁_{p} M_{s_{p}}, Q ≃ ⨁_{q} M_{r_{q}}.  (33)

¿From these decompositions we obtain

|P ⊗ Q| =^{(33)} | ⨁_{p,q} (M_{s_{p}} ⊗ M_{r_{q}})| = ∑_{p,q} |M_{s_{p}} ⊗ M_{r_{q}}|

⩽^{(31)} ∑_{p,q} |M_{s_{p}} ⊗ I^{⊕r_{q}}|=| ⨁_{p,q}(M_{s_{p}} ⊗ I^{⊕r_{q}})| =^{(33)}|P ⊗ I^{⊕ dim_{k} Q}|.

This proves Lemma 3. □

Since u ∈ G, and V and C are the kG-modules, then V and C are also kC-modules. Since u is a nonidentity and the kernel of the kGmodule V is trivial, the decomposition of the kC-module V as a direct sum of the indecomposable submodules contains at least one summand of dimension s ⩾ 2. Therefore, for some kC-module N we have

V ≃ M_{s} ⊕ N, s ⩾ 2.  (34)

Since the kG-modules C and (Sym^{2}V^{∗}) ⊗ V are isomorphic, from (34) and the self-duality of finite dimensional kC-modules we obtain

C ≃ (Sym^{2}(M_{s} ⊕ N)) ⊗ (M_{s} ⊕ N).  (35)

Using that Sym^{2}(A ⊕ B) and (Sym^{2}A) ⊕ (A ⊗ B) ⊕ (Sym^{2}B) are isomorphic for every modules A and B (see, e.g., [7, (B.2), p. 473]), from (35) we obtain

C ≃ ((Sym^{2}M_{s}) ⊗ M_{s}) ⊕ (M_{s}^{⊗2} ⊗ N) ⊕ ((Sym^{2}N) ⊗ M_{s})

⊕ ((Sym^{2}M_{s}) ⊗ N) ⊕ (N^{⊗2} ⊗ M_{s}) ⊕ ((Sym^{2}N) ⊗ N). (36)

Note also that Sym^{2}M_{s} ⊊ M_{s}^{⊗2} and (31) imply the inequality

|Sym^{2}M_{s}| ⩽ s − 1.  (37)

¿From Lemma 3, taking into account (37), (31) and the definition of | · |, we obtain

\begin{matrix}|(Sym^{2}M_{s}) ⊗ M_{s}| & ⩽^{(32)} |(Sym^{2}M_{s}) ⊗ I^{⊕s}| ⩽^{(37)} s(s − 1), \\ |M_{s}^{⊗2} ⊗ N| & ⩽^{(32)} |M_{s}^{⊗2} ⊗ I^{⊕(n−s)}| =^{(31)} s(n − s), \\ |(Sym^{2}N)⊗M_{s}| & ⩽^{(32)} |M_{s}⊗I^{⊕ \frac{(n−s+1)(n−s)}{2}} |= \frac{(n−s+1)(n−s)}{2}, \\ |(Sym^{2}M_{s}) ⊗ N| & ⩽^{(32)} |(Sym^{2}M_{s}) ⊗ I^{⊕(n−s)}| ⩽^{(37)}(n−s)(s−1), \\ |N^{⊗2} ⊗ M_{s}| & ⩽^{(32)} |M_{s} ⊗ I^{⊕(n−s)^{2}}| = (n − s)^{2}, \\ |(Sym^{2}N) ⊗ N| & ⩽ dim((Sym^{2}N) ⊗ N)= \frac{(n−s+1)(n−s)^{2}}{2} . \end{matrix}\} (38)

It follows from (36) and (38) that

\begin{aligned}|C| ⩽ s(s − 1) & + s(n − s) + \frac{(n − s + 1)(n − s)}{2} + (n − s)(s − 1) \\ & + (n − s)^{2} + \frac{(n − s + 1)(n − s)^{2}}{2} . \end{aligned} (39)

But (30) implies that for every finite dimensional kC-module N the equality N^{C} = |N| holds. Therefore, (39) means that dim_{k}C^{u} is not bigger than the right-hand side of inequality (39). As is known, dimension of the centralizer of the element u in G is not smaller than the rank of G that is equal to n (see, e.g., [19, Prop. 1, p. 94]). Since this centralizer lies in N_{G}(C^{u}), it now follows from Lemma 2 that dimension of the closure \overline{G · C^{u}} of the set G·C^{u} in C is not bigger than the number

\begin{aligned}s(s − 1) & + s(n − s) + \frac{(n − s + 1)(n − s)}{2} + (n − s)(s − 1) \\ & + (n − s)^{2} + \frac{(n − s + 1)(n − s)^{2}}{2} + n^{2} − n. \end{aligned} (40)

Twice the difference between dim C = (n^{3} + n^{2})/2 and number (40) is equal to

n^{2}(3s − 5) + n(−3s^{2} + 4s + 3) + (s^{3} − 2s^{2} + s).  (41)

Consider (41) is the polynomial in n. Its leading coefficient is positive for every s ⩾ 2, and the discriminant is equal to −3s^{3}(s−20/3)−54s(s− 22/27) + 9, which shows that the latter is negative for every s ⩾ 7. Direct calculation shows that it is negative also for s = 2, 3, 4, 5, 6. Therefore, for every s ⩾ 2 the specified difference is positive.

Hence C \ \overline{G · C^{u}} is a nonempty open subset of C such that the Gstabilizers of its points do not contain elements conjugate to u. But there are only finitely many conjugacy classes of unipotent elements in G in view of the finiteness of the set of possible Jordan normal forms of such elements (see also [19]). Therefore, if u runs through the set of representatives of nonidentity conjugacy classes of unipotent elements in G, then the intersection of the open subsets C \ \overline{G · C^{u}} is a nonempty open subsubset of C. The said implies that this subset may be taken as C_{(u)}.

This completes the proof of Theorem 3. □

Remark. Actually, Parts 2 and 3 of the proof give more than we used. Namely, the arguments used there show that for every constant α ∈ k and every element t ∈ T which does not lie in the center of the group G (i.e., which is not scalar), inequalitites (16) and (17), where h is given by formula (28), still hold if C^{t} in (16) is replaced by {m ∈ C | t · m = αm}. This means that we have proved the following

Theorem 6. For the natural action of the group G on the projectivization P C of the vector space C, the stabilizers of points in general position in P C coincide with the center of the group G.

Proof of Theorem 4.

The claim of simplicity is proved in [10, Thm. 4], and that of triviality follows from Theorem 3 in view of the first equality in (3). □

3. Proofs of Lemma 1, Theorems 1, 2, 5, and Corollary 3

The proof of Lemma 1 is based on one frequently used statement known as the No-name Lemma. We recall its formulation.

Let π: E → X be an algebraic vector bundle over a variety X. Assume that E and X are endowed with the actions of an affine algebraic group G such that π is the G-equivariant morphism and for every elements g ∈ G, x ∈ X the map of fibers π^{−1}(x) → π^{−1}(g · x) defined by the transformation g is k-linear.

Lemma 4 (No-name Lemma). Using the above notation, assume that the action of G on X is locally free and put d = dim E − dim X. Consider the action of G on X ×A^{d} via the first factor and let π_{1} : X × A^{d} → X be the natural projection. Then there exists a G-equivariant birational isomorphism φ: E −−→ X × A^{d} such that the following diagram is commutative

\xymatrix{E \ar[dr]_{π} \ar@{-->}[rr]^{φ} & & X × A^{d} \ar[dl]^{π_{1}} \\ & X} .

Proof. If the group G is finite, it is the classical “Speiser Lemma” [20]; in the general case, the proof, valid in arbitrary characteristic, see, e.g., in [14]. □

¿From now on we use the following notation. If X and Y are the irreducible varieties, then X ≈ Y means that X and Y are birationally isomorphic; if an algebraic group H acts on X and Y, then X ≈^{H} Y means that there is an H-equivariant birational isomorphism between X and Y.

Proof of Lemma 1.

Put d_{i} := dim L_{i}. The natural projection L_{2} ⊕ L_{3} → L_{2} is a vector bundle over L_{2}, to which, in view of (i), is applicable Lemma 4. It implies that

L_{2} ⊕ L_{3} ≈^{H} L_{2} × A^{d_{3}},  (42)

where H acts on L_{2}×A^{d_{3}} via the first factor. Similarly, applying Lemma 4 to the natural projections L_{1} ⊕ L_{2} → L_{1} and L_{1} ⊕ L_{2} → L_{2} and considering the actions of H on L_{1} × A^{d_{2}} and L_{2} × A^{d_{1}} via the first factors, we obtain

L_{1} × A^{d_{2}} ≈^{H} L_{1} ⊕ L_{2} ≈^{H} L_{2} × A^{d_{1}}.  (43)

It follows from (42) and (iii) that

(L_{2} ⊕ L_{3})---H ≈ (L_{2}---H) × A^{d_{3}} = (L_{2}---H) × A^{d_{1}} × A^{d_{3}−d_{1}},  (44)

and from (43) that

(L_{1}---H) × A^{d_{2}} ≈ (L_{1} ⊕ L_{2})---H ≈ (L_{2}---H) × A^{d_{1}}.  (45)

In view of (ii), we have (L_{1}---H) ≈ A^{dim L_{1}---H}, from where, in view of (44) and (45), we obtain

(L_{2} ⊕ L_{3})---H ≈ A^{d_{2}+d_{3}−d_{1}+dim L_{1}---H}.

This completes the proof. □

Proof of Theorem 1.

The claim is clear for n = 1, so further we assume that n ⩾ 2.

Let V^{⊕n} be the direct sum of n copies of the space V. One of the orbits of the diagonal action of G on V^{⊕n} is open in V^{⊕n}, therefore, k(V^{⊕n})^{G} = k. Besides, this action is locally free. It follows from this, (6), and Theorem 3 that for H = G, L_{1} = V^{⊕n}, L_{2} = C, L_{3} = A, and n ⩾ 3 the conditions of Lemma 1 hold (for n = 2 condition (iii) of this lemma does not hold). Hence for n ⩾ 3 the claim that we are proving follows from (3) and Lemma 1.

All multiplications m from the G-module C (respectively, A) such that, for every element v ∈ {V, m}, the linear operator V → V, a ↦ va has zero trace, constitute a submodule C_{0} (respectively, A_{0}). Besides, every element ℓ ∈ V^{∗} defines the multiplications m_{ℓ+} ∈ C, m_{ℓ−} ∈ A, for which the products of elements a, b ∈ V are given, respectively, by the formulas

ab := ℓ(a)b + ℓ(b)a,  (46)
ab := ℓ(a)b − ℓ(b)a.  (47)

The subsets C′ := {m_{ℓ+} | ℓ ∈ V^{∗}} and A′ := {m_{ℓ−} | ℓ ∈ V^{∗}} also are the submodules of, respectively, the G-modules C and A. Both of these submodules are isomorphic to the G-module V^{∗}. For every of the k-algebras {V, m_{ℓ+}} and {V, m_{ℓ+}}, the trace of the operator of left multiplication by an element v ∈ V is equal to nℓ(v). This implies that

C = C_{0} ⊕ C′ and A = A_{0} ⊕ A′ if char k does not divide n. (48)

Now let n = 2. Then A = A′ in view of (6), and it follows from (3), (48), and the condition char k ≠ 2 that

M = C_{0} ⊕ C′ ⊕ A′.  (49)

Since one of the G-orbits is open in V^{∗} ⊕V^{∗}, we have k(V^{∗} ⊕V^{∗})^{G} = k. Besides, the action of G on V^{∗} ⊕ V^{∗} is locally free. This implies that for H = G, L_{1} = L_{2} = C′ ⊕ A′, and L_{3} = C_{0} the conditions of Lemma 1 hold. Therefore, for n = 2, the claim under the proof follows from this lemma and (49). □

In view of Theorems 3 and 4, Theorems 2 and 5 result from the following general statement (Theorem 7 below) about special (in the sense of Serre) algebraic groups. Recall from [17, Sect. 4.1] that these are algebraic groups S such that every S-torsor is trivial in the Zariski topology (or, equivalently, for every field extension K/k the Galois cohomology H^{1}(K, S) is trivial).

Theorem 7. Let X be an irreducible variety endowed with a locally free action of a special algebraic group S. Then there exists an irreducible closed subset Z of X such that:

(i) the map k(M)^{G} → k(S), f ↦ f|_{S} is well-defined and is a kisomorphism of fields;

(ii) for every point x of an open dense subset of X, there exists a unique element s∈S such that s · x∈S.

Proof. Consider a rational quotient π_{S,X}: X −−→ X---S. By Rosenlicht’s theorem [15, Thm. 2] there exists an S-invariant open subset U of X lying in the domain of definition of π_{S,X} such that W := π_{S,X}(U) is open in X---S and every fiber of the morphism

π_{S,X}|_{U}: U → W  (50)

is an S-orbit. Since the action is locally free, replacing U by a suitable invariant open subset, we may assume that the S-stabilizer of every point of U is trivial. In turn, this implies (see [14, Remark 4]) that making another such a replacement, we may assume that (50) is a torsor. It then follows from the specialness of the group G that morphism (50), and hence the rational map π_{S,X}, admit a rational section σ: X---S −−→ X (i.e., π ∘ σ = id). This implies that one can take the closure of the set σ(X---S) in X as Z from the formulation of Theorem 2. □

Proof of Theorems 2 and 5.

According to [17, §4], the group G is special. Hence, in view of Theorem 7, Theorems 2 and 5 follows, respectively, from Theorems 4 and 3. □

Proof of Corollary 3.

This follows from the known fact (see, e.g., [11, Cor. 2(i)]) that if X in Theorem 7 is an affine space and the action is linear, then the field k(X)^{S} is stably rational over k (indeed, X is birationally isomorphic to G×X---G by Theorem 7, therefore, the claim follows from the rationality of the underlying variety of G). □

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Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia

National Research University, Higher School of Economics, Myasnitskaya 20, Moscow 101000, Russia

E-mail address: popovvl@mi.ras.ru