Noncommutative Bialynicki-Birula Theorem

In this short note we prove that every maximal torus action on the free algebra is conjugate to a linear action. This statement is the free algebra analogue of a classical theorem of A. Bia{\l}ynicki-Birula.

Let K be our ground field, which is assumed to be algebraically closed. Let Z = {z 1 , z 2 , . . .} = {z i : i ∈ I} be a finite or a countable set of variables (where I = {1, 2, . . .} is an index set), and let Z * denote the free semigroup generated by Z, Z + = Z * \{1}. Moreover let F I (K) = K Z be the free associative K-algebra andF I (K) = K Z be the algebra of formal power series in free variables.
Denote by W = Z the free monoid of words over the alphabet Z (with 1 as the empty word) such that |W| ≥ 1, for |W| the length of the word W ∈ Z + .
For an alphabet Z, the free associative K-algebra on Z is where the multiplication is K-bilinear extension of the concatenation on words, Z * denotes the free monoid on Z, and KW denotes the free K-module on one element, the word W. Any element of K Z can thus be written uniquely in the form where the coefficients a i 1 ,i 2 ,...,i k are elements of the field K and all but finitely many of these elements are zero. In our context, the alphabet Z is the same as the set of algebra generators, therefore the terms "monomial" and "word" will be used interchangeably.
In the sequel, we employ a (slightly ambiguous) short-hand notation for a free algebra monomial. For an element z, its powers are defined as usual. Any monomial z i 1 z i 2 . . . z i k can then be written in a reduced form with subwords zz . . . z replaced by powers.
We then write where by I we mean an assignment of i k to j k in the word z I . Sometimes we refer to I as a multi-index, although the term is not entirely accurate. If I is such a multi-index, its abosulte value |I| is defined as the sum i 1 + · · · + i k . For a field K, let K × = K\{0} denote the multiplicative group of its non-zero elements viewed as an algebraic K-group. Definition 1.3. An n-dimensional algebraic K-torus is a group T n ≃ (K × ) n (with obvious multiplication).
Denote by A n the affine space of dimension n over K.

Definition 1.4. A (left) torus action is a morphism
that fulfills the usual axioms (identity and compatibility): An action σ is effective if for every t = 1 there is an element x ∈ A n such that σ(t, x) = x.
In [1], Bia lynicki-Birula proved the following two theorems. Theorem 1.5. Any regular action of T n on A n has a fixed point. Theorem 1.6. Any effective and regular action of T n on A n is a representation in some coordinate system.
The term "regular" is to be understood here as in the algebro-geometric context of regular function (Bia lynicki-Birula also considered birational actions). The last theorem says that any effective regular maximal torus action on the affine space is conjugate to a linear action, or, as it is sometimes called, linearizable.
An algebraic group action on A n is the same as an action by automorphisms on the algebra K[x 1 , . . . , x n ] of global sections. In other words, it is a homomorphism

An action is effective iff Ker
The polynomial algebra is a quotient of the free associative algebra F n = K z 1 , . . . , z n by the commutator ideal I (it is the two-sided ideal generated by all elements of the form f g − gf ). From the standpoint of Noncommutative geometry, the algebra Γ(X, O X ) of global sections (along with the category of f.g. projective modules) contains all the relevant topological data of X, and various non-commutative algebras (PI-algebras included) may be thought of as global function algebras over "noncommutative spaces". Therefore, noncommutative analogue of the Bia lynicki-Birula theorem is a subject of legitimate interest.
In this short note we establish the free algebra version of the Bia lynicki-Birula theorem. The latter is formulated as follows.
Theorem 1.7. Suppose given an action σ of the algebraic n-torus T n on the free algebra F n . If σ is effective, then it is linearizable.
2 Proof of Theorem 1.7 The proof proceeds along the lines of the original commutative case proof of Bia lynicki-Birula.
If σ is the effective action of Theorem 1.7, then for each t ∈ T n the automorphism is given by the n-tuple of images of the generators z 1 , . . . , z n of the free algebra: . . , f n (t, z 1 , . . . , z n )).
Each of the f 1 , . . . , f n is a polynomial in the free variables.
There is a translation of the free generators such that (for all t ∈ T n ) the polynomials f i (t, z 1 − c 1 , . . . , z n − c n ) have zero free part.
Proof. This is a direct corollary of Theorem 1.5. Indeed, any action σ on the free algebra induces, by taking the canonical projection with respect to the commutator ideal I, an actionσ on the commutative algebra K[x 1 , . . . , x n ]. If σ is regular, then so isσ. By Theorem 1.5,σ (or rather, its geometric counterpart) has a fixed point, therefore the images of commutative generators x i underσ(t) (for every t) will be polynomials with trivial degree-zero part. Consequently, the same will hold for σ.
We may then suppose, without loss of generality, that the polynomials f i have the form where by z J we denote, as in the introduction, a particular monomial (a word in the alphabet {z 1 , . . . , z n } in the reduced notation; J is the multi-index in the sense described above); also, N is the degree of the automorphism (which is finite) and a ij , a ijl , . . . are polynomials in t 1 , . . . , t n . As σ t is an automorphism, the matrix [a ij ] that determines the linear part is non-singular. Therefore, without loss of generality we may assume it to be diagonal (just as in the commutative case If T 1 ⊂ T n is any one-dimensional torus, the restriction of τ to T 1 is non-trivial. Indeed, were it to happen that for some T 1 , τ (t)z = z, t ∈ T 1 , (z = (z 1 , . . . , z n )) then our initial action σ, whose linear part is represented by τ , would be identity modulo terms of degree > 1: Now, equality σ(t 2 )(z) = σ(t)(σ(t)(z)) implies and therefore a ijl (t) = 0. The coefficients of the higher-degree terms are processed by induction (on the total degree of the monomial). Thus which is a contradiction since σ is effective. Finally, if [m ij ] were singular, then one would easily find a one-dimensional torus such that the restriction of τ were trivial.
Proof. This lemma mirrors the final part in the proof in [1]. The conjugation is straightforward, since for every s, t ∈ T n one has Denote byF n the power series completion of the free algebra F n , and letσ, τ andβ denote the endomorphisms of the power series algebra induced by corresponding morphisms of F n . The endomorphismsσ,τ ,β come from (polynomial) automorphisms and therefore are invertible.
(just as before, z J is the monomial with multi-index J). Then Now, from the conjugation property we must havê here the notation G(z) J stands for a word in G i (z) with multi-index J, while the exponents j 1 , . . . , j n count how many times a given index appears in J (or, equivalently, how many times a given generator z i appears in the word z J ). Therefore, the coefficient ofσ(t)(z i ) at z J has the form Therefore, B i (z) are polynomials in the free variables. What remains is to notice that z i = B i (G 1 (z), . . . , G n (z)). Thus β is an automorphism.

From Lemma 2.3 it follows that
which is the linearization of σ. Theorem 1.7 is proved.

Discussion
The noncommutative toric action linearization theorem that we have proved has several useful applications. In the work [6], it is used to investigate the properties of the group Aut F n of automorphisms of the free algebra. As a corollary of Theorem 1.7, one gets Corollary 3.1. Let θ denote the standard action of T n on K[x 1 , . . . , x n ] -i.e., the action θ t : (x 1 , . . . , x n ) → (t 1 x 1 , . . . , t n x n ).
Letθ denote its lifting to an action on the free associative algebra F n . Thenθ is also given by the standard torus action.
This statement plays a part, along with a number of results concerning the induced formal power series topology on Aut F n , in the establishment of the following proposition (cf. [6]). Proposition 3.2. When n ≥ 3, any Ind-scheme automorphism ϕ of Aut(K x 1 , . . . , x n ) is inner.
One could try and generalize the free algebra version of the Bia lynicki-Birula's theorem to other noncommutative situations. Another way of generalization lies in changing the dimension of the torus. In a complete analogy with further work of Bia lynicki-Birula [2], we expect the following to hold. On the other hand, there is little reason to expect this statement to hold with further lowering of the torus dimension. In fact, even in the commutative case the conjecture that any effective toric action is linearizable, in spite of considerable effort (see [7]), proved negative (counterexamples in positive characteristic due to Asanuma, [8]).
Another direction would be to replace T by an arbitrary reductive algebraic group, however the commutative case also does not hold even in characteristic zero (cf. [9]).