CHARACTERIZATION OF n-DIMENSIONAL NORMAL AFFINE SLn-VARIETIES

We show that any normal irreducible affine n-dimensional SLn-variety X is determined by its automorphism group seen as an ind-group in the category of normal irreducible affine varieties. In other words, if Y is an irreducible affine normal algebraic variety such that Aut(Y) ≃ Aut(X) as an ind-group, then Y ≃ X as a variety. If we drop the condition of normality on Y , then this statement fails. In case n ≥ 3, the result above holds true if we replace Aut(X) by 𝒰(X), where 𝒰(X) is the subgroup of Aut(X) generated by all one-dimensional unipotent subgroups. In dimension 2 we have some interesting exceptions.


Introduction and main results
Our base field is the field of complex numbers C. For an affine variety X, the automorphism group Aut(X) has the structure of an ind-group. We will shortly recall the basic definitions and results in Section 2. The classical example is Aut(A n ), n > 1, the group of automorphisms of the affine n-space A n . Recently, Hanspeter Kraft proved the following result which shows that the affine n-space is determined by its automorphism group (see [Kr15]).
Theorem 1. 1. Let Y be a connected affine variety. If Aut(Y ) Aut(A n ) as indgroups, then Y A n as varieties.
Note that this result was generalised in [CRX19] (see also [KRvS19] and in a similar spirit, see [LRU20, Thm. 1]) where the authors proved Theorem 1.1 under a weaker condition: namely, that groups Aut(Y ) and Aut(A n ) are isomorphic only as abstract groups. Moreover, recently Theorem 1.1 was generalized in [LRU18, Thm. 1.4] (see also [RvS21,Main Thm. 1]) where it was proved that an affine toric variety different from the algebraic torus is determined by its automorphism group seen as an ind-group in the category of normal affine irreducible varieties. If we drop the normality condition in [LRU18,Thm. 1.4], the situation changes. In this paper, we show that for "most" n-dimensional affine normal varieties X endowed with a nontrivial regular SL n = SL n (C)-action, there are infinitely many affine varieties Y such that Aut(Y ) Aut(X) as an ind-group and we classify all such Y .
Let d > 1. Consider the action of µ d = {ξ ∈ C * | ξ d = 1} on A n by scalar multiplication and denote by A d,n the quotient of A n by µ d . Note that A d,n is normal. Denote also by π : A n → A d,n the quotient map. This means that A d,n is an affine variety with coordinate ring C[x 1 , . . . , x n ] dk , the algebra of invariants where C[x 1 , . . . , x n ] dk denotes the homogeneous polynomials of degree dk. Note that A d,n is indeed an orbit space, because µ d is finite. For d > 1, A d,n has an isolated singularity in π(0) and π induces anétale covering A n \ {0} → A d,n \ {p(0)} with Galois group µ d . Remark 1.2. We will see in Lemma 6.1 and Proposition 5.4 that any affine normal variety endowed with a regular nontrivial SL n -action is isomorphic to either SL 2 /T , SL 2 /N or to A d,n for some d ∈ N, where T ⊂ SL 2 is the standard subtorus and N ⊂ SL 2 is the normalizer of T . This implies that Theorem 1.3 and Theorem 1.4 below indeed provide the characterization of n-dimensional normal affine SL nvarieties. Then the induced morphism η : A d,n → A s d,n is the normalization and has the property that the induced map η : A d,n \ { } ∼ − → A s d,n \ { } is an isomorphism, where denotes the points corresponding to the homogeneous maximal ideals. In fact, η is SL n -equivariant, and A d,n \ { } is an SL n -orbit. We prove the following result.

Consider the affine variety
Theorem 1. 3. Let X be an irreducible affine variety. Then Aut(X) and Aut(A d,n ) are isomorphic as ind-groups if and only if X A s d,n as a variety for some s ∈ N. Theorem 1.3 and the following result shows that SL 2 /T and SL 2 /N are the only affine n-dimensional SL n -varieties (except A n ) that are determined by their automorphism groups in the category of affine irreducible varieties.
Theorem 1. 4. Let X be an irreducible variety such that Aut(X) Aut(SL 2 /T ) respectively Aut(X) Aut(SL 2 /N ) as ind-groups. Then X SL 2 /T respectively X SL 2 /N as varieties.
For an affine variety X we denote by U (X) ⊂ Aut(X) the subgroup generated by the one-dimensional unipotent subgroups. We do not know whether U (X) has the structure of an ind-subgroup (i.e., whether U (X) ⊂ Aut(X) is closed). That is why we introduce the definition of an algebraic isomorphism. This is an isomorphism φ : U (X) ∼ − → U (Y ) such that for any subgroup U ⊂ U (X), where U is a closed onedimensional unipotent subgroup of Aut(X), the image φ(U ) ⊂ Aut(Y ) is a closed one-dimensional unipotent subgroup and φ| U : U ∼ − → φ(U ) is an isomorphism of algebraic groups.
Theorem 1. 5. Let X be A d,n , SL 2 /T or SL 2 /N and Y be an irreducible affine variety. Assume that there is an algebraic isomorphism U (X) Acknowledgement. The author thanks both referees for important remarks and suggested improvements. The author would also like to thank Hanspeter Kraft for his support while writing this paper. In particular, Proposition 6.21 was pointed out to him by Hanspeter Kraft. Finally, the author is grateful to Michel Brion who suggested a number of improvements and Mikhail Zaidenberg for useful discussions.

Ind-groups
The notion of an ind-group goes back to Shafarevich who called such objects infinite dimensional groups (see [Sh66]). We refer to [Kum02] and [FK18] for basic notions in this context. Definition 2.1. By an ind-variety we mean a set V together with an ascending filtration V 0 ⊂ V 1 ⊂ V 2 ⊂ · · · ⊂ V such that the following holds: (1) V = k∈N V k , (2) each V k has the structure of an affine algebraic variety, (3) for all k ∈ N the subset V k ⊂ V k+1 is closed in the Zariski-topology.
A morphism from an ind-variety V = k∈N V k to an ind-variety W = m∈N W m is a map φ : V → W such that for any k there is an m such that φ(V k ) ⊂ W m and such that the induced map V k → W m is a morphism of algebraic varieties. Isomorphisms of ind-varieties are defined in the obvious way.
Two filtrations V = k∈N V k and V = k∈N V k are called equivalent if for every k there is an m such that V k ⊂ V m is a closed subvariety as well as V k ⊂ V m .
An ind-variety V has a natural topology: is open, (resp. closed), for all k. A locally closed subset S ⊂ V has the induced structure of an ind-variety. It is called an ind-subvariety. A subset S ⊂ V that is a closed subset of some V k is called an algebraic subset.
The product of two ind-varieties is defined in the usual way. This allows to give the following definition.
Definition 2.2. An ind-variety G is said to be an ind-group if the underlying set G is a group such that the map G × G → G, (g, h) → gh −1 , is a morphism.
An ind-group G is called connected if for every g ∈ G there is an irreducible curve C and a morphism C → G whose image contains the neutral element e and g.
A closed subgroup H of G (i.e., H is a subgroup of G and is a closed subset) is again an ind-group under the closed ind-subvariety structure on G. A closed subgroup H of an ind-group G is called an algebraic subgroup if and only if H is an algebraic subset of G.
. Let X be an affine variety. Then Aut(X) has the structure of an ind-group such that for any algebraic group G, there is a correspondence between regular G-actions on X and ind-group homomorphisms G → Aut(X).
If G is an algebraic group acting regularly and faithfully on X, then by Proposition 2.3, we can consider G as an algebraic subgroup of Aut(X). We will often switch between these two points of view.

Locally nilpotent derivations and G a -actions
Additive group actions on affine varieties can be described by a certain kind of derivations. We recall some of the basics here (see [Fre06] for details). Let λ : G a → Aut(X) be a G a -action on an affine variety X. Such an action induces a derivation on the level of regular functions O(X) by where G a = Spec(C[s]). We have that for every f ∈ O(X) there exists an k ∈ N with δ k λ (f ) = 0. Derivations that have such a property are called locally nilpotent. Moreover, every G a -action on X arises from a certain locally nilpotent derivation δ and the G a -action λ δ : G a × X → X is obtained from δ via

Automorphisms
Proposition 3.1. Let π : A n → A d,n . Then every automorphism of A d,n lifts to an automorphism of A n which commutes with each element of µ d .
Proof. The quotient map π : . . , x n ] dk denotes the homogeneous polynomials of degree dk. Let φ ∈ Aut(A d,n ). First we claim that . . , x n ], where i = j and φ * is the pull-back of φ. Let p be a common factor of p i and p j . Then p = g∈µ d gp divides p d i and p d j . By construction it is clear that p ∈ O(A d,n ). Hence, (φ * ) −1 ( p) is a common factor of (φ * ) −1 (p d i ) = x d 2 i and (φ * ) −1 (p d j ) = x d 2 j . Therefore, p ∈ C and then, p ∈ C.
We have Since for an automorphism φ : A d,n → A d,n we have that φ * (x d i ) = q d i for some q i ∈ C[x 1 , . . . , x n ], we define the morphism φ = (q 1 , . . . , q n ) : A n → A n given by the map (x 1 , . . . , x n ) → (q 1 , . . . , q n ).
If φ is the identity automorphism, then the restriction of φ * to O(A d,n ) is the identity and we have that In this case φ is an automorphism: i.e., the trivial automorphism of A d,n lifts to an automorphism of A n which we denote by ∆(w 1 , . . . , w n ).
Let now θ : A d,n → A d,n be the inverse automorphism of φ ∈ Aut(A d,n ). Since φ•θ is the trivial automorphism of A d,n , it lifts to an automorphism ∆(w 1 , . . . , w n ) of A n and so φ is an automorphism with the inverse θ • ∆(w 1 , . . . , w n ) −1 . To finish the proof, we need to show that φ commutes with µ d . Indeed, since φ * preserves for some l = 1, . . . , d − 1. Since polynomial q i has a linear summand it follows that l = 1. The proof follows.
Let X be an affine variety, H be a finite group that acts faithfully on X, and let π : X → X/H be the quotient morphism. Since H acts faithfully, H naturally embeds into Aut(X) and we identify H with its image in Aut(X). Denote by Aut H (X) ⊂ Aut(X) the subgroup of all automorphisms of X which normalize H: i.e., the subgroup of those automorphisms φ such that φ −1 • H • φ = H. (a) There is a canonical homomorphism of groups φ : Aut H (X) → Aut(X/H), (b) if X is normal and contains only finitely many fixed points of H then every C + -action on X/H lifts to a C + -action on X.
for some h ∈ H. Therefore φ * (f ) ∈ O(X) H , which means that φ induces an automorphism of X/H. Let us recall that a closed algebraic subgroup U of Aut(X) is a 1-dimensional unipotent subgroup if U C + .
Proof. The surjectivity of φ d follows from Proposition 3.1. The last claim of the statement follows from Lemma 3.2 (b). What remains is to compute the kernel of φ d .
It is clear that This means that f acts trivially on

Root subgroups
Let G be an ind-group, and let T ⊂ G be a closed torus.
Definition 4.1. A closed subgroup U ⊂ G isomorphic to C + and normalized by T is called a root subgroup with respect to T . The character of T on Lie U C (i.e., the algebraic action of T on Lie U ) is called the weight character of U .
Let X be an affine variety and consider a nontrivial algebraic action of C + on X, given by λ : for s ∈ C and x ∈ X. It is easy to see that this is again a C + -action. In fact, the corresponding locally nilpotent derivation to f · λ is f δ λ , where δ λ is the locally nilpotent derivation that correspond to λ (see Section 2.2 for details). It is clear that if X is irreducible and f = 0, then f · λ and λ have the same invariants. If U ⊂ Aut(X) is a closed subgroup isomorphic to C + and if f ∈ O(X) U is a U -invariant, then in a similar way we define the modification f · U of U . Choose an isomorphism λ : C + → U and set Note that Lie(f · U ) = f Lie U ⊂ Vec(X), where Vec(X) denotes the Lie algebra of (algebraic) vector fields on X, i.e., Vec(X) = Der(O(X)), the Lie algebra of derivations of O(X).
If a torus T acts linearly and rationally on a vector space V , then we call V multiplicity-free if the weight spaces V α are all of dimension less than or equal to 1. . Let X be an irreducible affine variety and let T ⊂ Aut(X) be a torus. Assume that there exists a root subgroup U ⊂ Aut(X) with respect to T such that the T -module O(X) U is multiplicity-free.
The next result is going to be of use in the sequel and can be found in [Lie11, Thm. 1]. We denote by U (A n ) the subgroup of Aut(A n ) of the following form and by T n a maximal subtorus of U (A n ) of the form The character ξ λ corresponding to the root subgroup U is the following: Remark 4.4. The last lemma can also be expressed in the following way (see [KS13, Remark 2]): there is a bijective correspondence between the T n -stable onedimensional unipotent subgroups U ⊂ Aut(A n ) and the characters of T n of the form λ = j λ j j where one λ i equals 1 and the others are ≤ 0. We will denote this set of characters by X u (T n ): If λ ∈ X u (T n ), then U λ denotes the corresponding one-dimensional unipotent subgroup normalized by T n .

A special subgroup of Aut(X)
For any affine variety X consider the normal subgroup U (X) of Aut(X) generated by closed one-dimensional unipotent subgroups. The group U (X) was introduced and studied in [AFK13], where the authors called it the group of special automorphisms of X. Following [Kr15], we introduce the notion of an algebraic homomorphism between these groups.
is closed, and φ| U : U → φ(U ) is a homomorphism of algebraic groups. We say that φ is an algebraic isomorphism if φ is an isomorphism of groups and Lemma 5.2. Let φ : U (X) → U(Y ) be an algebraic homomorphism. Then, for any algebraic subgroup G ⊂ U (X) generated by one-dimensional unipotent subgroups of Aut(X), the image φ(G) is an algebraic subgroup of U (Y ) and φ| G : G → φ(G) is a homomorphism of algebraic groups.
Let X be an affine variety and let η : X → X be a normalization map. It is well known that any automorphism of X lifts uniquely to an automorphism of X. Indeed, for a given automorphism φ : X → X, the composition φ • η : X → X is a morphism, which by the universal property of normalization factors through a morphism φ : X → X such that φ • η = η • φ. It remains to argue that φ is an automorphism. But for the same reason, the inverse φ −1 lifts to an automorphism ψ : X → X. Since η : X → X is birational, the compositions ψ • φ and φ • ψ are equal to the identity on a dense open subset of an irreducible variety, hence everywhere. This shows that η induces a well-defined injective homomorphism η : Aut(X) → Aut( X). Moreover, in [FK18, Prop. 12. 1.1] it is proved that η is a closed immersion of ind-groups: i.e., η(Aut(X)) ⊂ Aut( X) is a closed subgroup and η induces the isomorphism of ind-groups Aut(X) ∼ − → η(Aut(X)). Hence, we have the following statement.
Lemma 5. 3. Let X be an irreducible affine variety, and let η : X → X be its normalization. Then every automorphism of X lifts uniquely to an automorphism of X and the induced map η : Aut(X) → Aut( X) is a closed immersion of indgroups.
Proposition 5.4. Let n ≥ 3 and let X be an n-dimensional irreducible affine variety endowed with a nontrivial SL n -action. Then . . , d l ) = d and the normalization of X is isomorphic to A d,n . The same holds when n = 2 and the normalization of X is A d,2 for some d ∈ N.
Proof. First, let n ≥ 3. If X is normal, then from [KRZ20, Thm. 1.6 and Prop. 4.4(2); see also Example 4.5] it follows that X A d,n for some d ∈ N. It is well known that Now, consider any n-dimensional irreducible affine variety X endowed with a nontrivial SL n -action and a normalization morphism η : A d,n → X. Since any SL n -action on O(X) lifts to an SL n -action on O(A d,n ), it follows that O(X) is a SL n -submodule of O(A d,n ) and therefore where Ω is a submonoid of N under addition. Since O(X) is finitely generated, Ω ⊂ N is a finitely generated submonoid: i.e., there exist d 1 , . . . , d l ∈ N such that Ω = d 1 N + · · · + d l N. The claim follows.
Lemma 6.1. Let X be an affine normal irreducible variety of dimension two endowed with a nontrivial SL 2 -action. Then X is SL 2 -equivariantly isomorphic to one of the following varieties: (a) A d,2 for some d ∈ N, where the SL 2 -action on A d,2 is induced by the standard SL 2 -action on A 2 , (b) SL 2 /T , where T is the standard subtorus of SL 2 and SL 2 acts on SL 2 /T by left multiplication, (c) SL 2 /N , where N is the normalizer of T and SL 2 acts on SL 2 /N by left multiplication.
The SL 2 -action on SL 2 /T and on SL 2 /N from the Lemma above is transitive. The SL 2 -variety A d,2 is the union of a fixed point and the orbit ( Moreover, any closed subgroup of SL 2 of codimension less than or equal to 2 is conjugate to either T , or N , or U d for some d ≥ 1, or B = a t 0 a −1 t ∈ C, a ∈ C * (see, for example, [Pop73,p. 803] Proposition 6.2. If a reductive group G acts on an affine variety X and if the stabilizer of a point x ∈ X contains a maximal torus, then the orbit Gx is closed. Remark 6.4. Note that SL 2 /T P 1 × P 1 \ ∆, where ∆ is the diagonal, and SL 2 /N P 2 \ C, where C is a smooth conic (see [Pop73, Lem. 2]).

The structure of Aut(SL 2 /T )
The variety SL 2 /T is isomorphic to the following so-called Danielewski surface: i.e., the smooth 2-dimensional affine quadric V (xz − y 2 + y) ⊂ A 3 (see [DP09]) and the quotient map π : SL 2 → SL 2 /T is given by a b c d → (ab, ad, cd). It is not difficult to see that X = V (xz + y 2 − 1) V (xz − y 2 + y) ⊂ A 3 . From now on and until end of Section 6, we identify SL 2 /T with X = V (xz + y 2 − 1). Consider the orthogonal group O 3 = O(3, C) associated with the quadratic form y 2 + xz generated by τ : A 3 → A 3 given by the map (x, y, z) → (−x, −y, −z) and by the group SO 3 = SO(3, C) that is composed of the matrices Following [Lam05, Thm. 6] (see also [MM90]), Aut(X) is the amalgamated product of the orthogonal group O 3 = SO 3 × τ and J τ along their intersection C, where J is the subgroup of Aut(X) of the automorphisms of the form Note that SO 3 is generated by C + -actions. Define J to be the subgroup of J generated by C + -actions. The subgroup of Aut(X) generated by SO 3 and J coincides with the subgroup of Aut(X) generated by SO 3 and J and is a subgroup of U (X). Moreover, because τ normalizes J, SO 3 , we have that Aut(X) = J, SO 3 τ . This implies that Aut(X) is not connected, and τ / ∈ Aut • (X). Since the closure of U (X) ⊂ Aut(X) is connected as U (X) is generated by connected subgroups and since J, SO 3 ⊂ U (X) we have that U (X) ⊂ Aut(X) coincides with Aut • (X) and hence is closed. Moreover, Aut(X) = U (X) τ . The following proposition is going to be of use later (see [Neu48,Cor. 8.11]).
Proposition 6.5. In the amalgamated product G = A * C B with the unified subgroup C = A ∩ B, consider two subgroups A ⊂ A and B ⊂ B, and let Lemma 6.6. The group U (X) is the amalgamated product of SO 3 and J along their intersection. Proof. We know that Aut(X) is the amalgamated product of O 3 = SO 3 × τ and J τ along their intersection C. Moreover, since SO 3 ∩C = J ∩ SO 3 = J ∩C, we have by Proposition 6.5 that J, SO 3 ⊂ Aut(X) is the amalgamated product of J and SO 3 along their intersection. As U (X) = J, SO 3 the claim follows.
Lemma 6. 7. The subgroup Aut SO3 (X) ⊂ Aut(X) of those automorphisms that commute with SO 3 is τ . Moreover, X/ τ SL 2 /N . Proof. Let ϕ ∈ Aut SO3 (X). Since Aut(X) is the amalgamated product of O 3 and J along their intersection, we can write ϕ as the product where a i ∈ O 3 and b i ∈ J. Since a = ϕ • a • ϕ −1 for any a ∈ SO 3 , we have that b k ∈ O 3 ∩ J and ϕ can be written as a 1 • b 1 • · · · • a k for some a i ∈ O 3 and b i ∈ J. Assume first that k > 1. Hence, the expression Since this should hold for any a ∈ SO 3 , we get a contradiction. Therefore, k = 1 and hence ϕ ∈ O 3 . This implies that Aut SO3 (X) = τ as the centralizer of SO 3 in O 3 is τ .
To finish the proof we need to argue that X/ τ SL 2 /N . We first note that X/ τ is normal and since SL 2 -action on X is transitive, it follows that the induced action of SL 2 on X/ τ is transitive too. Hence, from Lemma 6.1 it follows that X/ τ SL 2 /N as X X/ τ .

6.3]), where Aut • (X) is the neutral component of Aut(X). Hence, U (X) is an ind-group.
Proposition 6.8. We have the following properties.
(a) All closed subgroups S ⊂ Aut(X) isomorphic to PSL 2 are conjugate.
(b) The root subgroups with respect to a maximal torus T of any S PSL 2 are multiplicity-free with weights 1, 2, 3, . . . up to an automorphism of T .
Proof. (a) Since Aut(X) is the amalgamated product of O 3 and J over their intersection, we have that by [Sr80] any closed subgroup S ⊂ Aut(X) isomorphic to S is conjugate to one of the factors O 3 or J. Since all unipotent subgroups of J commute, S can not be embedded into J and hence S is conjugate to a subgroup of O 3 : i.e., to SO 3 . The claim follows. Now we are going to prove (b). Let U ⊂ Aut(X) be a root subgroup with respect to T . This means that T U is an algebraic subgroup of Aut(X) and by [Sr80], T U is conjugate to a subgroup of either O 3 or J. If T U is conjugate to a subgroup of O 3 , then the weight of U with respect to T is either 1 or −1, i.e., up to an automorphism of T we can assume that the weight is 1. If T U is conjugate to a subgroup of J, then without loss of generality we can assume that T U is an algebraic subgroup of J ≤k generated by elements of the form for some natural k, where C[z] ≤k denotes the polynomials of degree less or equal than k. Moreover, since all tori in J ≤k are conjugate we can assume that By the following computation, (tx, y, t −1 z) • (x + 2yP (z) − zP 2 (z), (y − zP (z)), z) • (t −1 x, y, tz) it is easy to see that a root subgroup U ⊂ J ≤k should have the form Note that the root subgroup U i with respect to T has the weight i + 1. The claim follows. This condition for φ is equivalent to the condition that φ normalizes τ . Moreover, since τ has order two, φ commutes with τ . Recall that by Aut τ (X) we denote the subgroup of those elements of Aut(X) that normalize τ , but in this particular case Aut τ (X) is even the subgroup of those automorphisms of Aut(X) that commute with τ .
As we have mentioned above, φ ∈ Aut(X) induces an automorphism of Y SL 2 /N if and only if φ ∈ Aut τ (X). On the other hand, since X SL 2 /T is simply connected and the quotient map π : X → X/ τ = Y is anétale covering, every automorphism ϕ of Y can be lifted to a continous analytical automorphism of X and hence by [Sr58,Prop. 20], ϕ can be lifted to an automorphism ϕ of X: i.e., ϕ ∈ Aut τ (X). Hence, we have the surjective homomorphism Aut τ (X) → Aut(Y ) with the kernel τ and so Aut(Y ) Aut τ (X)/ τ . (1) Observe that SO 3 × τ is the subgroup of Aut τ (X). Define the subgroup J τ ⊂ Aut(X) of those automorphisms from J which normalize τ . It is not difficult to see that J τ is comprised of the following automorphisms: Cz 2l .
We have the following statement.
Lemma 6.9. The subgroup Aut τ (X) ⊂ Aut(X) is the direct product of τ and the amalgamated product of SO 3 and J τ along their intersection.
Proof. As we have mentioned above, Aut τ (X) is the subgroup of those automorphisms of Aut(X) that commute with τ . Assume φ ∈ Aut(X) commutes with τ . Since Aut(X) is the amalgamated product of SO 3 × τ and J τ , one can write φ as a product a 1 • b 1 • · · · • a k • b k , where a i ∈ SO 3 × τ and b i ∈ J τ . Further, because φ commutes with τ , τ φτ = φ or equivalently, Since τ commutes with SO 3 , one can rewrite this equation as follows: From the amalgamated product structure of Aut(X), it follows that τ But this can happen only if b i commutes with τ : i.e., b i ∈ J τ × τ . Therefore, Aut τ (X) is generated by SO 3 × τ and J τ × τ . Moreover, Aut τ (X) = SO 3 , J τ × τ and by Proposition 6.5, SO 3 , J τ is the amalgamated product of SO 3 and J τ over their intersection.
From Lemma 6.9 and (1) we have the following statement.
Lemma 6. 10. The automorphism group Aut(Y ) is isomorphic to the amalgamated product of SO 3 and J τ . In particular, Aut(Y ) = U (Y ).
Corollary 6.12. We have the following properties. Proof. (a) By Lemma 6.10, Aut(Y ) is the amalgamated product of SO 3 and J τ and by [Sr80] any algebraic subgroup of the amalgamated product is conjugate to one of the factors. Since J τ does not contain a copy of PSL 2 , it follows that S is conjugate to a subgroup of SO 3 : i.e., to SO 3 itself.
(b) Without loss of generality, we can assume that S equals SO 3 and T ⊂ SO 3 is the subtorus of the form Any root subgroup of Aut(Y ) with respect to T lifts to a root subgroup U of Aut τ (X) (see Lemma 3.2) with respect to the subtorus p −1 ( T ) • ⊂ Aut τ (X). As it follows from the proof of Proposition 6.8(b), U coincides with The weight of the root subgroup U 2i ⊂ Aut τ (X) with respect to p −1 ( T ) • is 2i + 1. Since the kernel of p −1 ( T ) → T is trivial, we have that the set of weights of root subgroups of Aut(Y ) with respect to T is {2i + 1 | i ∈ N}. This proves the first part of the statement. The second part follows because if there is an algebraic isomorphism ϕ : U (X) → U(Y ), then ϕ maps root subgroups of U (X) with respect to a subtorus T ⊂ U (X) to root subgroups of U (Y ) with respect to ϕ( T ) that have the same weights. But as follows from Proposition 6.8 and the first part of this proof, it is not the case.
Remark 6. 13. Analogously as in the proof of Lemma 6.7, using amalgamated product structure of Aut(Y ) described in Lemma 6.10, we can show that the subgroup Aut SO3 (X) ⊂ Aut(X) of those automorphisms that commute with SO 3 is trivial.

On the automorphism group of A d,2
Recall that by Proposition 3.3, there is a surjective homomorphism φ d : Aut µ d (A n ) → Aut(A d,n ) of groups. Consider now the maximal subtorus and recall that by T n we denote the subtorus of the form Lemma 6.14. Let U ⊂ Aut(A d,n ) be a root subgroup with respect to T d,n which has a character χ. Then U lifts to a root subgroup Proof. From Proposition 3.1, it follows that any root subgroup U of Aut(A d,n ) with respect to T d,n lifts to a unipotent subgroup U = (φ −1 d (U )) • of Aut µ d (A n ). Moreover, U is normalized by (φ −1 d (T d,n )) • = T n . Now, let u ∈ U be a nontrivial element and u = φ d ( u) ∈ U . We have group isomorphisms Now the proof follows from the formula Observe that the homomorphism φ d : Aut µ d (A n ) → Aut(A d,n ) induces the homomorphism φ d : U µ d (A n ) → U(A d,n ) which has the kernel µ (n,d) , where U µ d (A n ) ⊂ Aut µ d (A n ) is a subgroup generated by C + -actions.
In [BH03], it is proved that any faithful action of an (n − 1)-dimensional torus on an n-dimensional toric T Z -variety Z is conjugate to a subtorus of the big torus T Z . This result is used in order to prove the following lemma.
Lemma 6. 15. Let T be an algebraic subtorus of U (A d,n ) of dimension (n − 1). Then there exists an algebraic isomorphism F : n ) is an algebraic subtorus of dimension n−1, and since A d,n is toric, by [BH03, Thm. p. 2] there exists ϕ ∈ Aut(A d,n ) such that ϕ • T • ϕ −1 ⊂ T d,n . Moreover, since U (A d,n ) is a normal subgroup of Aut(A d,n ), ϕ • T • ϕ −1 ⊂ T d,n and hence since dim T d,n = n − 1, ϕ • T ϕ −1 = T d,n . This proves that an algebraic isomorphism F : Let Z be an irreducible affine variety of dimension n ≥ 2 and ψ : U (Z) ∼ → U (A d,n ) be an algebraic isomorphism. Let T be an (n − 1)-dimensional algebraic subtorus of U (Z). Then, after composing ψ with a suitable algebraic isomorphism F : U (A d,n ) → U(A d,n ) (see Lemma 6.15), we can assume that ψ(T ) = T d,n .
Lemma 6. 16. Root subgroups U and ψ(U ) have the same weight characters with respect to T and ψ(T ) = T d,n , respectively: i.e., if χ : T d,n → C * is the weight of ψ(U ), then the weight of U is χ • ψ.
Proof. Let U be a root subgroup of U (Z) with respect to T and Lie U = Cν, where ν is a generator. Then ψ(U ) is the root subgroup of U (A d,n ) with respect to T d,n . The algebraic isomorphism ψ induces an isomorphism dψ u e : Lie U → Lie ψ(U ). Note that the action of T on U induces the action of T on Lie U . Then ψ(t) • dψ u e (ν) • ψ(t −1 ) = χ(ψ(t))dψ u e (ν) = dψ u e (χ(ψ(t))ν) = dψ u e (χ • ψ(t)ν), where t ∈ T . On the other hand, . The claim follows.
Lemma 6.17. Let d be even. Then the set of weights of root subgroups of Aut(A d,2 ) with respect to T d,2 is {(kd + 2)/2 | k ∈ N ∪ {0}} up to an automorphism of T d,2 .
Proof. By Proposition 3.3, any root subgroup of Aut(A d,2 ) with respect to T d,2 lifts to a root subgroup of Aut µ d (A 2 ) with respect to φ −1 d (T d,2 ) • = T 2 . By Lemma 4.3, any root subgroup of Aut(A 2 ) with respect to T 2 is equal either to for some s, l ∈ N ∪ {0}. Root subgroups U s and U l belong to Aut µ d (A 2 ) if and only if s, l ∈ dN + 1. The weight of the action of T 2 = {(cx, c −1 y) | c ∈ C * } on U s by t • u • t −1 , t ∈ T 2 and u ∈ U s equals s + 1. Analogously, the weight of T 2 -action on U l is −l − 1. Therefore, the set of weights of root subgroups of Aut µ d (A 2 ) with respect to T 2 is {kd + 2 | k ∈ N ∪ {0}} up to an automorphism of T 2 . Moreover, since the kernel of the map φ d : T 2 → T d,2 is µ 2 as d is even, the statment follows from Lemma 6.14.
By the Jung-Van der Kulk Theorem (see [Ju42] and [Kul53]), Aut(A 2 ) = Aff 2 * C J, where Aff 2 is the group of affine transformations of A 2 and J = {(ax + c, by + f (x)) | a, b ∈ C * , c ∈ C, f (y) ∈ C[x]} and C = Aff 2 ∩ J. Now, the subgroup Aut µ d (A 2 ) ⊂ Aut(A 2 ) contains the standard GL 2 ⊂ Aut(A 2 ) and By [AZ13,Thm. 4.2], Aut(A d,2 ) Aut µ d (A 2 )/µ d is the amalgamated product of GL 2 /µ d and J d /µ d along their intersection. Moreover, as we will see in Lemma 6.18 below, such an amalgamated product structure induces the amalgamated product structure of U (A d,2 ). Denote by J d the subgroup of J d of the form and by T d,2 the one-dimensional subtorus of Aut(A d,2 ) induced by the C * -action on A 2 given by the maps {(x, y) → (cx, y) | c ∈ C * }. We have the following statement. Aut µ d (A 2 )/µ d , the subgroups {(x, y) → (cx, y) | c ∈ C * } and U (A 2 ) of Aut(A 2 ) also have a nontrivial intersection, which is not the case. Hence, we conclude that Aut(A d,2 ) = U (A d,2 ) T d,2 .
Recall that Aut(A d,2 ) is the amalgamated product of GL 2 /µ d and J d /µ d along their intersction and GL 2 / where C is the intersection of A = SL 2 /(µ d ∩ SL 2 ) and B = J d /(µ d ∩ J d ). As both A and B are generated by unipotent subgroups it follows that A * C B ⊂ U (A d,2 ). The other inclusion follows from (2). The claim follows.
Remark 6. 19. Define the homomorphism of abstract groups by projection onto the first factor. Such a homomorphism is a morphism of indgroups which implies that U (A d,2 ) ⊂ Aut(A d,2 ) is a closed subgroup.
Remark 6.20. By Lemma 6.18, U (A d,2 ) is the amalgamated product of the groups SL 2 /(µ d ∩ SL 2 ) and J d /( J d ∩ µ d ) along their intersection. Note that if d is even, then SL 2 /(µ d ∩ SL 2 ) is isomorphic to PSL 2 . If d is odd, then SL 2 /(µ d ∩ SL 2 ) is isomorphic to SL 2 .
The following result was pointed out to me by Hanspeter Kraft. Proof. Let X be isomorphic either to SL 2 /T or to SL 2 /N . Then U (X) contains a copy of PSL 2 (see Lemma 6.6 and Lemma 6.10, respectively). Hence, by Lemma 6.1 Z is isomorphic either to SL 2 /T , to SL 2 /N , or to A d,2 for some d ∈ N. We claim that Z can be isomorphic to A d,2 only if d is even. Indeed, assume that Aut(A d,2 ) contains an algebraic subgroup S isomorphic to PSL 2 . Since S ⊂ U (A d,2 ) it follows from Lemma 6.18 that S is conjugate either to a subgroup of SL 2 /(µ d ∩ SL 2 ) or J d /(µ d ∩ J d ) (see [Sr80]). Moreover, since J d /(µ d ∩ J d ) does not contain a copy of PSL 2 , S should be conjugate to a subgroup of SL 2 /(µ d ∩ SL 2 ). Hence, by Remark 6.20 we conclude that d is even. By Corollary 6.12 we have U (SL 2 /T ) U (SL 2 /N ). Hence, to prove (a) we first need to show that an algebraic isomorphism φ : U (A d,2 ) ∼ − → U (SL 2 /T ) implies that d = 2. By Lemma 6.17, the set of weights of root subgroups of U (A d,2 ) with respect to T d,2 is {(kd + 2)/2 | k ∈ N ∪ {0}} up to an automorphism of T d,2 . Since T d,2 is a subgroup of some S ⊂ U (A d,2 ) isomorphic to PSL 2 we have by Proposition 6.8 that the set of weights of root subgroups of U (X SL 2 /T ) with respect to φ(T d,2 ) is {1, 2, 3, . . . } up to an automorphism of φ(T d,2 ). By Lemma 6.16, the set of weights of root subgroups of U (A d,2 ) with respect to T d,2 and of U (SL 2 /T ) with respect to φ(T d,2 ) are equal. Therefore, d indeed equals 2. To finish the proof of (a) we need to show that U (A 2,2 ) and U (X SL 2 /T ) are algebraically isomorphic. To do so, we first note that by Lemma 6.18 and Lemma 6.6 the first factors SL 2 /µ 2 and SO 3 from the amalgamated product structure of U (A 2,2 ) and of U (X) respectively are isomorphic to PSL 2 . Moreover, J 2 and J are algebraically isomorphic, as both J 2 and J are direct limits of isomorphic algebraic groups. Finally, the intersections SL 2 /µ 2 ∩ J 2 ⊂ Aut(A 2,2 ) and SO 3 ∩ J ⊂ Aut(X) are also isomorphic as algebraic groups as they are both isomorphic to a Borel subgroup of PSL 2 .
Define a homomorphism ϕ : U (A d,2 ) → U(X) that sends isomorphically the first factor SL 2 /µ 2 of the amalgamated product of U (A d,2 ) to the first factor SO 3 of the amalgamated product of U (X) in a way that ϕ(SL 2 /µ 2 ∩ J 2 ) = SO 3 ∩ J ⊂ Aut(X) and the second factor J 2 of the amalgamated product of U (A d,2 ) to the second factor J of the amalgamated product of U (X). Such a map is well defined and is an isomorphism as follows from the amalgamated product structure of U (A 2,2 ) and U (X SL 2 /T ). The proof of (a) follows.
To prove (b) we first need to show that an algebraic isomorphism φ : U (A d,2 ) ∼ − → U (SL 2 /N ) implies that d = 4. As we have already mentioned above in this proof, Therefore, S d,n is isomorphic to S l,n as an algebraic group. Hence, from Lemma 7.1 it follows that (d, n) = (l, n). The second part of the statement follows from the first one directly since U (A d,n ) contains a copy of S d,n . Proposition 7.3. Let X be A d,n , SL 2 /T or SL 2 /N and Y be an irreducible affine variety. Assume that there is an algebraic isomorphism U (X) ∼ − → U (Y ). Then dim Y ≤ dim X. Moreover, if additionally Y is normal, then (a) if X SL 2 /T , then Y A 2,2 or Y SL 2 /T , (b) if X A 2,2 , then Y A 2,2 or Y SL 2 /T , (c) if X SL 2 /N , then Y A 4,2 or Y SL 2 /N , (d) if X A 4,2 , then Y A 4,2 or Y SL 2 /N , (e) if X = A d,n , where (d, n) / ∈ {(2, 2), (2, 4)}, Y X.
Proof. Fix an algebraic isomorphism ψ : U (X) ∼ → U (Y ) and denote by T the image of T d,n if X = A d,2 or the image of a maximal subtorus T of U (X) if X = SL 2 /T or SL 2 /N . By Lemma 6.14, Proposition 6.8, and Corollary 6.12, all root subgroups U ⊂ U (Y ) with respect to T have different weights. In particular, the root subgroups O(Y ) U · U ⊂ U (Y ) have different weights, which implies that O(Y ) U is multiplicity-free because the map O(Y ) U → O(Y ) U · U is injective. Hence, by Lemma 4.2 we have that dim Y ≤ dim T + 1 = n, which proves the first part of the proposition. Now (a), (b), (c) and (d) follow from Proposition 6.21.
To prove (e), we note that U (A d,n ) contains a copy of SL n /µ (n,d) , which implies that SL n acts nontrivially on Y and thus by Proposition 5.4, Lemma 6.1 and Proposition 7.3(a)-(d), Y A l,n for some l ∈ N. Hence, ψ : U (A d,n ) ∼ → U(A l,n ). By Lemma 6.15, there exists an algebraic isomorphism F : U (A l,n ) ∼ → U(A l,n ) such that F (ψ(T d,n )) = T l,n . Therefore, we can assume that ψ(T d,n ) = T l,n .
It induces the C + -action U on A d,n which is normalized by T d,n . Hence, ψ(U ) ⊂ U (A l,n ) is a root subgroup with respect to ψ(T d,n ) = T l,n . By Proposition 3.3, ψ(U ) lifts to a C + -action on A n normalized by T n . Since (n, d) = (n, l) by Lemma 7.2 and because U and ψ(U ) have the same weight characters with respect to T d,n and T l,n respectively (see Lemma 6.16), Lemma 6.14 implies that ψ(U ) lifts to a root subgroup (x 1 + cx d+1 2 , x 2 , . . . , x n ) | c ∈ C of Aut(A n ) with respect to T l . Therefore, l ≤ d. Analogously, d ≤ l, i.e., d = l.
The proof follows.
Proof of Theorem 1.5. Let ψ : U (X) ∼ − → U (Y ) be an algebraic isomorphism. Proposition 7.3 implies that dim Y ≤ dim X. Since SL n acts regularly and nontrivially on X, SL n also acts nontrivially and regularly on Y .
where N ≥k = {m ∈ N | m ≥ k}. Now assume that Y is isomorphic to SL 2 /T or to SL 2 /N , then by Proposition 6.3, Y = Y . Then (e) follows from Proposition 6. 21.
Let now X SL 2 /T . Then by Lemma 6.1, Y can only be isomorphic to SL 2 /T , SL 2 /N or A 2,2 . By Proposition 6.21, Y is isomorphic to SL 2 /T or to A 2,2 . If Y SL 2 /T , from Proposition 6.3, it follows that Y = Y . Hence, (b) follows from the first part of the proof. Statement (d) follows analogously. Proof of Theorem 1.4. Let Z be isomorphic either to SL 2 /T or to SL 2 /N . Then an isomorphism Aut(X) ∼ − → Aut(Z) induces an algebraic isomorphism U (X) ∼ − → U (Z). By Theorem 1.5, X is isomorphic either to Z or to A s 2k,2 for some s ∈ N and k ∈ {1, 2}. To finish the proof we need to show that Aut(Z) can not be isomorphic to Aut(A s 2k,2 ). But this is clear as all A s 2k,2 admit an action of a two-dimensional torus and varieties SL 2 /T and SL 2 /N do not admit such an action.