aa r X i v : . [ m a t h . R A ] M a y ON THE STOCHASTIC LIE ALGEBRA
MANUEL GUERRA, ANDREY SARYCHEV
Abstract.
We study the structure of the Lie algebra s ( n, R ) corre-sponding to the so-called stochastic Lie group S ( n, R ). We obtain theLevi decomposition of the Lie algebra, classify Levi factor and clas-sify the representation of the factor in R n . We discuss isomorphism of S ( n, R ) with the group of invertible affine maps Aff ( n − , R ). We provethat s ( n, R ) is generated by two generic elements. Stochastic Lie group and stochastic Lie algebra
Let S +0 ( n, R ) denote the space of transition matrices of size n , i.e., thespace of real n × n matrices with all entries non-negative and row sums equalto 1.One important motivation for the study of such matrices is their relationto Markov processes: It is easy to see that for any Markov process X with n possible states, the family P ( s, t ) = [ p i,j ( s, t )] ≤ i,j ≤ n , ≤ s ≤ t < + ∞ , where p i,j ( s, t ) is the probability of X t = j , conditional on X s = i , is afamily of transition matrices such that(1.1) P s,t = P u,t P s,u , ∀ ≤ s ≤ u ≤ t < + ∞ . Conversely, the Kolmogorov extension theorem (see e.g. [2], TheoremIV.4.18), states that for every family (cid:8) P ( s, t ) ∈ S +0 ( n, R ) (cid:9) ≤ s ≤ t< + ∞ satisfy-ing (1.1), there exists a Markov process X such that p i,j ( s, t ) =Pr { X t = j | X s = i } for every i, j ≤ n and every 0 ≤ s ≤ t < + ∞ .Let S + ( n, R ) denote the space of nonsingular transition matrices. It isclear that S +0 ( n, R ) is a semigroup with respect to matrix multiplication,and S + ( n, R ) is a subsemigroup. However, S + ( n, R ) is not a group, sincethe inverse of a transition matrix is not, in general, a transition matrix.The smallest group containing S + ( n, R ) is denoted by S ( n, R ). Due tothe considerations above, this is called the stochastic group [5]. It can beshown that S ( n, R ) = (cid:8) P ∈ R n × n : Det( P ) = 0 , P = (cid:9) , Manuel Guerra was partly supported by FCT/MEC through the project CEMAPREUID/MULTI/00491/2013. where is the n -dimensional vector with all entries equal to 1. It followsthat S ( n, R ), provided with the topology inherited from the usual topologyof R n × n , is a n × ( n −
1) dimensional analytic Lie group.The Lie algebra of S ( n, R ) is called stochastic Lie algebra , and is denotedby s ( n, R ). Notice that s ( n, R ) is isomorphic to the tangent space of S ( n, R )at the identity s ( n, R ) ∼ T Id S ( n, R ) = (cid:8) A ∈ R n × n : A = 0 (cid:9) , s ( n, R ) is provided with the matrix commutator [ A, B ] = AB − BA .We introduce the subset s + ( n, R ) = { A ∈ s ( n, R ) : a i,j ≥ , ∀ i = j } . It is clear that s + ( n, R ) is not a subalgebra of s ( n, R ), but it is a convexcone with nonempty interior in s ( n, R ). Since S + ( n, R ) is invariant underthe flow by ODE’s of type ˙ P t = P t A, with A ∈ s + ( n, R ), it follows that S + ( n, R ) has nonempty interior in S ( n, R ).In [1], it is stated that the Levi decomposition(1.2) s ( n, R ) = l ⊕ r , has the following components:a) The radical r is the linear subspace generated by the matrices(1.3) ˆ R i = E i ( n ) − E n ( n ) , i = 1 , . . . , n − , ˆ Z = Id − n J n , where E i ( n ) are the matrices with the elements in the i -th columnequal to 1 and all other elements equal to zero, J n is the matrix withall elements equal to 1;b) The Levi subalgebra l is the linear subspace of real traceless matriceswith all row and column sums equal to zero.The result is correct but the respective proof of [1, Proposition 3.3] seemsto contain a logical gap in what regards the semisimplicity of l and themaximality of r .In what follows, we present an orthonormal basis for s ( n, R ) which hasinteresting properties with respect to the Lie algebraic structure of s ( n, R ).In particular, it allows for the explicit computation of the Killing form andtherefore we prove semisimplicity of l by application of Cartan criterion.We also obtain the Dynkin diagram of l , showing that it is isomorphic to sl ( n − , R ). 2. Basis for the Lie algebra s ( n, R )Choose an orthonormal basis v , . . . , v n − of the hyperplaneΠ n = { x ∈ R n : x + . . . + x n = 0 } , TOCHASTIC LIE ALGEBRA 3 and set v = √ n (1 , . . . , ∈ R n . Recall that for a, b ∈ R n , the dyadicproduct a ⊗ b is the matrix: a ... a n ⊗ (cid:0) b · · · b n (cid:1) = a b · · · a b n ... ... a n b · · · a n b n . The matrices Z = 1 √ n − I n − v ⊗ v ) , (2.1) R i = v ⊗ v i , i = 1 , . . . , n − n − n − A = span { A ij , i = 1 , . . . n − , j = 1 , . . . , n − , i = j } , spanned by the rank-1 matrices(2.3) A ij = v i ⊗ v j . Since v i ∈ Π, there holds v ∗ ( v i ⊗ v j ) = ( v · v i ) v ∗ j = 0. Similarly,( v i ⊗ v j ) v = 0. Hence the matrices A ij have zero row and column sums.Since Tr( v i ⊗ v j ) = v i · v j = 0, the matrices A ij are traceless.Now, consider the linear subspace(2.4) H = ( H = n − X ℓ =1 γ ℓ ( v ℓ ⊗ v ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X ℓ =1 γ ℓ = 0 ) . The row and column sums of each ( v ℓ ⊗ v ℓ ) are zero, and the trace of H ∈ H equals n − P ℓ =1 γ ℓ = 0.We set(2.5) l = A ⊕ H . We introduce a basis of H :(2.6) H k = n − X ℓ =1 γ kℓ ( v ℓ ⊗ v ℓ ) , k = 1 , . . . , ( n − , where γ k = ( γ k , . . . , γ kn − ) , k = 1 , . . . , ( n − n − = (cid:8) x ∈ R n − : x + . . . + x n − = 0 (cid:9) . Using the definition of dyadic product and elementary properties of thetrace, it is straightforward to check that the matrices
Z, R i ( i = 1 , . . . , n − ,A ij ( i, j = 1 , . . . , n − , i = j ) , H i ( i = 1 , . . . , n − MANUEL GUERRA, ANDREY SARYCHEV form an orthonormal system with respect to the matrix scalar product h A, B i = Tr( AB ∗ ).The following Lemma presents the multiplication table for our basis. Itsproof is accomplished by a direct computation. Lemma 2.1.
For meaningful values of the indexes i, j, k, ℓ there holds: [ Z, R i ] = − n − R i ; [ Z, A ij ] = 0 ; [ Z, H i ] = 0 ; [ R i , R j ] = 0 ; [ R i , A j,k ] = (cid:26) R k , if i = j, , if i = j ;[ R i , H j ] = γ ji R i ; [ A ij , A kℓ ] = ( v i ⊗ v i ) − ( v j ⊗ v j ) = n − P r =1 (cid:16) γ ri − γ rj (cid:17) H r , if i = ℓ, j = k,A iℓ , if i = ℓ, j = k, − A kj , if i = ℓ, j = k, , if i = ℓ, j = k ;[ A ij , H k ] = (cid:16) γ kj − γ ki (cid:17) A ij ; [ H i , H j ] = 0 . (cid:3) Remark . Lemma 2.1 shows that the orthogonal subspaces A , H possessremarkable properties:1. H is a Cartan subalgebra of l .2. [ H , A ] ⊂ A . The adjoint action of H on A is diagonal, for H ∈ H :ad HA ij = ( γ i − γ j ) A ij .
3. [ A ij , A ji ] = v i ⊗ v i − v j ⊗ v j = H ij ∈ H .4. For i = j , { A ij , A ji , [ A ij , A ji ] } spans a 3-dimensional Lie subalgebra:[ H ij , A ij ] = 2 A ij , [ H ij , A ji ] = − A ji .
5. For any ( ij ) , ( kℓ ) the commutator [ A ij , A kℓ ] = ad A ij A kℓ is orthogo-nal to A kℓ with respect to the matrix scalar product. (cid:3) Semisimplicity of l In this section, we prove semisimplicity of l by direct computation of theKilling form B . Proposition 3.1.
The Killing form B satisfies: i) B ( A , H ) = 0 , ii) B ( H i , H j ) = 2( n − h H i , H j i , for i, j = 1 , . . . , n − , iii) B ( A ij , A kℓ ) = (cid:26) , if ( i, j ) = ( ℓ, k ) , n − , if ( i, j ) = ( ℓ, k ) . (cid:3) According to Cartan criterion for semisimplicity, we get
TOCHASTIC LIE ALGEBRA 5
Corollary 3.2.
The Killing form B is non-degenerate and the algebra l issemisimple. (cid:3) Proof of Proposition 3.1. (i)
Take A ij , H k from the basis of A and H , re-spectively.Since H is Abelian, (ad A ij ad H k ) H = 0.Due to Lemma 2.1, for any A ℓm , ad A ij ad H k A ℓm = C ad A ij A ℓm . Byproperty 5 in Remark 2.2, the last matrix is orthogonal to A ℓm and thereforethe trace of the restriction (ad A i,j ad H k ) | A is null, and we can conclude that B ( A , H ) = 0. (ii) Choose H k , H ℓ ∈ H . As far as (ad H k ad H ℓ ) | H = 0, we only need tocompute the trace of (ad H k ad H ℓ ) | A .By Lemma 2.1, ad H k ad H ℓ A ij = ad H k ( γ ℓi − γ ℓj ) A ij = ( γ ℓi − γ ℓj )( γ ki − γ kj ) A ij .Hence, B ( H k , H ℓ ) = X i,j ( γ ℓi − γ ℓj )( γ ki − γ kj ) ==( n − X i γ ℓi γ ki − X i γ ℓi X j γ kj − X j γ ℓj X i γ ki + ( n − X j γ ℓj γ kj . Since P i γ ki = 0, it follows that B ( H k , H ℓ ) = 2( n − X i γ ℓi γ ki = 2( n − h H k , H ℓ i . (iii) Pick A ij , A kℓ . For every H m (3.1) ad A ij ad A kℓ H m = ad A ij ( γ mℓ − γ mk ) A kℓ , lies in A whenever ( k, ℓ ) = ( j, i ). This impliesTr (ad A ij ad A kℓ ) | H = 0 , for ( k, ℓ ) = ( j, i ) . To compute Tr (ad A ij ad A kℓ ) | A , notice that h A αβ , ad A ij ad A kℓ A αβ i = v ∗ α ( A ij ad A kℓ A αβ − (ad A kℓ A αβ ) A ij ) v β ==( v α · v i ) v ∗ j ( A kℓ A αβ − A αβ A ij ) v β − ( v β · v j ) v ∗ α ( A kℓ A αβ − A αβ A kℓ ) v i . Since i = j and k = ℓ , v ∗ j A αβ A ij v β = v ∗ α A kℓ A αβ v i = 0, and therefore h A αβ , ad A ij ad A kℓ A αβ i ==( v j · v k )( v i · v α )( v ℓ · v α ) + ( v i · v ℓ )( v j · v β )( v k · v β ) , (3.2)which is zero whenever ( k, ℓ ) = ( j, i ).For ( k, ℓ ) = ( j, i ), the equality (3.1) and Lemma 2.1 yield h H m , ad A ij ad A ji H m i =( γ mi − γ mj ) h H m , ad A ij A ji i ==( γ mi − γ mj ) h H m , v i ⊗ v i − v j ⊗ v j i = ( γ mi − γ mj ) , and Tr (ad A ij ad A ji ) | H = n − P m =1 ( γ mi − γ mj ) . MANUEL GUERRA, ANDREY SARYCHEV
To compute the last expression, let us form the matrix(3.3) Γ = γ · · · γ n − ... ... γ n − · · · γ n − n − . Then ΓΓ ∗ is the matrix of the orthogonal projection of R n − onto the sub-space Π n − . Take a standard basis e , . . . , e n − in R n − , and note that e i − e j ∈ Π n − . ThenTr (ad A ij ad A ji ) | H = n − X m =1 ( γ mi − γ mj ) == ( e i − e j ) ∗ ΓΓ ∗ ( e i − e j ) = ( e i − e j ) ∗ ( e i − e j ) = 2 . In what regards Tr (ad A ij ad A ji ) | A , then by (3.2): h A αβ , ad A ij ad A ji A αβ i = ( v i · v α ) + ( v j · v β ) . Hence,Tr (ad A ij ad A ji ) | A = X α, β ≤ n − α = β (( v i · v α ) + ( v j · v β )) = 2( n − , and therefore Tr (ad A ij ad A ji ) = 2( n − (cid:3) Classification of the Levi subalgebra l Now we wish to prove the following result concerning the type of thesemisimple subalgebra l . Theorem 4.1.
The Levi subalgebra l is isomorphic to the special linear Liealgebra sl ( n − , R ) . (cid:3) Proof.
As stated in Remark 2.2, H is a Cartan subalgebra of l . From Lemma2.1, we see that the nonzero characteristic functions of l with respect to H are the linear functionals α ij : H 7→ R such that α ij ( H k ) = γ ki − γ kj , for 1 ≤ k ≤ n − , ≤ i, j ≤ n − , i = j, and the corresponding characteristic spaces are A ij = { tA ij : t ∈ R } ≤ i, j ≤ n − , i = j. Thus, l is split as l = H ⊕ M i = j A ij . Hence the set, R = { α ij : 1 ≤ i ≤ n − , ≤ j ≤ n − , i = j } is a rootsystem of l . TOCHASTIC LIE ALGEBRA 7
Since the Killing form restricted to H is diagonal, the dual space H ∗ isprovided with the inner product uniquely defined by h α ij , α ℓ,m i = n − X k =1 (cid:16) γ ki − γ kj (cid:17) (cid:16) γ kℓ − γ km (cid:17) = ( e i − e j ) ∗ ( e ℓ − e m )for every α ij , α ℓm ∈ R . Thus, R is isomorphic to the root system E = { e i − e j : 1 ≤ i ≤ n − , ≤ j ≤ n − , i = j } on the hyperplane Π n − . Since e ℓ − e m = m − P i = ℓ ( e i − e i +1 ) , if ℓ < m, ℓ − P i = m − ( e i − e i +1 ) , if ℓ > m, it follows that the set ∆ = (cid:8) α , α , α , . . . , α ( n − n − (cid:9) is a system ofpositive simple roots. Further, (cid:10) α i ( i +1) , α i ( i +1) (cid:11) = 2 1 ≤ i ≤ n − , (cid:10) α i ( i +1) , α j ( j +1) (cid:11)(cid:10) α i ( i +1) , α i ( i +1) (cid:11) = (cid:26) − | i − j | = 1 , | i − j | > . Thus, the Dynkin diagram of l is of type A n − , and therefore, l is isomorphicto sl ( n − , R ) (see, e.g., [6, Chapter 14]). (cid:3) ❝ α ❝ α ❝ α ❝ α ( n − n − ❝ α ( n − n − Figure 1.
Dynkin diagram of l Representation of the Levi factor l in V = R n Considering l as a subalgebra of the stochastic (matrix) algebra s ( n, R )defines its representation φ : l gl ( n ) in V = R n . To characterize it, letus pick the basis v , v , . . . , v n − , introduced in Section 2, and consider thematrix M ∈ R n × n : M = (cid:0) v v · · · v n − (cid:1) .By construction, M is orthogonal and the mapping ∀ y ∈ l : y M ∗ φ ( y ) M, defines an isomorphic representation of l in V = R n .Note that the subspace V = span { v , . . . , v n − } is invariant under φ ( l )and therefore we get:(5.1) ∀ y ∈ l : M ∗ φ ( y ) M = (cid:18) M ∗ φ ( y ) M (cid:19) , where M = (cid:0) v v · · · v n − (cid:1) ∈ R n × ( n − . MANUEL GUERRA, ANDREY SARYCHEV
The mapping y φ ( y ) = M ∗ φ ( y ) M is a faithful representation of l in V = R n − .Formula (5.1) identifies the representation of the semisimple Levi factor l in R n by stochastic matrices with a direct sum of the faithful representation φ in R n − and the null 1-dimensional representation.Besides M ∗ A ij M = e i ⊗ e j for i, j ∈ { , , . . . , n − } , i = j,M ∗ H i M = diag( γ i ) for i = 1 , , . . . , n − . Therefore φ maps isomorphically the Cartan subalgebra H onto the spaceof traceless diagonal ( n − × ( n −
1) matrices, while φ ( A ) coincides withthe space of ( n − × ( n −
1) matrices with vanishing diagonal.6.
Affine group and affine Lie algebra
It is noticed in [5] that the group of S ( n, R ) is isomorphic to the group Aff ( n − , R ) of the affine maps S : x → Ax + B, x ∈ R n − . We wishto discuss this relation, in the light of the results obtained above. We alsodiscuss the relation between the elements of S ( n, R ) and finite state spaceMarkov processes outlined in Section 1.Let ( R n ) ∗ be the dual of R n . As usual, elements of R n are identifiedwith column vectors, and elements of ( R n ) ∗ are identified with row vectors.Further, we identify any vector x = ( x , x , . . . , x n ) ∈ R n with the function x : i x i , with the domain D n = { , , . . . , n } , and identify any dual vector p = ( p , p , . . . , p n ) ∈ ( R n ) ∗ with the (signed) measure p on the set D n suchthat p { i } = p i , for i = 1 , , . . . , n . Thus, the product px is identified withthe integral R D n x dp .Each S ∈ S ( n, R ) can be identified either with the linear endomorphismof R n , S : x Sx or with the linear endomorphism of ( R n ) ∗ , S : p pS .Let Y be a D n -valued Markov process and S ∈ S + ( n, R ) be defined by s ij = Pr { Y t = j | Y s = i } for every i, j ∈ D n (0 ≤ s ≤ t < + ∞ , fixed). Thenthe vector Sx is identified with the function i E [ x ( Y t ) | Y s = i ], whilethe covector pS is identified with the probability law of Y t assuming theprobability law of Y s is p .For every S ∈ S ( n, R ), the map p pS preserves each affine space ofthe form { p ∈ ( R n ) ∗ : p = C } ( C ∈ R , fixed), which is the space of signedmeasures on D n such that p ( D n ) = C . Note that, { t : t ∈ R } is the uniqueaffine (linear) proper subspace of R n which is preserved by all the maps x Sx with S ∈ S ( n, R ). TOCHASTIC LIE ALGEBRA 9
Now, consider the group of invertible affine maps S : q qA + B , q ∈ (cid:0) R n − (cid:1) ∗ . The group can be identified with the subgroup A (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) of GL (( R n ) ∗ ): A (cid:16)(cid:0) R n − (cid:1) ∗ (cid:17) = (cid:26) (cid:18) B A (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) A ∈ R ( n − × ( n − is nonsingular (cid:27) . The Lie algebra a (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) of A (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) consists of matrices (cid:18) B A (cid:19) . Now, fix S ∈ S ( n, R ). By the results of Section 2, S can be written as S = β Z + n − X i =1 β i R i + A, with β , β , . . . , β n − ∈ R , A ∈ l . Taking into account that Zv = 0 ,Zv i = 1 √ n − v i , R i v j = (cid:26) v , if j = i, , if j = i, for i = 1 , , . . . , n − , we get M ∗ SM = β ∗ M ∗ AM + β √ n − Id ! , where β ∗ = ( β , β , . . . , β n − ). Thus, the similarity S M ∗ SM is anisomorphism from S ( n, R ) into a (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) . In particular, the radical of a (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) is the linear space of matrices (cid:18) β ∗ β Id (cid:19) , β , β , . . . , β n − ∈ R , while the Levi subalgebra of a (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) consists of matrices (cid:18) A (cid:19) , A ∈ sl ( n − , R ) . Thus, the Levi splitting of a (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) corresponds to two connected Liesubgroups of A (cid:0)(cid:0) R n − (cid:1) ∗ (cid:1) : The subgroup generated by the translations andrescalings of (cid:0) R n − (cid:1) ∗ , and the subgroup of orientation and volume preservinglinear transformations in (cid:0) R n − (cid:1) ∗ . The mapping q = ( q , q , . . . , q n − ) C − n − X i =1 q i , q , q , . . . , q n − ! coordinatizes the affine subspace { p ∈ ( R n ) ∗ : p = C } . Minimal number of generators of s ( n, R )Finally we prove Theorem 7.1.
The Lie algebra s ( n, R ) is generated by two matrices. (cid:3) The argument in our proof is an adaptation of the argument used in [4] toprove that every semisimple Lie algebra is generated by two elements. Wewill use the following lemma:
Lemma 7.2.
For every integer n ≥ there is a vector γ ∈ R n such that a) n P i =1 γ i = 0 ; b) γ i = 0 , i = 1 , . . . , n ; c) γ i = γ j , ∀ i, j ∈ { , . . . , n } , i = j ; d) γ i − γ j = γ k − γ ℓ , ∀ i, j, k, ℓ ∈ { , . . . , n } , i = j, k = ℓ, ( i, j ) =( k, ℓ ) .For every γ satisfying (a)–(d) and every λ ∈ R \ { } , λγ satisfies (a)–(d). (cid:3) Proof.
For n = 2, the Lemma holds with γ = (1 , − n ≥
2, and fix γ ∈ R n satisfying (a)–(d) . Let ˜ γ = ( γ , . . . , γ n − , γ n − ε, ε ) . Since there are only finitely many values of ε such that ˜ γ fails at least onecondition (a)–(d) , we see that the Lemma holds for n + 1.The last statement in the Lemma is obvious, since the equations in con-ditions (a)–(d) are homogeneous. (cid:3) Proof of Theorem 7.1.
Pick a vector γ ∈ R n − satisfying conditions (a)–(d) of Lemma 7.2, let Γ be the matrix (3.3), and β = ( β , . . . , β n − ) = γ T Γ. Let
Z, R i , A ij , H i be elements of our basis of s ( n, R ), and consider the matrices X = Z + n − X k =1 β k H k , Y = R + X i = j A ij . Using the Lemma 2.1, we obtainad XY =[ Z, R ] + X i = j [ Z, A ij ] + n − X k =1 β k [ H k , R ] + n − X k =1 β k [ H k , A ij ] == − n − R + 0 − γ R + X i = j ( γ i − γ j ) A ij == − (cid:18) n − γ (cid:19) R + X i = j ( γ i − γ j ) A ij . TOCHASTIC LIE ALGEBRA 11
Multiplying γ by an appropriate non zero constant we can make γ = − n − ,and thus ad XY = X i = j ( γ i − γ j ) A ij . Iterating, we see thatad k XY = X i = j ( γ i − γ j ) k A ij ∀ k ∈ N . Let m = ( n − n − · · ·
00 1 1 · · ·
10 0 γ − γ · · · γ n − − γ n − γ − γ ) · · · ( γ n − − γ n − ) ... ... ... ...0 0 ( γ − γ ) m · · · ( γ n − − γ n − ) m = 0 , we see that the matrices X, Y, ad XY, . . . , ad m XY span the same subspaceas the matrices X, R , A ij , i, j ≤ n − , i = j , and this subspace lies in Lie { X, Y } , the Lie algebra generated by X, Y .By the Lemma 2.1, [ R , A i ] = R i , for i = 1 , , . . . , n −
1. Hence { R , . . . , R n − } ⊂ Lie { X, Y } . Finally, also by the Lemma 2.1, [ A ij , A ji ] = n − P r =1 ( γ ri − γ rj ) H r . This impliesthat [ A j , A j ] , j = 2 , . . . , n − n − H . Hence, H ⊂
Lie { X, Y } and Z ∈ Lie { X, Y } . (cid:3) Acknowledgement
The authors are grateful to A.A.Agrachev for stimulating remarks.
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ISEG and CEMAPRE, ULisboa, Rua do Quelhas 6, 1200-781 Lisboa, Por-tugal, University of Florence, DiMaI, v. delle Pandette 9, Firenze, 50127,Italy
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