Testing isomorphism of complex and real Lie algebras
TTESTING ISOMORPHISM OF COMPLEX AND REAL LIE ALGEBRAS
TUAN A. NGUYEN, VU A. LE, AND THIEU N. VO
Abstract.
In this paper, we give algorithms for determining the existence of isomorphismbetween two finite-dimensional Lie algebras and compute such an isomorphism in the affir-rmative case. We also provide algorithms for determining algebraic relations of parameters inorder to decide whether two parameterized Lie algebras are isomorphic. All of the consideredLie algebras are considred over a field F , where F = C or F = R . Several illustrative examplesare given to show the applicability and the effectiveness of the proposed algorithms. Introduction
In this paper, we will consider a computer-based approach for solving the following twoproblems:
Problem 1.
Given two F -Lie algebras L and L (cid:48) of same dimension. Deciding whether L andL0 are isomorphic or not, and determine an isomorphism in the affirmative case. Problem 2.
Given two F -Lie algebras L ( c ) and L (cid:48) ( d ) depending on r -tuple c = ( c , . . . , c r ) ∈ F r and s -tuple d = ( d , . . . , d s ) ∈ F s , respectively. Find conditions of parameters c and d such that L ( c ) and L (cid:48) ( d ) are isomorphic.By definition, an F -Lie algebra , say L , is an F -vector space endowed with a skew-symmetricbilinear map [ · , · ] : L × L → L which obeys the Jacobi identity:[[ X, Y ] , Z ] + [[ Y, Z ] , X ] + [[ Z, X ] , Y ] = 0 , for all X, Y, Z ∈ L. If L is n -dimensional with a basis { X , . . . , X n } then we have[ X i , X j ] = n (cid:88) k =1 a kij X k ; 1 ≤ i < j ≤ n. We call a kij ∈ F the structure constants of L .An F -linear isomorphism φ between two F -Lie algebras ( L, [ · , · ] L ) and ( L, [ · , · ] L (cid:48) ) is calledan isomorphism if it preserves Lie brackets, i.e.,(1.1) φ ([ X, Y ] L ) = [ φ ( X ) , φ ( Y )] L (cid:48) , for all X, Y ∈ L. The problem of classifying Lie algebras up to isomorphism is a fundamental problem of LieTheory. Invariants (such as ideals in characteristic series, the nilradical, the center) containspartial informations of a Lie algebra. Invariants of isomorphic Lie algebras are the same.However, it is still an open problem to determine a complete list of invariants such that theyare strong enough to characterize a Lie algebra. Therefore, it is impossible to decide theisomorphism between Lie algebras just by means of their invariants.
Mathematics Subject Classification.
Key words and phrases.
Lie algebras, testing isomorphism, triangular decomposition, algorithms. a r X i v : . [ m a t h . R A ] F e b TUAN A. NGUYEN, VU A. LE, AND THIEU N. VO
To the best of our knowledge, Gerdt and Lassner [8] were the first authors consideringProblem 1 under a view from computer algebra. In their algorithm, condition (1.1) is trans-formed into a system of polynomial equations, therefore, the problem of testing Lie algebraisomorphism is reduced to the problem of testing the existence of a solution of a polynomialsystem. Gr¨obner basis technique is then used to solve the latter problem. However, since thecomplexity of computing Gr¨obner bases is very costly, the algorithm is impractical when thedimension pass 6. Furthermore, it is not clear whether this algorithm is applicable for solvingProblem 2.We provide new algorithms for solving Problems 1 and 2 in cases F = C and R . Inheritedfrom the idea by Gerdt and Lassner [8], we also rewrite the considered problems in terms ofpolynomial equations. However, we will use the so-called triangular decomposition instead ofusing Gr¨obner bases to deal with the polynomial systems. There are two main advantages ofusing triangular decomposition. On the one hand, the algorithm for checking the existenceof a solution of a polynomial system by using triangular decomposition runs much fasterthan that using Gr¨obner bases. On the other hand, triangular decomposition can be usedto deal with polynomial systems with parameters and over the real fields. Details aboutthe construction of triangular decomposition for polynomial systems with implementation inMaple were presented in [1, 2, 3, 4, 5, 6].We recall necessary definitions and algorithms for triangular decomposition for polynomialsystems over fields of characteristics zero in Sections 2, and for semi-algebraic system over thereal field in Section 3. Algorithms for the projection operator are recalled in Section 4. InSection 5, we present algorithms for solving Problem 1 by using triangular decomposition. InSection 6, we construct algorithms for solving Problem 2 by using triangular decompositionand projection. Several illustrative examples are given in Section 7 to show the applicabilityand effectiveness of the proposed algorithms.2. Triangular decomposition of polynomial systems
Here, we recall the basic ideas of Triangular decomposition of polynomial systems. Moredetailes on the theory and applications of triangular decomposition appear in [1, 5].In this section, k is a field with algebraic closure K . The notation R := k [ x ] indicates thepolynomial ring k [ x , . . . , x n ] with ordered variables x = x < · · · < x n .Let p ∈ k [ x ] \ k . The greates variable of p is called the main variable of p and denotedby mvar( p ). If mvar( p ) = x i then we can consider p as a univariate polynomial by x i , i.e. p = k [ x , . . . , x i − ][ x i ], and the greatest coefficient of p is called the initial of p and denote itby init( p ).For F ⊂ k [ x ], we denote by (cid:104) F (cid:105) the ideal in k [ x ] spanned by F , and V ( F ) the zero set(solution set or algebraic variety) of F in K n .Let I ⊂ k [ x ] be an ideal. • A polynomial p ∈ k [ x ] is called a zerodivisor modulo I if there exists q ∈ k [ x ] suchthat pq ∈ I and p or q belongs to I . If p is neither 0 nor zerodivisor modulo I thenwe call p ∈ k [ x ] is regular modulo I • For h ∈ k [ x ], the saturated ideal of I with respect to (hereafter, w.r.t.) h is an idealin k [ x ] as follows: I : h ∞ := { q ∈ k [ x ] : ∃ m ∈ N such that h m q ∈ I } . ESTING ISOMORPHISM OF COMPLEX AND REAL LIE ALGEBRAS 3
A subset T ⊂ k [ x ] \ k consists of polynomials with pairwise distinct main variables is calleda triangular set . For triangular set T ⊂ k [ x ]: • We denote by sat( T ) ⊂ k [ x ] the saturated ideal of T defined as follows: if T = ∅ thensat( T ) is the trivial ideal { } , otherwise it is the ideal (cid:104) T (cid:105) : h ∞ T . • Let p, q ∈ k [ x ]. If either p or q is not constant and has main variable v , then wedefine res( p, q, v ) as the resultant of p and q w.r.t. v . We define res( p, T ) inductivelyas follows: if T = ∅ , then res( p, T ) = p ; otherwise let v be greatest variable appearingin T , then res( p, T ) = res (res( p, T v , v ) , T Triangularize ( F ) to compute a tri-angular decomposition of V ( F ). It has been implemented in MAPLE. Example 2.4. Consider the following system: x + y + z = 4 ,x + 2 y = 5 ,xz = 1 . Set F := (cid:8) x + y + z − , x + 2 y − , xz − (cid:9) ⊂ R := R [ x, y, z ]. Then, Triangularize ( F, R )returns three regular chains as follows: T = (cid:8) xz − , y − , x − (cid:9) ,T = (cid:8) z + 1 , y − , x + 1 (cid:9) ,T = (cid:8) z − , y − , x − (cid:9) . The first regular chain has h T = x , so W ( T ) = V ( T ) \ V ( x ). For two remaining ones, wehave W ( T ) = V ( T ) and W ( T ) = V ( T ). Since V ( F ) = (cid:83) i =1 W ( T i ) (note that V ( F ) ⊂ C ),we need to solve three systems as follows: xz − y − x − x (cid:54) = 0 , z = − y = 2 x = 1 , z = 1 y = 2 x = 1 . Example 2.5. Consider the following system: x + y + z = 1 ,x + y + z = 1 ,x + y + z = 1 . TUAN A. NGUYEN, VU A. LE, AND THIEU N. VO Due to Cox et al. [7, Chapter 3, § I = (cid:10) x + y + z − , x + y + z − , x + y + z − (cid:11) w.r.t. lex order reduces to solve the following system: x + y + z − y − y − z + z = 02 yz + z − z = 0 z − z + 4 z − z = 0 . Set F := (cid:8) x + y + z − , x + y + z − , x + y + z − (cid:9) ⊂ R := R [ x, y, z ]. Triangularize ( F, R )reduces to solve four systems as follows: z − x = 0 y − x = 0 x + 2 x − , z = 0 y = 0 x − , z = 0 y − x = 0 , z − y = 0 x = 0 . Remark . A disadvantage of Gr¨obner bases is that they do not necessarily have a triangularset shape. Consequently, solving Gr¨obner bases to construct isomorphims is much harder, andin general, it seems to be impossible.3. Triangular decomposition of semi-algebraic systems In this section, we recall a little bit about triangular decomposition of semi-algebraic sys-tems. For more details, we refer the readers to [1, 3, 6] and references therein.In this section, k is a field of characteristic 0 and K is its algebraic closure. For p ∈ k [ x ] \ k ,we denote by der( p ) the derivative of p w.r.t. mvar( p ).Let T ⊂ k [ x ] be a triangular set. Denote by mvar( T ) the set of main variables of thepolynomials in T . A variable v ∈ x is called algebraic w.r.t. T if v ∈ mvar( T ), otherwise it issaid free w.r.t. T . We shall denote by u = u , . . . , u d and y = y , . . . , y m respectively the freeand the main variables of T . We let d = 0 whenever T has no free variables.Let T ⊂ k [ x ] be a regular chain and H ⊂ k [ x ]. The pair [ T, H ] is a regular system if eachpolynomial in H is regular modulo sat( T ). If H = { h } then we write [ T, h ] for short. Regularchain T or regular system [ T, H ] is squarefree if der( t ) is regular w.r.t. sat( T ) for all t ∈ T .Let [ T, H ] be a squarefree regular system of k [ u , y ]. Let bp be the primitive and squarefree part of the product of all res(der( t ) , T ) and all res( h, T ) for h ∈ H and t ∈ T . We call bp the border polynomial of [ T, H ].Let us consider four finite subset of Q [ x , . . . , x n ] as follows: F = { f , . . . , f s } , N = { n , . . . , n t } , P = { p , . . . , p r } , H = { h , . . . , h l } . We denote by N ≥ and P > the sets of inequalities { n ≥ , . . . , n t ≥ } and { p > , . . . , p r > } , respectively; and H (cid:54) = the set of inequations { h (cid:54) = 0 , . . . , h l (cid:54) = 0 } . Definition 3.1 (Semi-algebraic systems) . We denote by S := [ F, N ≥ , P > , H (cid:54) = ] the semi-algebraic system (SAS), that is the conjunction of the following conditions: f = · · · = f s = 0, N ≥ , P > and H (cid:54) = . Definition 3.2 (Pre-regular semi-algebraic system) . Let [ T, P ] be a squarefree regular systemof Q [ u , y ] with border polynomial bp . Let B ⊂ Q [ u ] be a polynomial set such that bp divides ESTING ISOMORPHISM OF COMPLEX AND REAL LIE ALGEBRAS 5 the product of polynomials in B . We call the triple [ B (cid:54) = , T, P > ] a pre-regular semi-algebraicsystem (PRSAS) of Q [ x ]. Zero set of [ B (cid:54) = , T, P > ], denoted by Z R ( B (cid:54) = , T, P > ), is the set( u, y ) ∈ R n such that b ( u ) (cid:54) = 0 for all b ∈ B , t ( u, y ) = 0 for all t ∈ T , and p ( u, y ) > p ∈ P . Lemma 3.3 ([3, Lemma 1]) . Let S be a SAS of Q [ x ] . Then there exists finitely many PRSASs [ B i (cid:54) = , T i , P i> ] , i = 1 , . . . , e , such that Z R ( S ) = e (cid:83) i =1 Z R ( B i (cid:54) = , T i , P i> ) . Definition 3.4 (Regular semi-algebraic system) . Let T ⊂ Q [ x ] be a squarefree regular chain.Let P ⊂ Q [ x ] be finite and such that each polynomial in P is regular w.r.t. sat( T ), i.e. [ T, P ]is a regular system. Define P > := { p > | p ∈ P } . Let Q be a quantifier-free formula over Q [ x ] involving only the u variables. Let S = Z R ( Q ) ⊂ R d be the semi-algebraic subset of R d defined by Q . When d = 0, the 0-ary Cartesian product R d is treated as a singleton set. Wesay that R := [ Q , T, P > ] is a regular semi-algebraic system (RSAS) if:(1) S is a non-empty open subset in R d ,(2) The regular system [ T, P ] specializes well at every u ∈ S ,(3) At each u ∈ S , specialized system [ T ( u ) , P ( u ) > ] admits real solutions.The zero set of R , denoted by Z R ( R ), is the set of points ( u, y ) ∈ R d × R n − d such that Q ( u )holds, t ( u, y ) = 0 for all t ∈ T and p ( u, y ) > p ∈ P . Lemma 3.5 ([3, Lemma 3]) . Let [ B (cid:54) = , T, P > ] be a PRSAS of Q [ u , y ] . One can decide whetherits zero set is empty or not. If it is not empty, then one can compute a RSAS [ Q , T, P > ] whosezero set is the same as that of [ B (cid:54) = , T, P > ] . Proposition 3.6 ([3, Theorem 2]) . Let S be a SAS of Q [ x ] . Then one can compute a (full)triangular decomposition of S , that is finitely many RSASs such that the union of their zerosets is the zero set of S .Remark . Chen et al. [3, Section 7] presented an algorithm to compute a triangulardecomposition of a semi-algebraic system S = [ F, N ≥ , P > , H (cid:54) = ] which was denoted by Real-Triangularize ( S ). It has been implemented to MAPLE. Example 3.8. Consider the following system: (cid:40) x + y + z + 2 = 03 x + 4 y + 4 z + 5 = 0 . Put F := { x + y + z + 2 , x + 4 y + 4 z + 5 } , N ≥ = P > = H (cid:54) = := ∅ . Set S := [ F, ∅ , ∅ , ∅ ]be a SAS in R := R [ x, y, z ]. Since RealTriangularize ( S , R ) returns ∅ , the given system has noreal root. Note that Triangularize ( F, R ) returns one regular chain (cid:8) z + y − , x + 3 (cid:9) whichimplies that the given system has complex roots.4. Projections For many purposes, we need to find solutions of a polynomial system F as well as of aSAS S which consist of r parameters and n − r unknowns. In particular, we want to find outthe values of parameters in which the given system admits solutions since it concerns directlywith Problem 2. In this situation, the projections are very useful (see [2, 4]). TUAN A. NGUYEN, VU A. LE, AND THIEU N. VO Polynomial systems. Let k be a field of characteristic zero with algebraic closure K . Definition 4.1 (Constructible set) . Let F = { f , . . . , f s } and H = { h , . . . , h l } be two finiteset of k [ x ]. The conjunction of f = · · · = f s = 0 and h (cid:54) = 0 , . . . , h l (cid:54) = 0 is called a constructiblesystem of k [ x ], and denoted by [ F, H ]. Its zero set in K n , i.e., V ( F, H ) := V ( F ) \ V ( H ), iscalled a basic constructible set of k [ x ]. A constructible set of k [ x ] is a finite union of basicconstructible sets of k [ x ].Now, we set R := k [ x , u ] = k [ x > · · · > x n − r > u > · · · > u r ] be the polynomialring. For F ⊂ R , we consider x and u respectively as unknowns and parameters, that is,its ( x , . . . , x n − r )-solutions are multivariate functions of ( u , . . . , u r ). In other words, R := k [ u ][ x ] = k [ u > · · · > u r ][ x > · · · > x n − r ]. Denote by π r K : K n = K n − r × K r → K r theprojection which maps ( x , u ) in the entire space to u in the last r -dimensional parametersubspace. There are two situations as follows. • For F ⊂ R , we want to find the image π r K ( V ( F )) of its zero set V ( F ) ⊂ K n under π r K . • For F, H ⊂ R , we want to find the image π r K ( V ( F, H )) of the constructible set V ( F, H ) ⊂ K n under π r K .Note that these images are constructible sets of k [ u ]. In MAPLE, the commands to find π r K ( V ( F )) and π r K ( V ( F, H )) are Projection ( F, r, R ) and Projection ( F, H, r, R ), respectively. Example 4.2. Consider F := { x + y − } and H := { x + y − } in R := R [ y, x ] ≡ R [ x ][ y ],i.e., y and x are respectively the unknown and the parameter. We want to find the values of x such the system [ F, H ] admits complex roots. To this end, we need to find π C ( V ( F, H )). Projection ( F, H, , R ) returns a constructible set of R [ x ] as follows: (cid:40) x (cid:54) = 0 x − (cid:54) = 0 , x = 0 . Thus, the system [ F, H ] admits complex roots iff x (cid:54) = 1.4.2. Semi-algebraic systems. We set R := R [ x , u ] = R [ x > · · · > x n − r > u > · · · > u r ]be the polynomial ring, where x are unknowns and u are parameters. Denote by π r R : R n = R n − r × R r → R r the projection which maps ( x , u ) in the entire space to u in the last r -dimensional parameter subspace. For a SAS S = [ F, N, P, H ] of R , we want to find the image π r R ( Z R ( S )) of its zero set Z R ( S ) ⊂ R n under π r R .In this case, the image π r R ( Z R ( S )) is RSASs of R [ u ] which can be found by the command Projection ( F, N, P, H, r, R ) in MAPLE. Example 4.3. Consider the following quadratic equation: ax + bx + c = 0; a, b, c ∈ R , a (cid:54) = 0 . In this example, we reexamine the well-known result that this quadratic equation admits realroots iff its discriminant ∆ := b − ac is non-negative. First, we put F := (cid:8) ax + bx + c (cid:9) , N = P := ∅ and H := { a } . Afterwards, we set S := [ F, N, P, H ] be a RSAS in R := R [ a, b, c ][ x ]. Then, Projection ( F, N, P, H, , R ) returns three RSASs of R [ a, b, c ] as follows: b = 0 c = 0 a (cid:54) = 0 , (cid:40) ac − b = 0 b (cid:54) = 0 and c (cid:54) = 0 , b − ac > and a (cid:54) = 0 . It is obvious that these results can be sum up by ∆ ≥ ESTING ISOMORPHISM OF COMPLEX AND REAL LIE ALGEBRAS 7 Algorithms for Problem 1 Let L = span { X , . . . , X n } and L (cid:48) = span { Y , . . . , Y n } be two n -dimensional F -Lie algebraswith structure constants a kij and b kij , respectively. An isomorphism φ : L → L (cid:48) must satisfy: • φ ([ X i , X j ]) = [ φ ( X i ) , φ ( X j )] for 1 ≤ i < j ≤ n , • det [ φ ] (cid:54) = 0, where [ φ ] is the matrix of φ .Assume that the matrix [ φ ] of φ is as follows:[ φ ] = z · · · z n ... . . . ... z n · · · z nn . The condition φ ([ X i , X j ]) = [ φ ( X i ) , φ ( X j )] is equivalent to n (cid:80) k =1 z ks a kij − n (cid:80) k,l =1 z ik z jl b skl = 0 , ≤ i < j ≤ n, s = 1 , . . . , n. The condition det [ φ ] (cid:54) = 0 is equivalent to 1 − z det[ φ ] = 0, where z ∈ F is a new unknown.Therefore, the isomorphic conditions to the following system of equations(5.1) n (cid:80) k =1 z ks a kij − n (cid:80) k,l =1 z ik z jl b skl = 0 , ≤ i < j ≤ n, s = 1 , . . . , n, − z det[ φ ] = 0 , which consists of ≤ n (cid:0) n (cid:1) +1 polynomials in F [ z, z , . . . , z n , . . . , z n , . . . , z nn ] of degree ≤ n +1with n + 1 unknowns z, z ij ∈ F . Hence, we have Theorem 5.1. L and L (cid:48) are isomorphic iff the zero set of (5.1) is non-empty. Since (5.1) is a polynomial system, we can use triangular decompositions. To this end, wefirst set the polynomial ring R := F [ z, z , . . . , z n , . . . , z n , . . . , z nn ] and F be all of polynomialson the left-hand side of (5.1). Afterwards, determining whether the zero set of (5.1) is empty ornot is based on the triangular decomposition. If F = C then we find a triangular decompositionof V ( F ) ⊂ C n +1 . If F = R then we find a triangular decomposition of Z R ( F, ∅ , ∅ , ∅ ) ⊂ R n +1 .These procedures are given in Algorithms 1 and 2. Algorithm 1: Testing isomorphism of complex Lie algebras Input: Structure constants a kij ∈ C of L and b kij ∈ C of L (cid:48) Output: Yes ( L ∼ = L (cid:48) ) or No ( L (cid:29) L (cid:48) ) R := C [ z, z , . . . , z n , . . . , z n , . . . , z nn ]; F := { Polynomials determining equations of (5.1) } ; V := Triangular decomposition of V ( F ); / Triangularize ( F, R )/ if V = ∅ then ouput No else output Yes TUAN A. NGUYEN, VU A. LE, AND THIEU N. VO Algorithm 2: Testing isomorphism of real Lie algebras Input: Structure constants a kij ∈ R of L and b kij ∈ R of L (cid:48) Output: Yes ( L ∼ = L (cid:48) ) or No ( L (cid:29) L (cid:48) ) R := R [ z, z , . . . , z n , . . . , z n , . . . , z nn ]; F := { Polynomials determining equations of (5.1) } ; S := [ F, ∅ , ∅ , ∅ ]; Z := Triangular decomposition of Z R ( S ); / RealTriangularize ( S , R )/ if Z = ∅ then ouput No else output Yes 6. Algorithms for Problem 2 Given two n -dimensional parametric F -Lie algebras L ( c ) = span { X , . . . , X n } and L (cid:48) ( d ) =span { Y , . . . , Y n } whose structure constants are a kij ∈ F [ c ] and b kij ∈ F [ d ], respectively. Notethat two tuples of parameters c and d may satisfy some additional conditions. Our objectiveis to determine the values of c and d such that L ( c ) ∼ = L (cid:48) ( d ). We divide into two cases asfollows. A. F = C . Assume that two tuples c , d satisfy additionally c ∈ C = V ( F , H ) ⊂ C r ; F , H ∈ C [ c ] , d ∈ C = V ( F , H ) ⊂ C s ; F , H ∈ C [ d ] . Assume that φ : L → L (cid:48) is an isomorphism with matrix:[ φ ] = z · · · z n ... . . . ... z n · · · z nn . Since c ∈ C and d ∈ C , the isomorphic conditions is equivalent to(6.1) n (cid:80) k =1 z ks a kij − n (cid:80) k,l =1 z ik z jl b skl = 0 , ≤ i < j ≤ n, s = 1 , . . . , n, − z det[ φ ] = 0 ,f = 0 , for all f ∈ F ∪ F ,h (cid:54) = 0 , for all h ∈ H ∪ H , which consists of ≤ n (cid:0) n (cid:1) + 1 + | F | + | F | + | H | + | H | polynomials in C [ z, z , . . . , z n , . . . , z n , . . . , z nn , c , d ]of degree ≤ max (cid:26) n + 1 , max f ∈ F ∪ F deg f, max h ∈ H ∪ H deg h (cid:27) with n + 1 unknowns z, z ij and r + s parameters c , d . Set F := { Polynomials determining equations of (6.1) } and H := H ∪ H . Then, we have that: Theorem 6.1. L ( c ) and L (cid:48) ( d ) are isomorphic iff ( c , d ) ∈ π r + s C ( V ( F, H )) , where π r + s C isthe projection to the ( c , d ) -parameters space C r + s . ESTING ISOMORPHISM OF COMPLEX AND REAL LIE ALGEBRAS 9 Proof. As we have seen above L (cid:48) ( c ) ∼ = L (cid:48) ( d ) ⇔ (6.1) admits complex roots ( z, z ij , c , d ) ⇔ ( z, z ij , c , d ) ∈ V ( F, H ) ⇔ ( c , d ) ∈ π r + s C ( V ( F, H )) . (cid:3) We can sum up the procedure by Algorithm 3. Algorithm 3: Testing isomorphism of parametric complex Lie algebras Input: Structure constants a kij ∈ C [ c ] of L ( c ), b kij ∈ C [ d ] of L (cid:48) ( d ) where c ∈ C ⊂ C r , d ∈ C ⊂ C s and C i = V ( F i , H i ) Output: (cid:116) i A i ⊂ C := V ( F ∪ F , H ∪ H ) ⊂ C r + s s.t. L ( c ) ∼ = L ( d ) iff c , d ∈ A i forsome i R := C [ z, z , . . . , z n , . . . , z n , . . . , z nn , c , d ]; F := { Polynomials determining equations of (6.1) } ; H := H ∪ H ; π r + s C ( V ( F, H )); / Projection ( F, H, r + s, R )/ B. F = R . Assume that two tuples c , d satisfy additionally c ∈ C = Z R ( F , N , P , H ) ⊂ R r ; F , N , P , H ∈ R [ c ] , d ∈ C = Z R ( F , N , P , H ) ⊂ R s ; F , N , P , H ∈ R [ d ] . Similarly, an isomorphism φ : L → L (cid:48) with matrix[ φ ] = z · · · z n ... . . . ... z n · · · z nn . is equivalent to(6.2) n (cid:80) k =1 z ks a kij − n (cid:80) k,l =1 z ik z jl b skl = 0 , ≤ i < j ≤ n, s = 1 , . . . , n, − z det[ φ ] = 0 ,f = 0 , for all f ∈ F ∪ F ,n ≥ , for all n ∈ N ∪ N ,p > , for all p ∈ P ∪ P ,h (cid:54) = 0 , for all h ∈ H ∪ H , which consists of ≤ n (cid:0) n (cid:1) +1+ | F | + | F | + | N | + | N | + | P | + | P | + | H | + | H | polynomialsin R [ z, z , . . . , z n , . . . , z n , . . . , z nn , c , d ] of degree ≤ max (cid:26) n + 1 , max f ∈ F ∪ F deg f, max n ∈ N ∪ N deg n, max p ∈ P ∪ P deg p, max h ∈ H ∪ H deg h (cid:27) with n + 1 unknowns z, z ij and r + s parameters c , d . Put F := { Polynomials determining equations of (6.2) } N := N ∪ N ,P := P ∪ P ,H := H ∪ H , and set S := [ F, N, P, H ] be a SAS of R [ z, z , . . . , z n , . . . , z n , . . . , z nn , c , d ]. Then, wehave that: Theorem 6.2. L ( c ) and L (cid:48) ( d ) are isomorphic iff ( c , d ) ∈ π r + s R ( Z R ( S )) , where π r + s R isthe projection to the ( c , d ) -parameters space R r + s . We can sum up the procedure by Algorithm 4. Algorithm 4: Testing isomorphism of parametric real Lie algebras Input: Structure constants a kij ∈ R [ c ] of L ( c ), b kij ∈ R [ d ] of L (cid:48) ( d ) where c ∈ R ⊂ R r , d ∈ R ⊂ R s , R i = Z R ( F i , N i , P i , H i ) Output: (cid:116) i A i ⊂ R := Z R ( F ∪ F , N ∪ N , P ∪ P , H ∪ H ) ⊂ R r + s s.t. L ( c ) ∼ = L (cid:48) ( d ) iff c , d ∈ A i for some i R := R [ z, z , . . . , z n , . . . , z n , . . . , z nn , c , d ]; F := { Polynomials determining equations of (6.2) } ; N := N ∪ N ; P := P ∪ P ; H := H ∪ H ; S := [ F, N, P, H ]; π r + s R ( Z R ( S )); / Projection ( F, N, P, H, r + s, R )/7. Experimentations In this section, we present examples to demonstrate how Algorithms 1, 2, 3 and 4 can beapplied. Let us start with a simple case in dimension 3. Example 7.1. Consider L = span { X , X , X } and L (cid:48) = span { Y , Y , Y } with[ X , X ] = 2 X + X , [ X , X ] = − X + 2 X , [ Y , Y ] = 3 Y + 2 Y , [ Y , Y ] = − Y + Y . Assume that φ : L → L (cid:48) is an isomorphism with matrix[ φ ] = z z z z z z z z z . ESTING ISOMORPHISM OF COMPLEX AND REAL LIE ALGEBRAS 11 By simple computations, system (5.1) consists of 9 equations as follows: f := 1 − z det[ φ ] = 0 f := − z + 2 z = 0 f := 2 z + z = 0 f := − z z + z z + 3 z z − z z = 0 f := − z z − z z + 2 z z + z z = 0 f := − z z + z z + 3 z z − z z + 2 z + z = 0 f := − z z − z z + 2 z z + z z + 2 z + z = 0 f := − z z + z z + 3 z z − z z − z + 2 z = 0 f := − z z − z z + 2 z z + z z − z + 2 z = 0First of all, we put F := { f , f , . . . , f } . Afterwards, we set S := [ F, ∅ , ∅ , ∅ ] be a SAS in R [ z, z , z , z , z , z , z , z , z , z ]. Then, RealTriangularize ( S , R ) returns two RSASsas follows: (cid:0) z − z z + z (cid:1) z − z + z − z = 0 z + 2 z − z = 0 z = 0 z = 0 z − z (cid:54) = 0 , z z − z + z = 0 z + 2 z = 0 z = 0 z = 0 z = 0 z − z (cid:54) = 0Since the output is non-empty, L and L (cid:48) are isomorphic over R . So are they over C . Fromthese RSASs, an isomorphism φ : L → L (cid:48) can be easily constructed, namely, we have[ φ ] = − − . Example 7.2. Consider two 4-dimensional real Lie algebras given in [10]: g , : [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e ; g , : [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e . System (5.1) determining an isomorphism φ : g , → g , consists of 23 equations f = · · · = f = 0. We first put F := { f , . . . , f } and then set S := [ F, ∅ , ∅ , ∅ ] be a SAS in R := R [ z, z , . . . , z , . . . , z , . . . , z ]. Then, RealTriangularize ( S , R ) returns ∅ , i.e., Z R ( S ) = ∅ ,and thus g , (cid:29) g , over R . Note that if we consider g , and g , over C then the polynomialring is R := C [ z, z , . . . , z , . . . , z , . . . , z ]. In this case, Triangularize ( F, R ) returns one regular chain which reduces to the following system: z z z z + 1 = 0 z − z z z = 0 z , z , z = 0 z + ( z − z z ) z = 0 z − z z = 0 z = 0 z − ( z + z z ) z = 0 z + z z = 0 z = 0 z + 1 = 04 z z z (cid:54) = 0and thus g , ∼ = g , over C . Solving this system gives us an isomorphism as follows:[ φ ] = i i − i 00 1 1 00 0 0 i ; ( i is the imaginary unit) . Example 7.3. In this example, we will optimize parameters β and γ of the 5-dimensionalreal Lie algebras g βγ , given in [11]:[ e , e ] = e , [ e , e ] = e + e , [ e , e ] = βe , [ e , e ] = γe ( βγ (cid:54) = 0) . Our objective is equivalent to find out the conditions of two pairs of real numbers ( β, γ )and ( δ, σ ) for which g βγ , and g δσ , are isomorphic. Now, system (6.2) consists of 45 equations f = · · · = f = 0. First, we put F := { f , . . . , f } , N = P := ∅ and H := { β, γ, δ, σ } . Next,we set S := [ F, N, P, H ] be a SAS in R := R [ z, z , . . . , z n , . . . , z , . . . , z , β, γ, δ, σ ]. Then, Projection ( F, N, P, H, , R ) returns two RSASs as follows: β − δ = 0 γ − σ = 0 δ (cid:54) = 0 and σ (cid:54) = 0 , β − σ = 0 γ − δ = 0 δ (cid:54) = 0 and σ (cid:54) = 0 . The first RSAS is trivial since two Lie algebras coincide. The second one indicates that twopairs ( β, γ ) and ( γ, β ) are equivalent, i.e., g βγ , ∼ = g γβ , . Since ( β, γ ) and ( γ, β ) are symmetricover the line y = x in the punctured Oxy -plane (see Figure 1), we can choose the pair ( β, γ )below the line y = x including this line except the origin. To sum up, the desired optimalconditions for parameters of g βγ , is β ≥ γ and βγ (cid:54) = 0. Example 7.4. Consider two 6-dimensional Lie algebras with basis { e , . . . , e } as follows: L , : [ e , e ] = e + e , [ e , e ] = e , [ e , e ] = e ,L c , : [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = ce , where c is a non-zero parameter. These are two families of 6-dimensional nilpotent Lie algebrasover a field of characteristic zero which were classified by Morozov [9, Section 2]. In thisexample, we reexamine Morozov’s results, i.e., we want to check if L , and L c , belong to ESTING ISOMORPHISM OF COMPLEX AND REAL LIE ALGEBRAS 13 y x y = x x > yx < y ( γ , β )( β , γ )( β , γ ) ( γ , β ) Figure 1. Pairs of real numbers ( β, γ ) and ( γ, β ).two non-isomorphic classes. To this end, we will find out all values of parameter c for which L , ∼ = L c , .In this case, both two systems (6.1) and 6.2 consist of 55 equations f = · · · = f = 0. Set F := { f , . . . , f } , N = P := ∅ and H := { c } . • Over R , we set R := R [ z, z , . . . , z n , . . . , z , . . . , z , c ]. Projection ( F, N, P, H, , R )returns one RSAS which is c > 0. This means that all real Lie algebras L c , with c > L , . • Over C , we set R := C [ z, z , . . . , z n , . . . , z , . . . , z , c ]. Projection ( F, H, , R ) returnsone constructible set which is c (cid:54) = 0. This means that all complex Lie algebras L c , are isomorphic to L , .To sum up, Morozov’s classification of 6-dimensional Lie algebras over a field of characteristiczero is redundant, and we can refine his results appropriately. References [1] C. Cheng, Solving Polynomial Systems via Triangular Decomposition , PhD Thesis, University of WesternOntario, London, Ontario, Canada, 2011. 2, 4[2] C. Chen, J. D. Davenport, M. M. Maza, B. Xia, R. Xiao, Computing with semi-algebraic sets representedby triangular decomposition , Proceedings of 2011 International Symposium on Symbolic and AlgebraicComputation (ISSAC 2011), ACM Press, 2011, pp. 75–82. 2, 5[3] C. Cheng, J. H. Davenport, J. P. May, M. M. Maza, B. Xia, R. Xiao, Triangular decomposition of semi-algebraic systems , J. Symbolic Comput. (2013) 3–26. 2, 4, 5[4] C. Chen, O. Golubitsky, F. Lemaire, M. M. Maza, W. Pan, Comprehensive Triangular Decomposition ,Proc. CASC 2007, LNCS, Vol. 4770, Springer, 2007, pp. 73–101. 2, 5[5] C. Cheng, M. M. Maza, Algorithms for computing triangular decomposition of polynomial systems , J.Symbolic Comput. (2012) 610–642. 2, 3[6] C. Cheng, J. H. Davenport, M. M. Maza, B. Xia, R. Xiao, Computing with semi-algebraic sets: Relaxationtechniques and effective boundaries , J. Symbolic Comput. (2013) 72–96. 2, 4[7] D. A. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to ComputationalAlgebraic Geometry and Commutative Algebra , Springer Switzerland, 2015. 4[8] V. P. Gerdt, W. Lassner, Isomorphism verification for complex and real Lie algebras by Gr¨obner basistechnique , in: N. H. Ibragimov, M. Torrisi, A. Valenti (Eds.), Modern Group Analysis: Advanced Analyticaland Computational Methods in Mathematical Physics, Springer Netherlands, 1993, pp. 245–254. 2 [9] V. V. Morozov, Classification of nilpotent Lie algebras of dimension 6 , Izv. Vyssh. Uchebn. Zaved. Mat. (1958) 161–171. 12[10] G. M. Mubarakzyanov, On solvable Lie algebras , Izv. Vyssh. Uchebn. Zaved. Mat. (1963) 114–123. 11[11] G. M. Mubarakzyanov, Classification of real structures of Lie algebras of fifth order , Izv. Vyssh. Uchebn.Zaved. Mat. (1963) 99–106. 12 Tuan A. Nguyen, Faculty of Political Science and Pedagogy, University of Physical Educa-tion and Sports, Ho Chi Minh City, Vietnam. Email address : [email protected] Vu A. Le, Department of Economic Mathematics, University of Economics and Law, VietnamNational University - Ho Chi Minh City, Vietnam. Email address : [email protected] Thieu N. Vo, Fractional Calculus, Optimization and Algebra Research Group, Faculty ofMathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Email address ::