Weak polynomial identities for a vector space with a symmetric bilinear form
aa r X i v : . [ m a t h . R A ] M a y WEAK POLYNOMIAL IDENTITIES FOR A VECTOR SPACEWITH A SYMMETRIC BILINEAR FORM
VESSELIN S. DRENSKY, PLAMEN E. KOSHLUKOV
Abstract.
Let V k be a k -dimensional vector space with a non-degeneratesymmetric bilinear form and let C k be the Clifford algebra on V k . The weakpolynomial identities of the pair ( C k , V k ) are investigated. It is proved thatall they follow from [ x , x ] = 0 when k = ∞ and from [ x , x ] = 0 and S k +1 ( x , . . . , x k +1 ) = 0 when k < ∞ . The Specht property of the weakidentity [ x , x ] = 0 is established as well. Introduction
The weak polynomial identities have been introduced by Razmyslov [6] in hisstudying of the 2 × K of characteristic 0. Let R be an associative algebra and let G be a vector subspace of R generating R asan algebra. The polynomial f ( x , . . . , x n ) of the free associative algebra K h X i iscalled a weak identity for the pair ( R, G ) if f ( g , . . . , g n ) = 0 for all g , . . . , g n ∈ G .The weak identities of ( R, G ) form an ideal T ( R, G ) of K h X i .There are various possibilities for defining consequences of a weak identity de-pending on the properties of G . Generally let Ω ⊂ K h X i be a family of polynomialssuch that ω ( g , . . . , g n ) ∈ G for every ω ( x , . . . , x n ) ∈ Ω and for all g , . . . , g n ∈ G .The weak identity f ( x , . . . , x m ) is an Ω-consequence of f ( x , . . . , x n ) if f belongsto the ideal of K h X i generated by { f ( ω , . . . , w n ) | ω i ∈ Ω , i = 1 , . . . , n } . Examples:
1. When G = R and Ω = K h X i we obtain the ordinary polynomialidentities with the usual rules for finding consequences of an identity.2. Let [ G, G ] ⊆ G , i.e., let G ⊂ R be a subalgebra of the adjoint Lie algebra R ( − ) of R and let Ω = L ( X ) ⊂ K h X i be the free Lie algebra canonically embeddedinto K h X i . Then the weak Lie identities are obtained [6].3. Assume G is closed with respect to the operation g ◦ h = 12 ( gh + hg ), i.e., G is a subalgebra of the Jordan algebra R (+) . In this case we obtain the weak Jordanidentities.4. Let Ω = sp( X ) be the vector subspace of K h X i spanned by X . Theideal T ( R, G ) of the weak identities for every pair (
R, G ) is Ω-closed. Hence if f ( x , . . . , x n ) ∈ T ( R, G ) then f (cid:16)X α j x j , . . . , X α nj x j (cid:17) ∈ T ( R, G ) for every
Published as Vesselin S. Drensky, Plamen E. Koshlukov, Weak polynomial identities for a vec-tor space with a symmetric bilinear form (Bulgarian summary), Mathematics and MathematicalEducation, 1987. Proceedings of the Sixteenth Spring Conference of the Union of Bulgarian Math-ematicians, Sunny Beach, April 6-10, 1987, 213-219, Publishing House of the Bulgarian Academyof Sciences, Sofia, 1987. Zbl 0658.16012, MR949941 (89j:15043). α ij ∈ K . The free algebra K h X i is isomorphic to the tensor algebra of sp( X ) andthe general linear group GL = GL (sp( X )) acts canonically on K h X i . Obviously T ( R, G ) is a GL -invariant ideal. Assume H = { h i ( x , . . . , x n i ) | i ∈ I } is a subsetof K h X i . The ideal generated by { h i ( ω , . . . , ω n i ) | i ∈ I, ω j ∈ sp( X ) } is called a GL -ideal generated by H and its elements are GL -consequences of H .Let G = V k be a k -dimensional vector space with a non-degenerate symmetricbilinear form h , i , and let R = C k be the Clifford algebra of V k . Furthermore weset V = V ∞ and C = C ∞ .The investigation of the weak identities for the pair ( C k , V k ) has been initiatedby Il’tyakov [4]. Our main purpose is to find a basis for the identities of ( C, V ). Theorem 1.
The GL -ideal T ( C, V ) of all weak identities in the pair ( C, V ) isgenerated by [ x , x ] . The proof of this result uses the technique of the representations of the generallinear group and it is in the spirit of [2].
Corollary 1.
The weak identities for the pair ( C k , V k ) are GL -consequences of (1) [ x , x ] and of the standard polynomial S k +1 ( x , . . . , x k +1 ) = X ( − σ x σ (1) · · · x σ ( k +1) . Assume M is the 2 × sl is its Lie subalgebra of all tracelessmatrices. Razmyslov [6] found a basis for the weak Lie identities of ( M , sl ). As aconsequence of Theorem 1 we obtain Corollary 2. (i)
The GL -ideal T ( M , sl ) is generated by [ x , x ] and S ( x , x , x , x ) . (ii) [6] The weak Lie identities of ( M , sl ) follow from (1).Applying ideas from [5] we prove that the weak identity (1) satisfies the Spechtproperty. Theorem 2.
Every GL -ideal of K h X i containing the polynomial [ x , x ] isfinitely generated. The results of this note are very close related to the problems for finding thebases of the ordinary and the weak identities for the Jordan algebra of a symmetricbilinear form. We hope they represent a step to the solution of these importantproblems in the theory of Jordan algebras with polynomial identities.2.
Identities in the pair ( C, V )Let us denote by U the GL -ideal of K h X i generated by the polynomial [ x , x ], F = K h X i /U , and F m = K h x , . . . , x m i / ( K h x , . . . , x m i ∩ U ). In the sequel weshall work in the algebras F and F m . We shall also use other letters, e.g., y , y , . . . for the generators of F . Lemma 1.
For each n > an equality of the form (2) x y · · · y n x − x y · · · y n x = X i A n i [ x , x ] B n i holds for F . Here A n i and B n i are homogeneous polynomials in the variables y , . . . , y n and deg A n i > . In particular for n = 2(3) x y y x − x y y x = 12 { ( y y + y y )[ x , x ] + y [ x , x ] y − y [ x , x ] y } holds.Proof. The linearization of (1) is f ( x , x , y ) = [ x ◦ x , y ] = 0 . Hence we obtain0 = f ( x , x , y ) + 2 f ( x , y, x ) = x yx − x yx + [ x , x ] ◦ y, i.e.,(4) x yx − x yx = − [ x , x ] ◦ y. Bearing in mind that f ( x , x , y , . . . , y n ) = [ x ◦ y , y ] y · · · y n x = 0we deduce2 { f ( x , x , y , . . . , y n ) − f ( x , x , y , . . . , y n ) } = x y · · · y n x − x y · · · y n x + y ( x y · · · y n x − x y · · · y n x ) − y ( x y y · · · y n x − x y y · · · y n x ) − y y ( x y · · · y n x − x y · · · y n x ) = 0 . Hence we can express x y · · · y n x − x y · · · y n x as a linear combination of poly-nomials beginning with y or y and skew-symmetric in x and x . Using (4) weestablish (3). The equality (2) follows by obvious induction on n . (cid:3) Lemma 2.
Weak identities of the following form are GL -consequences of (1): X ( − σ x σ (1) · · · x σ ( k ) yx σ ( k +1) · · · x σ ( n ) = α nk yS n ( x , . . . , x n ) + β nk S n ( x , . . . , x n ) y, (5) where α nk , β nk ∈ K , α nk = β n,n − k , k = 1 , . . . , n − , furthermore α n = − n − n , β n = ( − n − n ;(6) x i S n ( x , . . . , x n ) = ( − n − S n ( x , . . . , x n ) x i for i = 1 , . . . , n. Proof. (i) An involution ∗ is defined in the algebra K h X i by (cid:16)X α i x i · · · x i n (cid:17) ∗ = X α i x i n · · · x i . Since [ x , x ] ∗ = − [ x , x ], the action of ∗ is carried over to F . Thus if we apply ∗ to (5) we obtain X ( − σ x σ ( n ) · · · x σ ( k +1) yx σ ( k ) · · · x σ (1) = X ( − σ { α nk x σ ( n ) · · · x σ (1) y + β nk yx σ ( n ) · · · x σ (1) } . Replacing the positions of σ ( n ) and σ (1), σ ( n −
1) and σ (2), etc. we changesimultaneously the sign of all summands. Hence we obtain that α nk = β n,n − k holds in (5). First we shall prove (5) by induction on n for k = 1 (and therefore for k = n − k . For n = 2 and k = 1 theequality (4) gives X ( − σ x σ (1) yx σ (2) = − yS ( x , x ) − S ( x , x ) y. For n = 3, k = 1 we obtain from (3) X ( − σ x σ (1) ( yx σ (2) ) x σ (3) = 12 X ( − σ { ( yx σ (2) + x σ (2) y ) x σ (1) x σ (3) + yx σ (1) x σ (2) x σ (3) − x σ (1) x σ (2) x σ (3) y } = − X ( − σ x σ (1) yx σ (2) x σ (3) − yS ( x , x , x ) + 12 S ( x , x , x ) y. Therefore it holds X ( − σ x σ (1) yx σ (2) x σ (3) = − yS ( x , x , x ) + 13 S ( x , x , x ) y. Let n > k = 1. Then X ( − σ ( x σ (1) yx σ (2) · · · x σ ( n − ) x σ ( n ) = X ( − σ (cid:26) − n − n − yx σ (1) · · · x σ ( n ) + ( − n − n − x σ (1) ( x σ (2) · · · x σ ( n − yx σ ( n ) ) (cid:27) = − n − n − yS n ( x , . . . , x n )+ ( − n − n − X ( − σ (cid:26) ( − n − n − x σ (1) yx σ (2) · · · x σ ( n ) − n − n − x σ (1) · · · x σ ( n ) y (cid:27) and (cid:18) − n − (cid:19) X ( − σ x σ (1) yx σ (2) · · · x σ ( n ) = − n − n − yS n ( x , . . . , x n ) − ( − n − ( n − n − S n ( x , . . . , x n ) y, X ( − σ x σ (1) yx σ (2) · · · x σ ( n ) = − n − n yS n ( x , . . . , x n ) + ( − n − n S n ( x , . . . , x n ) y. Now we assume 1 < k < n . We shall suppose that (5) holds for all smaller valuesof n . Then we have X ( − σ x σ (1) ( x σ (2) · · · x σ ( k ) yx σ ( k +1) · · · x σ ( n ) )= X ( − σ α n − ,k − x σ (1) yx σ (2) · · · x σ ( n ) + β n − ,k − S n ( x , . . . , x n ) y = ( α n − ,k − α n yS n ( x , . . . , x n ) + ( α n − ,k − β n + β n − ,k − ) S n ( x , . . . , x n ) y. (ii) It suffices to prove the equality (6) for i = 1 only. Obviously x S n ( x , . . . , x n ) = x X σ (1)=1 ( − σ x σ (2) · · · x σ ( n ) + x X σ (1) =1 ( − σ x σ (1) · · · x σ ( n ) . Similarly, S n ( x , . . . , x n ) x = X τ ( n )=1 ( − τ x τ (1) · · · x τ ( n − x + X τ ( n ) =1 ( − τ x τ (1) · · · x τ ( n ) x . By (1) x lies in the center of F , hence( − σ x x σ (2) · · · x σ ( n ) = ( − σ x σ (2) · · · x σ ( n ) x = ( − n − ( − τ x τ (1) · · · x τ ( n − x , where σ (1) = τ ( n ) = 1, τ (1) = σ (2) , . . . , τ ( n −
1) = σ ( n ). Now it follows from (5): x X ( − σ x σ (1) · · · x σ ( k − x x σ ( k +1) · · · x σ ( n ) = ( − k x { α n − ,k x S n − ( x , . . . , x n ) + β n − ,k S n − ( x , . . . , x n ) x } , X ( − τ x τ (1) · · · x τ ( n − k − x x τ ( n − k +1) · · · x τ ( n ) x = ( − n − k − { α n − ,n − − k x S n − ( x , . . . , x n ) x + β n − ,n − − k S n − ( x , . . . , x n ) x } and we establish (6) from the conditions α n − ,n − − k = β n − ,k and β n − ,n − − k = α n − ,k . (cid:3) Lemma 3. (i)
Let Y , . . . , Y n − be monomials in y , y , . . . . Then there existmonomials D n i , E n i in y , y , . . . such that it holds in F (7) X ( − σ x σ (1) Y x σ (2) Y · · · Y n − x σ ( n ) = X i D n i S n ( x , . . . , x n ) E n i . (ii) Let Y , . . . , Y n − be monomials in x , . . . , x n . Then there exists a polynomial D ( x , . . . , x n ) such that in F (8) X ( − σ x σ (1) Y x σ (2) Y · · · Y n − x σ ( n ) = S n ( x , . . . , x n ) · D ( x , . . . , x n ) . Proof. (i) We shall carry out an induction on n and on d = deg Y + · · · + deg Y n − .The basis for the induction n = 2 follows from (2). For each n and for d = 1 (7)is a consequence of (5). We assume that (7) holds for n − d .When d > X ( − σ x σ (1) Y ( x σ (2) Y · · · Y n − x σ ( n ) )= X i,σ ( − σ x σ (1) Y D n − ,i x σ (2) · · · x σ ( n ) E n − ,i . We apply the induction to the summands with deg E n − ,i >
0. When deg E n − ,i =0 we use (2) for x σ (1) Y D n − ,i x σ (2) − x σ (2) Y D n − ,i x σ (1) . Then some variable in Y D n − ,i appears on the left-hand side of the sum and we can apply again theinduction on d .(ii) The equality (8) is a direct consequence of (6) and (7). (cid:3) Proof of Theorem 1.
First we note that the pair (
C, V ) satisfies the identity (1)since for each vector v ∈ V , v = h v, v i ∈ K and v lies in the center of C . Nowlet λ = ( λ , . . . , λ r ), λ ≥ · · · ≥ λ r >
0, be a partition and let the columns ofthe diagram [ λ ] have lengths r = r , r , . . . , r p , respectively. We choose an integer m ≥ r . Since the ideal U of K h X i is GL -invariant, the algebra F m is a GL m -module. Assume N m ( λ ) is an irreducible GL m -submodule of F m correspondingto the partition λ . It is known that N m ( λ ) is generated by a multihomogeneouspolynomial f λ ( x , . . . , x r ) of degree λ i in x i . Furthermore f λ ( x , . . . , x r ) = X Y X σ ( − σ Y x σ (1) Y x σ (2) · · · Y r − x σ ( r ) Y r , where Y , Y , . . . , Y r are monomials in x , . . . , x r . By (8) f λ ( x , . . . , x r ) = S r ( x , . . . , x r ) · D ( x , . . . , x r )holds in F m and the polynomial D ( x , . . . , x r ) generates the irreducible GL m -module N m ( λ − , λ − , . . . , λ r − Applying several times the equality (8) we obtain that in F m f λ ( x , . . . , x r ) = α Y S r i ( x , . . . , x r i ) , α ∈ K. Therefore all isomorphic irreducible submodules of F m are glued together. Hence F m is a submodule of P N m ( µ ) where the summation is over all partitions µ =( µ , . . . , µ r ), µ ≥ · · · ≥ µ r ≥
0. Assume e , e , . . . is a basis of V such that h e i , e i i 6 = 0 , h e i , e j i = 0 , i = j. Then S n ( e , . . . , e n ) = n ! e · · · e n and Q S r i ( e , . . . , e r i ) = 0 in C . The polynomial Q S r i ( x , . . . , x r i ) generates a submodule N m ( λ ) of F m and it does not vanish on( C, V ). Hence all submodules N m ( λ ) do participate in F m and U coincides with the GL -ideal of the weak identities for ( C, V ). The proof of the theorem is completed. (cid:3)
Proof of Corollary 1.
It follows from the proof of Theorem 1 that F m = P N m ( λ , . . . , λ m ). Each of the modules N m ( λ , . . . , λ m ) is generated by a poly-nomial f λ ( x , . . . , x r ) = Y S r i ( x , . . . , x r i ) , where r = r , r , . . . , r p are the lengths of the columns of [ λ ]. Since dim V k = k , for r > k we have that S r ( x , . . . , x r ) = 0 is a weak identity for the pair ( C k , V k ). Onthe other hand S r ( e , . . . , e r ) = 0 in C r for r ≤ k and for every basis e , . . . , e k of V k with h e i , e i i 6 = 0, h e i , e j i = 0, i = j . Therefore K h x , . . . , x m i / ( K h x , . . . , x m i ∩ T ( C k , V k )) = X N m ( λ , . . . , λ k )and the GL -ideal T ( C k , V k ) is generated by [ x , x ] and S k +1 ( x , . . . , x k +1 ). (cid:3) Proof of Corollary 2.
It is known [6] that the pair ( M , sl ) satisfies the weakidentity (1). Besides (see [1]) K h x , . . . , x m i / ( K h x , . . . , x m i ∩ T ( M , sl ) = X N m ( λ , λ , λ ) . The elements of the submodule N m ( λ , . . . , λ m ) of F m follow from S ( x , x , x , x ) =0 when λ = 0. This gives the assertion (i). The assertion (ii) follows immediatelyfrom (i) since one can easily obtain that the Lie weak identity S ( x , x , x , x ) = 0is a consequence of (1). (cid:3) Proof of Theorem 2.
It follows easily from (3) that the Young diagrams forma partially well-ordered set with respect to the inclusion. Therefore each set P ofdiagrams has a finite subset P = { [ λ (1) ] , . . . , [ λ ( s ) ] } such that for every [ λ ] ∈ P thereexists a [ λ ( i ) ] ∈ P with [ λ ( i ) ] ⊆ [ λ ]. Assume I is a GL -ideal of K h X i containing[ x , x ] and I is the homomorphic image of I under the canonical homomorphism K h X i → F = K h X i /U . Let N ( λ ) be the irreducible GL -module related to λ . Itfollows from the decomposition F = P N ( λ ) that I = P N ( λ ), where [ λ ] rangesover some set P .Denote by P the finite subset of the elements of P minimal with respect to theinclusion. The theorem will be proved if we establish that all elements of I areconsequences of f λ ∈ N ( λ ) where [ λ ] ∈ P .As in [5] it suffices to show that f µ follows from f λ in F when [ µ ] is obtained from[ λ ] by adding a box. Let λ = ( λ , . . . , λ r ) and m > r . We define a homomorphismof GL -modules ϕ : N m ( λ ) ⊗ N m (1) → F in the following way: ϕ (cid:16)X f i ⊗ x i (cid:17) = X f i x i , f i ∈ N m ( λ ) . By the Branching Theorem N m ( λ ) ⊗ N m (1) = P N m ( µ ), where [ µ ] is obtainedfrom [ λ ] by adding a box. The explicit form of the generators f µ ( x , . . . , x m ) of N m ( µ ) is found in [5]. As in [5] we can verify that f µ ( e , . . . , e m ) = 0 for everybasis e , e , . . . of V , h e i , e i i 6 = 0, h e i , e j i = 0, i = j . Thus we obtain that f µ is aconsequence of f λ in F . (cid:3) References [1]
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E-mail address : [email protected] Plamen Koshlukov: Department of Mathematics, IMECC, UNICAMP, S´ergio Buarquede Holanda 651, Campinas, SP 13083-859, Brazil
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