aa r X i v : . [ m a t h . R T ] N ov Abstract
We develop the general Theory of Cayley Hamilton algebras usingnorms and compare with the approach, valid only in characteristic 0,using traces and presented in a previous paper [20]. orms and Cayley Hamilton algebras Claudio procesiNovember 11, 2020
To the memory of Edoardo Vesentini
Contents n –dimensional representations . . . . . . . . . . . . . . . . . 41.4 Generic matrices and invariants . . . . . . . . . . . . . . . . 51.7 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 n –Cayley–Hamilton algebras 9 n –Cayley–Hamilton algebras . . . . . . . . . . . . . . . . . 112.8 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 122.12 Symbolic approach . . . . . . . . . . . . . . . . . . . . . . . 142.21 The free n –Cayley–Hamilton algebra . . . . . . . . . . . . . 182.26 Azumaya algebras . . . . . . . . . . . . . . . . . . . . . . . 202.34 Σ–algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.42.1 The first and second fundamental Theorem for ma-trix invariants revisited . . . . . . . . . . . . . . . . . 272.47.1 The main Theorem of Cayley–Hamilton algebras . . 29 T –ideals and relatively free algebras . . . . . . . . . . . . . 44 Foreword
A basic fact for an n × n matrix a (with entries in a commutative ring)is the construction of its characteristic polynomial χ a ( t ) := det( t − a ), t avariable, and the Cayley Hamilton theorem χ a ( a ) = 0.The notion of Cayley Hamilton algebra (CH algebras for short), seeDefinition 2.45, was introduced in 1987 by Procesi [16] as an axiomatictreatment of the Cayley Hamilton theorem. This was done in order toclarify the Theory of n –dimensional representations, cf. Definition 1.2, ofan associative and in general noncommutative algebra R . With 1, unlessotherwise specified, (from now on just called algebra ).The theory was developed only in characteristic 0, for two reasons; thefirst being that at that time it was not clear to the author if the charac-teristic free results of Donkin [9] and Zubkov [33] were sufficient to foundthe theory in general. The second reason was mostly because it looked notlikely that the main theorem Definition 0.1.
Given an algebra R over a commutative ring A and n ∈ N ,an n –norm is a multiplicative polynomial law, as A –modules, N : R → A homogeneous of degree n (see Definition 2.2).An algebra R over a commutative ring A with an n –norm N : R → A is a Cayley Hamilton algebra if, for every commutative A algebra B , each a ∈ B ⊗ A R satisfies its characteristic polynomial χ a ( t ) := N ( t − a ), that is χ a ( a ) = 0 . The original definition over Q is through the axiomatization of a trace ,and closer to the Theory of pseudocharacters see [20]. The definition via3race is also closer to the languange un Universal algebra, while the oneusing norms is more categorical in nature.This paper is a continuation of [20] where we have developed the the-ory in characteristic 0 using the notion of trace algebra. Here instead theapproach is characteristic free and through the axiomatization of Norms.The first section of this paper forms an exposition of known results withone or two new facts or proofs. We suggest the reader to start at § n –Cayley–Hamilton algebras. In § T –ideals. n –dimensional representations Let us recall some basic facts which are treated in detail in the forthcomingbook with Aljadeff, Giambruno and Regev [1].For a given n ∈ N and a ring A by M n ( A ) we denote the ring of n × n matrices with coefficients in A , by a symbol ( a i,j ) we denote a matrix withentries a i,j ∈ A, i, j = 1 , . . . , n .In particular we will usually assume A commutative so that the con-struction A M n ( A ) is a functor from the category C of commutative ringsto that R of associative rings. To a map f : A → B is associated a map M n ( f ) : M n ( A ) → M n ( B ) in the obvious way M n ( f )(( a i,j )) := ( f ( a i,j )). Definition 1.2.
By an n –dimensional representation of a ring R we meana homomorphism f : R → M n ( A ) with A commutative.The set valued functor A hom R ( R, M n ( A )) is representable. That is,there is a commutative ring T n ( R ) and a natural isomorphism j A hom R ( R, M n ( A )) j A ≃ hom C ( T n ( R ) , A ) , j A : f ¯ f given by the universal map j R : R → M n ( T n ( R )) and a commutativediagram f = M n ( ¯ f ) ◦ j R : R j R / / f $ $ ❏❏❏❏❏❏❏❏❏❏ M n ( T n ( R )) M n ( ¯ f ) (cid:15) (cid:15) M n ( A ) . (1)4he map j R : R → M n ( T n ( R )) is called the universal n –dimensional rep-resentation of R or the universal map into n × n matrices .Of course it is possible that R has no n –dimensional representations, inwhich case T n ( R ) = { } .Of course the same discussion can be performed when R is in the cat-egory R F of algebras over a commutative ring F. In this case the functor A hom R F ( R, M n ( A )) is on commutative F algebras and T n ( R ) is an F algebra.The construction of j R is in two steps. First one easily sees that when R = F h x i i i ∈ I is a free algebra then: Proposition 1.3. T n ( R ) = F [ ξ ( i ) h,k ] is the polynomial algebra over F in thevariables ξ ( i ) h,k , i ∈ I, h, k = 1 , . . . , n and j R ( x i ) = ξ i := ( ξ ( i ) h,k ) the genericmatrix with entries ξ ( i ) h,k . For a general algebra R one may present it as a quotient R = F h x i i /I of a free algebra. Then j F h x i i ( I ) generates in M n ( F [ ξ ( i ) h,k ]) an ideal whichis, as any ideal in a matrix algebra, of the form M n ( J ) , with J an ideal of F [ ξ ( i ) h,k ]. Then the universal map for R is given by F h x i i −−−−→ M n ( F [ ξ ( i ) h,k ]) y y R j R −−−−→ M n ( F [ ξ ( i ) h,k ] /J ) . By the universal property this is independent of the presentation of R . Definition 1.5.
The subalgebra F h ξ i | i ∈ I i of M n ( F [ ξ ( i ) h,k ]) , i ∈ I, h, k =1 , . . . , n generated by the matrices ξ i is called the algebra of generic matri-ces. A classical Theorem of Amitsur states that, if F is a domain then F h ξ i | i ∈ I i is a domain. If I has ℓ elements we also denote F h ξ i | i ∈ I i = F n ( ℓ ).If ℓ ≥ F n ( ℓ ) has a division ring of quotients D n ( ℓ ) which is of dimen-sion n over its center Z n ( ℓ ). These algebras have been extensively studied.One defines first the commutative subalgebra T n ( ℓ ) ⊂ Z n ( ℓ ) generated by5he coefficients σ i ( a ) of the characteristic polynomial det( t − a ) = t n + P ni =1 ( − i σ i ( a ) t n − i , ∀ a ∈ F n ( ℓ ). Next define S n ( ℓ ) = F n ( ℓ ) T n ( ℓ ) ⊂ D n ( ℓ ) , one can understand S n ( ℓ ) and T n ( ℓ ) by invariant theory, see Remark 2.23.The invariant theory involved is presented in [14] when F is a field ofcharacteristic 0, and may be considered as the first and second fundamentaltheorem of matrix invariants. For a characteristic free treatment see thebook [8]. In general, for simplicity of exposition assume that F is an infinitefield: Theorem 1.6.
The algebra T n ( ℓ ) is the algebra of polynomial invariantsunder the simultaneous action of GL ( n, F ) by conjugation on the space M n ( F ) ℓ of ℓ –tuples of n × n matrices.The algebra S n ( ℓ ) is the algebra of GL ( n, F ) –equivariant polynomialmaps from the space M n ( F ) ℓ of ℓ –tuples of n × n matrices to M n F ) . As usual together with a first fundamental theorem one may ask fora second fundamental theorem which was proved independently by Procesi[14] and Razmyslov [21] when F has characteristic 0 and by Zubkov [33]in general. In this paper it will appear as characterization of free Cayley–Hamilton algebras, Theorem 2.22.For a general algebra, quotient of the free algebra again one may addto R the algebra T n ( R ) generated by the coefficients of the characteristicpolynomial σ i ( a ) , ∀ a ∈ j R ( R ). The functor hom R ( R, M n ( A )) has a group of symmetries: the projectivelinear group P GL ( n ).It is best to define this as a representable group valued functor on thecategory C of commutative rings. The functor associates to a commutativering A the group G n ( A ) := Aut A ( M n ( A )) of A –linear automorphisms ofthe matrix algebra M n ( A ). One has a natural homomorphism of the generallinear group GL ( n, A ) to G n ( A ) which associates to an invertible matrix X the inner automorphism a XaX − .The functor general linear group GL ( n, A ) is represented by the Hopfalgebra Z [ x i,j ][ d − ] , i, j = 1 , . . . , n with d = det( X ) , X := ( x i,j ) withthe usual structure given compactly by comultiplication δ , antipode S andcounit ǫ : δ ( X ) = X ⊗ X, S ( X ) := X − , ǫ : X → n . G n ( A ) is represented by the sub Hopf algebra, of GL ( n, A ), P n ⊂ Z [ x i,j ][ d − ] formed of elements homogeneous of degree 0. It has abasis, over Z , of elements ad − h where a is a doubly standard tableaux withno rows of length n and of degree h · n . For a proof see [1] Theorem 3.4.21. Remark . Of course if we work in the category of commutative F algebraswe replace P n with P n ⊗ Z F ⊂ F [ x i,j ][ d − ].The identity map 1 P n : P n → P n induces a generic automorphism of M n ( P n ) which can be given by conjugation by the generic matrix X : J : M n ( P n ) → M n ( P n ) , J : a XaX − ∈ M n ( P n ) . (2)Observe that X is not a matrix in P n only the entries of XaX − are in P n i.e. are homogeneous of degree 0 in the entries of X . If we denote by y i,j the i, j entry of X − (given by the usual formula as the cofactor dividedby the determinant), we have Xe i,j X − = ( z ( i,j ) h,k ) , z ( i,j ) h,k = x h,i y j,k . In fact the entries z ( i,j ) h,k = x h,i y j,k of the matrices Xe i,j X − generate P n asalgebra and the ideal of relations is generated by the relations expressing thefact that the map e i,j ( z ( i,j ) h,k ) is a homomorphism (and then automaticallyan isomorphism).Then given an automorphism g ∈ Aut A ( M n ( A )) its associated classify-ing map ¯ g : P n → A fits in the commutative diagram: M n ( P n ) J / / M n (¯ g ) (cid:15) (cid:15) M n ( P n ) M n (¯ g ) (cid:15) (cid:15) M n ( A ) g / / M n ( A ) (3)Finally we have an action of G n ( A ) on hom R ( R, M n ( A )) by composing amap f with an automorphism g . One has a commutative diagram R j R / / f (cid:15) (cid:15) M n ( T n ( R )) M n ( g ◦ f ) (cid:15) (cid:15) M n ( A ) g / / M n ( A ) . (4)Assume now that R is an F algebra so also T n ( R ) is an F algebra and M n ( T n ( R )) = T n ( R ) ⊗ F M n ( F ). An automorphism g of M n ( F ) induces7n automorphism 1 ⊗ g of M n ( T n ( R )). Take A = T n ( R ) and f = j R inFormula (4) and set ˆ g := 1 ⊗ g ◦ j R , an automorphism of T n ( R ), so that1 ⊗ g ◦ j R = M n (ˆ g ) ◦ j R . If g , g are two automorphisms of M n ( F ) we have: M n ( \ g ◦ g ) ◦ j R = 1 ⊗ ( g ◦ g ) ◦ j R = 1 ⊗ g ◦ M n (ˆ g ) ◦ j R = M n (ˆ g ) ◦ ⊗ g ◦ j R = M n (ˆ g ) ◦ M n (ˆ g ) ◦ j R implies \ g ◦ g = ˆ g ◦ ˆ g . Which implies that the map g ˆ g is an antiho-momorphism from Aut F ( M n ( F )) to Aut ( T n ( R )) . Finally
Proposition 1.9.
The map g g ⊗ ˆ g − is a homomorphism from thegroup Aut F M n ( F ) to the group of all automorphisms of M n ( T n ( R )) . Theimage of R under j R is formed of invariant elements. This is particularly simple when F is an infinite field. In this case thegroup Aut F M n ( F ) = P GL ( n, F ) is Zariski dense in P GL ( n, ¯ F ) with ¯ F an algebraic closure of F . Otherwise this can be set in the language ofpolynomial laws. Proposition 1.10.
For every commutative F algebra B , an automorphism g ∈ Aut B M n ( B ) induces an automorphism of the B algebra T n ( R ) ⊗ F B .The map g g ⊗ ˆ g − is a homomorphism from the group Aut B M n ( B ) to thegroup of all automorphisms of M n ( B ) ⊗ B ( T n ( R ) ⊗ F B ) = M n ( T n ( R ) ⊗ F B ) .The image of B R under j R ⊗ B is formed of invariant elements. The functor G n ( A ) × hom R ( R, M n ( A ) is represented by P n ⊗ T n ( R ).Given f ∈ hom R ( R, M n ( A )) associated to ¯ f : T n ( R ) → A and an auto-morphism g of M n ( A ) associated to ¯ g : P n → A the pair is associatedto P n ⊗ T n ( R ) ¯ g ⊗ ¯ f −→ A ⊗ A m −→ A, m ( a ⊗ b ) := ab. The natural transformation G n ( A ) × hom R ( R, M n ( A )) → hom R ( R, M n ( A ))induces the coaction on the classifying rings η : T n ( R ) → P n ⊗ T n ( R ).Thus the composition g ◦ f ∈ hom R ( R, M n ( A )) is associated to g ◦ f : T n ( R ) η −→ P n ⊗ T n ( R ) ¯ g ⊗ ¯ f −→ A ⊗ A m −→ A, m ( a ⊗ b ) := ab. In particular for R = Z h x i i i ∈ I we have T n ( Z h x i i i ∈ I ) is the polynomialring in the entries ξ ( k ) i,j of the generic matrices ξ k and the action is via theformula ξ k Xξ k X − , ξ ( k ) i,j X a,b x i,a ξ ( k ) a,b y b,j , X − = ( y i.j ) .
8e have x i,a y b,j ∈ P n . The induced map T n ( Z h x i i i ∈ I ) → P n ⊗ T n ( F h x i i i ∈ I )can be identified to the coaction. η : T n ( R ) → P n ⊗ T n ( R ) , η ( ξ ( k ) i,j ) = n X a,b =1 x i,a ξ ( k ) a,b y b,j . (5) Remark . In general the invariants are the elements invariant underthe generic automorphism . In our case: T n ( R ) G n := { a ∈ T n ( R ) | η ( a ) = 1 ⊗ a } . (6) n –Cayley–Hamilton algebras Given a commutative ring F , an F module M , and a commutative F algebra B one has the base change functor from F –modules to B –modules: B M := B ⊗ F M. (7)Recall that, in [25] and [26], Roby defines: Definition 2.2. A polynomial law between two F modules M, N is a nat-ural transformation of the two set valued functors on the category C F ofcommutative F algebras: f B : B M → B N, B ∈ C F . (8)Such a law is homogeneous of degree n if: f B ( ba ) = b n f B ( a ) , ∀ b ∈ B, ∀ a ∈ B M, ∀ B ∈ C F . Given M let us denote, for N any module, by P n ( M, N ) the polynomiallaws homogeneous of degree n from M to N . This, by Roby’s Theory, is aset valued functor on modules N and it is representable.This is done by constructing the divided powers Γ n ( M ) = Γ n,F ( M )(over the base F ) together with the map i M : m m [ n ] of M to Γ n ( M ).The divided powers Γ n ( M ) are constructed by generators and relations.The construction is compatible with base change, that is given a commu-tative F algebra B we have:Γ n,B ( B M ) = B Γ n,F ( M ) .
9n fact in most applications there is a more concrete description ofΓ n ( M ). For instance if M is a free (or just projective) F module one de-scribes the divided power as symmetric tensors:Γ n ( M ) ≃ ( M ⊗ n ) S n , M ⊗ n = M ⊗ F M ⊗ F M ⊗ · · · ⊗ F M. (9)From now on we assume to be in this case. Notice that by change of coeffi-cient rings if F → A is a map of commutative rings we have A M := A ⊗ F M and A M ⊗ n = A ⊗ F M ⊗ n , A Γ n ( M ) = Γ n ( A M ) . Remark . In general a polynomial law is not determined by the map f : M → N . Under further hypothesis this is the case, the simplest beingthat the base commutative ring F contains an infinite field. If we have two F –algebras R, S we have the notion of multiplicative poly-nomial law d : R → S that is d ( ab ) = d ( a ) d ( b ) , ∀ a, b ∈ B R, ∀ B. One proves that, if R is an algebra, then Γ n ( R ) is also an algebra, whichwe call the n th –Schur algebra of R , see [8], and i R is a universal multiplica-tive polynomial map, homogeneous of degree n . That is any multiplicativepolynomial law, homogeneous of degree n from R to an algebra S factorsthrough i R and a homomorphism of Γ n ( R ) to S .Denoting by M n ( R, S ) the set of multiplicative polynomial laws homo-geneous of degree n from R to S we have the isomorphism: ϕ : d d ◦ i R , hom R (Γ n ( R ) , S ) ϕ ≃ M n ( R, S ) , S ∈ R . (10)When we have Γ n ( R ) = ( R ⊗ n ) S n , Formula (9), the algebra structure onΓ n ( R ) is induced by the tensor product of algebras.If S is any algebra we denote by S ab its abelianization , that is S modulothe ideal C S generated by all commutators [ x, y ] = x · y − y · x .When we restrict, in Formula (10), to S commutative, this functor, oncommutative algebras, is represented by π : Γ n ( R ) → Γ n ( R ) ab that is theabelianization of Γ n ( R ). M n ( R, A ) ≃ hom C (Γ n ( R ) ab , A ) , A ∈ C . (11)This is an object studied by Roby in [26], and by Ziplies in [32] and discussedin some detail in my book with De Concini [8].10 .4 n –Cayley–Hamilton algebras Consider an algebra R , over a commutative ring Z . Definition 2.5.
A multiplicative polynomial law, of Z algebras, homoge-neous of degree n , N : R → Z wlll be called a norm .One may apply Roby’s theory. First N ( r ) = ¯ N ( r ⊗ n ) then, by Formula(11), the map ¯ N factors through the abelianization Γ n ( R ) ab of Γ n ( R ) (westill denote it by ¯ N ). R r r ⊗ n / / N " " ❊❊❊❊❊❊❊❊❊ Γ n ( R ) ¯ N (cid:15) (cid:15) π / / Γ n ( R ) ab ¯ N (cid:15) (cid:15) Z / / Z Γ n ( R ) the n th divided power . (12)For a ∈ B ⊗ Z R and a commutative variable t we have t − a ∈ B [ t ] ⊗ Z R and can define the characteristic polynomial of a as χ a ( t ) := N ( t − a ) ∈ B [ t ]. Definition 2.6.
We say that R is an n –Cayley–Hamilton algebra if one hasthe analogue of the Cayley–Hamilton Theorem χ a ( a ) = 0 , ∀ a ∈ B R, ∀ B .Sometimes we will use the short notation CH–algebra for Cayley–Hamil-ton algebra. Remark . In [19] we treated the Theory in characteristic 0. In this caseit is better to use instead of the norm the trace. This at the same timesimplifies the treatment but also yields stronger results due to the linearreductivity of the linear group.We start with an important example.If F is an infinite field and V a vector space of some finite dimension m over F one has for R = End F ( V ) that( End F ( V ) ⊗ n ) S n = End F ( V ⊗ n ) S n = End F [ S n ] ( V ⊗ n ) . The algebra
End F [ S n ] ( V ⊗ n ) is spanned by the elements g ⊗ n where g ∈ GL ( V ).If the characteristic of F is 0 one has the decomposition V ⊗ n = ⊕ λ ⊢ n | ht ( λ ) ≤ m M λ ⊗ S λ ( V )11here M λ is the irreducible representation of S n corresponding to λ and S λ ( V ), a Schur functor, the corresponding irreducible representation of GL ( V ). One proves that End F ( V ⊗ n ) S n = ⊕ λ ⊢ n | ht ( λ ) ≤ m End ( S λ ( V )) . (13)Each summand of this decomposition is a simple algebra and abelian onlywhen S λ ( V ) is 1–dimensional. This happens only if n = im and S λ ( V ) = V m ( V ) ⊗ i . In this case( End F ( V ) ⊗ n ) S n ab = End ( m ^ ( V ) ⊗ i ) , N ( a ) = det( a ) i , a ∈ End F ( V ) . (14)By the remarkable theory of good filtrations or combinatorially of standarddouble tableaux, in positive characteristic or over the integers the previousformula can be replaced by a canonical filtration of which Formula (13) isthe associated graded representation. From this one could deduce a generalform of Formula (14). Given an algebra R the composition of a homomorphism j : R → M n ( A )with the determinant det ◦ j : R j −→ M n ( A ) det −→ A is a multiplicative poly-nomial law, homogeneous of degree n . One then obtains a natural transfor-mation of functors: hom R F ( R, M n ( A )) → M n ( R, A ) , (15)and a commutative diagram. R j R / / f $ $ ❏❏❏❏❏❏❏❏❏❏ M n ( T n ( R )) M n ( ¯ f ) (cid:15) (cid:15) det / / T n ( R ) ¯ f (cid:15) (cid:15) M n ( A ) det / / A. (16)Since T n ( R ) is commutative, det ◦ j R factors as R i R −→ Γ n ( R ) π −→ Γ n ( R ) ab D −→ T n ( R ) , D ◦ π ◦ i R = det ◦ j R . The homomorphism D : Γ n ( R ) ab −→ T n ( R ) is clearly invariant under thegroup of automorphisms G n so it factors through T n ( R ) G n .12 emark . [Problem] One of the main problems of the theory is to un-derstand when is that D : Γ n ( R ) ab → T n ( R ) G n is an isomorphism.This happens in several cases. In particular we have a fundamentalresult, Theorem 20.24 of [8]. Theorem 2.10.
Consider the free algebra A h X i in some set of variables X = { x i } i ∈ I with A a field or the integers.Then D : Γ n ( A h X i ) ab → T n ( A h X i ) G n is an isomorphism. Notice that T n ( A h X i ) G n is the ring of invariants of X -tuples of n × n matrices .Theorem 2.10 is based on a Theorem of Procesi [14] and Razmyslov [23]characterizing trace identities of matrices (cf. Theorem 2.43), and provedby Zieplies [31] and Vaccarino [28], when A = Q .The general case is fully treated in [8]. It is based on the characteristicfree results of Donkin [9] and Zubkov [33] on the invariants of matrices anda careful combinatorial study of Γ n ( A h X i ) inspired by the work of Zieplies.In fact since A h X i is a free A module, its divided power is more conve-niently described as the symmetric tensors:Γ n ( A h X i ) ≃ ( A h X i ⊗ n ) S n . Since, using the basis of monomials, the space A h X i ⊗ n is a permutationrepresentation of S n , one has a combinatorial description of ( A h X i ⊗ n ) S n . The abelian quotient, isomorphic to the ring of invariants of matrices, doesnot have a combinatorial description and it is a rather hard object to study.
Example 2.11. If X = { x } is a single variable we have that A h X i = A [ x ]is the commutative ring of polynomials, A [ x ] ⊗ n = A [ x , . . . , x n ] so Γ n ( A [ x ])is the algebra of symmetric polynomials in n –variables, commutative.Consider the elementary symmetric function e i defined by:( t − x ) ⊗ n = n Y i =1 ( t − x i ) = t n − e t n − + e t n − − . . . + ( − n e n . (17)The map D maps ( t − x ) ⊗ n to det( t − ξ ) = t n + P ni =1 ( − i σ i ( ξ ) t n − i , thecharacteristic polynomial of a generic matrix ξ = ( ξ i,j ).So the elementary symmetric function e i maps to σ i ( ξ ), the generatorsof invariants of a single matrix.This is a very special case of Theorem 2.10.13lthough usually one deduces the functions σ i ( x ) for an n × n matrix x from the determinant of t − x one sees immediately that embedding the n × n matrices into n + 1 × n + 1 matrices as upper left corner i n ( x ) := (cid:12)(cid:12)(cid:12)(cid:12) x
00 0 (cid:12)(cid:12)(cid:12)(cid:12) , det( t − i n ( x )) = det (cid:12)(cid:12)(cid:12)(cid:12) t − x t (cid:12)(cid:12)(cid:12)(cid:12) = det( t − x ) t implies σ i ( x ) = σ i ( i n ( x )) so the functions σ i ( x ) are in fact also defined forinfinite matrices with only finitely many non 0 entries S n M n ( F ). It is thusimportant to understand the symbolic calculus on these functions and thisis the theme of the next section. The construction R Γ n ( R ) is functorial in R so given r ∈ R the map A [ x ] → R, x → r induces a map Γ n ( A [ x ]) → Γ n ( R ) under which e i → τ i ( r ) , (1 + r ) ⊗ n = 1 + τ ( r ) + τ ( r ) + . . . + τ n ( r ) . Proposition 2.13.
It is proved, in [8] Lemma 20.12, that, given a basis a i of R , the elements τ i ( a j ) generate ( R ⊗ n ) S n . Once we pass to the abelian-ization and set σ i ( a ) to be the class of τ i ( a ) we have σ i ( ab ) = σ i ( ba ) , ∀ a, b ,see [8] Proposition 20.20. We apply this construction to the free algebra. The grading of the al-gebra A h X i induces a grading of Γ n ( A h X i ). Recall that a monomial M ofpositive length, is called primitive if it is not a power N k , k > τ i ( M ) , i = 1 , . . . , n as M runs over the primitive monomials generateΓ n ( A h X i ). The elements τ i ( M ) satisfy complicated relations which are notfully understood. Definition 2.14.
Denote by S n,A the abelian quotient of Γ n ( A h X i ) andby σ i ( M ) the class of τ i ( M ) in S n,A .One can prove, Proposition 20.20 of [8], that if M = AB one has σ i ( AB ) = σ i ( BA ), one says that AB and BA are cyclically equivalent. A Lyndon word , is a primitive monomial minimal, in the lexicographicorder, in its class of cyclic equivalence.As for the theory of symmetric functions one can pass to the limit, as n → ∞ , of the algebras Γ n ( A h X i ) and their abelian quotients.14f ǫ : A h X i → A is the evaluation of X in 0, we have the map π n : A h X i ⊗ n +1 → A h X i ⊗ n , π n ( a ⊗ . . . ⊗ a n ⊗ a n +1 ) = a ⊗ . . . ⊗ a n ⊗ ǫ ( a n +1 ) . This induces a map, still called π n : Γ n +1 ( A h X i ) → Γ n ( A h X i ). We have π n ( τ i ( M )) = ( τ i ( M ) if i ≤ n i = n + 1 . Definition 2.15.
One can then define a limit algebra Γ ∞ ( A h X i ) generatedby the elements τ i ( M ) , i = 1 , . . . , ∞ as M runs over the primitive mono-mials and its abelian quotient S A h X i denoted often just S A , generated bythe classes σ i ( M ) of τ i ( M ).The maps π n give rise to limit maps: τ n : Γ ∞ ( A h X i ) → Γ n ( A h X i ) , τ n : S A h X i → S n,A h X i . (18)In [8] we have proved: Corollary 20.15
The algebra Γ ∞ ( Q h X i ) is the universal envelopingalgebra of Q + h X i considered as a Lie algebra.As for the structure of S n,A , Theorem 20.22 of [8] (due to Zieplies)states that, in the commutative algebra S A one has σ i ( AB ) = σ i ( BA ) forall monomials A, B and finally that S A = A [ σ i ( M )] , M a Lyndon word,is the free polynomial ring in the variables σ i ( M ) as M varies among theLyndon words.Let T A ( X ) denote the monoid of endomorphisms of A h X i given bymapping each variable x i ∈ X to some element f i ∈ A h X i + , the idealkernel of ǫ of elements with no constant term, this condition means thatthese endomorphisms commute with the map ǫ . Each such endomorphisminduces an endomorphism of each Γ n ( A h X i ) compatible with the maps π n and hence an endomorphism on S n,A := Γ n ( A h X i ) ab and on Γ ∞ ( A h X i ) and S A . Definition 2.16. A T –ideal of A h X i or of Γ n ( A h X i ) , or S n,A , n = 1 , . . . , ∞ is a multigraded ideal I closed under all endomorphisms induced by T A ( X ). Remark . The condition of I to be multigraded can be replaced by thecondition that, for every commutative A algebra B , the ideal B I := B ⊗ A I is closed under all endomorphisms induced by T B ( X ). This is in the spiritof polynomial laws. 15or each i = 1 , , . . . we have the maps f τ i ( f ) , A h X i + → Γ ∞ ( A h X i ); f σ i ( f ) , A h X i + → S A . (19)They are both polynomial laws homogeneous of degree i which commutewith the action of the endomorphisms T ( X ).There is an explicit Formula, see [8] Theorem 4.15 p. 37, which allowsus to compute these laws for S A . It is due to Amitsur, [2], (who stated it formatrix invariants), and later independently by Reutenauer and Sch¨utzen-berger [24]. Theorem 2.18.
Given an n ∈ N , non commutative variables x i and com-mutative parameters t i : σ n ( X i t i x i ) = X ( p < . . . < p k ) ⊂ W ,j , . . . , j k ∈ N , P j i ℓ ( p i ) = n ( − n − P j i t P ki =1 j i ν ( p i ) σ j ( p ) . . . σ j k ( p k )(20)Here W denotes the set of Lyndon words ordered by the degree lexico-graphic order. For a word p , ν ( p ) is the vector ( a , . . . , a n ) with a i countinghow many times the variable x i appears in p . Finally t ( a ,...,a n ) := Q ni =1 t a i i . In particular one can collect, in Formula (20) the terms of the samedegree in the variables x i and have an explicit expression of the polarized forms of σ n ( x ): σ n ( X i t i x i ) = X ( a ,...,a n ) | P i a i = n n Y i =1 t a i i σ n ; a ,...,a n ( x , . . . , x n ) . (21)Substituting for a variable x a linear combination P j t j M j of monomi-als and applying Formula (20) to σ n ( P j t j M j ) one obtains an element of S A provided one has a further law. In fact a primitive word computed inmonomials need no more be primitive so we also need the expression of theelements σ i ( x j ) in terms of the σ k ( x ) , k ≤ i · j. These Formulas arise fromExample 2.11 as the (stable) universal polynomial formulas in the algebraof symmetric functions expressing e i ( x j , . . . , x jn ) = P i,j ( e , . . . , e i · j ) (22)in terms of the elementary symmetric functions e k ( x , . . . , x n ) for n > i · j .One has 16 heorem 2.19. The Kernels of the maps Γ ∞ ( A h X i ) → Γ n ( A h X i ) , respec-tively S A → S n,A are the T –ideals generated by all the elements τ i ( f ) , i > n ,respectively the T –ideal generated by all the elements σ i ( f ) , i > n, f ∈ A h X i . In other words, in the case S n,A , the Kernel of π n is the ideal generatedby all the polarized forms σ m ; a ,...,a n ( p , . . . , p m ) , m > n with p , . . . , p m monomials.The Theorem of Zubkov then states that the ring of invariants of ma-trices has the same generators and relations as S n,A , when A = Z or a field,hence the isomorphism of Theorem 2.10.The following example shows for n = 2 an explicit deduction of themultiplicative nature of the determinant from these relations. σ , , ( a, b, ba ) =+ σ ( a ) σ ( b ) σ ( ab ) − σ ( a ) σ ( ab ) − σ ( b ) σ ( a b ) + σ ( a b ) − σ ( ab ) σ , ( a, b ) = − σ ( a ) σ ( b ) σ ( ab ) + σ ( a ) σ ( ab ) + σ ( b ) σ ( a b ) − σ ( a b )+ σ ( ab ) + σ ( a ) σ ( b ) σ , , ( a, b, ba ) + σ , ( a, b ) = σ ( ab ) − σ ( a ) σ ( b ) . One can in fact take the basic identity for invariants of n × n matrices.det( ab ) = det( a ) det( b ) ⇐⇒ σ n ( ab ) − σ n ( a ) σ n ( b ) = 0 . (23)Consider the polynomial ring A [ σ i ( p )] , p ∈ W , i ≤ n . Using Formula (20)define, for each f = P i t i M i ∈ A h X i , the element σ k ( f ) , k ≤ n as follows.If one substitutes each x i with M i in Formula (20) one has a formalexpression containing symbols σ i ( M ) , i ≤ k where M may be an arbi-trary monomial (including 1). Then M is cyclically equivalent to somepower N j with N ∈ W a Lyndon word. One then considers in the alge-bra of symmetric functions in exactly n variables the Formula (22) with e i = 0 , ∀ i > n i.e. e i ( x j , . . . , x jn ) = P i,j ( e , . . . , e n , , . . . ,
0) in terms of theelementary symmetric functions e k ( x , . . . , x n ) , k ≤ n . One then sets σ i ( N j ) = P i,j ( σ ( N ) , . . . , σ n ( N )) , σ i (1) := (cid:18) ni (cid:19) . (24)Given f = P i u i M i , g = P i v i M i ∈ A h X i one may consider σ n ( f g ) − σ n ( f ) σ n ( g ) = X h,k u h v k ϕ h,k , ϕ h,k ∈ A [ σ i ( p )] . (25)17valuating the variables ξ i ∈ X in the generic n × n matrices one hasa homomorphism ρ : A h X i → A [ ξ i ] to the algebra of generic matriceswhich extends to a homomorphism of the symbolic algebra ρ : A [ σ i ( p )] → A [ ξ ki,j ] P GL ( n ) to the ring of invariants of matrices. By the Theorem of Donkinthis is surjective. Moreover clearly the identity given by (24) holds for thecorresponding matrix invariants. As for (25) we have that σ n ( ρ ( f g )) − σ n ( ρ ( f )) σ n ( ρ ( g )) = det( ρ ( f ) ρ ( g )) − det( ρ ( f )) det( ρ ( g )) = 0 so all the ele-ments ϕ h,k map to 0. Theorem 2.20.
The Kernel of ρ is the ideal K of A [ σ i ( p )] generated bythe elements ϕ h,k of Formula (25) , when computed using Formula (24) and (20) .Proof. The previous relations express the identity σ n ( f g ) = σ n ( f ) σ n ( g ).Consider the algebra A [ σ i ( p )] /K , and the map A h X i → A [ σ i ( p )] /K map-ping f ∈ A h X i to the class ¯ σ n ( f ) of σ n ( f ) modulo K .By construction this is a multiplicative map homogeneous of degree n soit factors through a map A h X i → Γ n ( A h X i ) ab ¯ ρ → A [ σ i ( p )] /K . On the otherhand Γ n ( A h X i ) ab is generated by the elements σ i ( p ) and the generators of K are 0 in Γ n ( A h X i ) ab hence ¯ ρ is an isomorphism and so the claim followsfrom Theorem 2.10. n –Cayley–Hamilton algebra n –Cayley–Hamilton algebras over a commutative ring F form a category,where a map is assumed to commute with the norm.If the base ring F is either Z or a field, one has a particularly usefuldescription of the free n –Cayley–Hamilton algebra in any set of variables X = { x i } i ∈ I . It will be given in Corollary 2.24.Recall the definition 1.5 of F h ξ i i ⊂ M n ( F [ ξ ( i ) h,k ]) , i ∈ I, h, k = 1 , . . . , n the algebra of generic matrices, and the action of
P GL on M n ( F [ ξ ( i ) h,k ]) andon F [ ξ ( i ) h,k ]. The algebra F [ ξ ( i ) h,k ] P GL is by definition the algebra of invariantsof X –tuples of n × n matrices. The algebra M n ( F [ ξ ( i ) h,k ]) P GL is by definitionthe algebra of equivariant maps (polynomial laws) from X –tuples of n × n matrices to n × n matrices.One starts from the free algebra F h x i i i ∈ I and the map to the genericmatrices j := j F h x i i : F h x i i → F h ξ i i ⊂ M n ( F [ ξ ( i ) h,k ]), and then compose this18ith the determinant F h x i i j F h xi i −−−−→ F h ξ i i i −−−−→ M n ( F [ ξ ( i ) h,k ]) det −−−−→ F [ ξ ( i ) h,k ] P GL
Theorem 2.22.
1. In the commutative diagram F h x i i i ∈ I r r ⊗ n / / j F h xi i (cid:15) (cid:15) Γ n ( F h x i i i ∈ I ) N (cid:15) (cid:15) π / / Γ n ( F h x i i i ∈ I ) ab ≃ ¯ N (cid:15) (cid:15) F h ξ i i det / / F [ ξ ( i ) h,k ] P GL / / F [ ξ ( i ) h,k ] P GL (26) the last map Γ n ( F h x i i i ∈ I ) ab ¯ N → F [ ξ ( i ) h,k ] P GL is an isomorphism.2. Extend the map det ◦ j F h x i i to a norm F h x i i i ∈ I ⊗ F F [ ξ ( i ) h,k ] P GL → F [ ξ ( i ) h,k ] P GL . Then this induces a norm compatible homomorphism ¯ j F h x i i : F h x i i⊗ F F [ ξ ( i ) h,k ] P GL j ⊗ → F h ξ i i⊗ F F [ ξ ( i ) h,k ] P GL m → M n ( F [ ξ ( i ) h,k ]) P GL , with m the multiplication.3. ¯ j F h x i i is surjective and its kernel K is generated by the evaluation ofthe n characteristic polynomial of all its elements. This is proved in [8],Theorem 18.17 and Remark 18.18 based on thetheorems of Donkin [9] and Zubkov, [33].
Remark .
1. If the set of variables has ℓ elements, then the algebra F [ ξ ( i ) h,k ] P GL equals the algebra T n ( ℓ ) and the algebra M n ( F [ ξ ( i ) h,k ]) P GL equals S n ( ℓ ) of page 6.2. Using Theorem 2.19 part 3. can be equivalently stated as follows:Consider the map ρ n : S F h x i i i ∈ I τ n → S n,F h x i i i ∈ I ˜ N → F [ ξ ( i ) h,k ] P GL and: F h x i i i ∈ I ⊗ F S F h x i i i ∈ I τ n ⊗ ρ n → F h ξ i i i ∈ I ⊗ F F [ ξ ( i ) h,k ] P GL m → M n ( F [ ξ ( i ) h,k ]) P GL . (27)This map is surjective and its kernel is the ideal generated by theevaluation of the m characteristic polynomials of all its elements forall m > n and all the corresponding evaluations of the σ i in this ideal.As a Corollary one has 19 orollary 2.24. The algebra M n ( F [ ξ ( i ) h,k ]) P GL is a free algebra on the gen-erators ξ i in the category of n Cayley–Hamilton F –algebras.Proof. Let R be an n Cayley–Hamilton F –algebra, with norm algebra A and consider a set r i , i ∈ I of elements of R .Then one deduces a homomorphism f : F h x i i i ∈ I → R, x i r i . Nextone has a commutative diagram F h x i i f −−−−→ R y y Γ n ( F h x i i ) ¯ f −−−−→ Γ n ( R ) N −−−−→ A From this one has a homomorphism F [ ξ ( i ) h,k ] P GL → A and one of normalgebras ¯ f : F h x i i i ∈ I ⊗ F F [ ξ ( i ) h,k ] P GL → R .Clearly (by Theorem 2.22 3)) ¯ f vanishes on the kernel K of the quotientmap ¯ π n : F h x i i i ∈ I ⊗ F F [ ξ ( i ) h,k ] P GL → M n ( F [ ξ ( i ) h,k ]) P GL so ¯ f factors througha norm compatible map M n ( F [ ξ ( i ) h,k ]) P GL → R . Corollary 2.25.
Every n Cayley–Hamilton F –algebra R is the quotient, inthe category of n Cayley–Hamilton F –algebras, of an algebra M n ( F [ ξ ( i ) h,k ]) P GL in some generators ξ i , i ∈ I .In particular every n Cayley–Hamilton F –algebra R satisfies all poly-nomial identities with coefficients in F of n × n matrices. An important class of CH–algebras are Azumaya algebras, [5], [4]. For ourpurpose we may take as definition:
Definition 2.27.
An algebra R with center Z is Azumaya of rank n over Z if there is a faithfully flat extension Z → W so that W ⊗ Z R ≃ M n ( W ).Then it is easily seen that the determinant, of the matrix algebra M n ( W ) restricted to R maps to Z giving rise to a multiplicative map called the reduced norm N : R → Z. Then R , with this norm, is an n –CH algebra.We want to see that this is essentially the only norm on R .The next Theorem is due in part to Ziplies (but his proof is quite com-plicated and very long) [30]. 20 heorem 2.28. Let R be a rank n Azumaya algebra over Z . ( R ⊗ m ) S m ab = 0 if m is not a multiple of n . If m = in then the map ¯ N i : ( R ⊗ m ) S m ab → Z ,induced by the reduced norm N i : R → Z, is an isomorphism.Proof. By faithfully flat descent we may reduce to the case R = M n ( Z ).We first show the statement for ( M n ( Z ) ⊗ n ) S n ab . Apply Proposition 2.13to the basis of elementary matrices e i,j . We want to prove that for all e i,j we have σ h ( e i,j ) ∈ Z . If i = j we have σ h ( e i,j ) = σ h ( e i,i e i,j ) = σ h ( e i,j e i,i ) = 0 ,σ h ( e i,i ) = σ h ( e i,j e j,i ) = σ h ( e j,i e i,j ) = σ h ( e j,j ) . Apply now Amitsur’s Formula (20) to the σ j ( C ), C the permutation matrix C := e , + e , + . . . + e n − ,n + e n, , of the full cycle (1 , , . . . , n ) ∈ S n .The only non zero monomials in the x i = e i,i +1 have either value some e i,j , j = i or some e h,h , but of these the only Lyndon word is x x . . . x n = e , of degree n .Since σ j ( a ) = 0 , ∀ j > n we deduce σ i ( e , ) = σ i ( x x . . . x n ) = ( − ( n − i ) σ i · n ( e , + . . . + e n − ,n + e n, ) = 0 , ∀ i > . We claim that σ ( e , ) = ( − n − σ n ( C ) = 1, which completes the compu-tation.Now σ n is a multiplicative map and so it is 1 on the alternating group A n ⊂ S n . If n is odd then C ∈ A n so σ ( e h,h ) = 1. If n = 2 k then C ∈ A n .Set a := σ n ( C ), so σ ( e , ) = − a , we have a = σ n ( C ) = 1.Let A = − e , + P ni =2 e i,i Apply to A Formula (20), a primitive monomial in the terms of A van-ishes unless it is one e j,j . So σ n ( A ) = − k Y h =1 σ ( e h,h ) = − a k = − . Now det( AC ) = 1 so if n ≥ σ n ( AC ) = − a = ⇒ σ ( e h,h ) = 1 . For n = 2 one may check directly that A = (1 + e , )(1 + e , )(1 − e , )(1 − e , ) = 3 e , − e , + e ,
21s a commutator and deduce from Amitsur’s Formula for 1 = σ ( A ):1 = − σ ( − e , e , ) = σ ( e , ) . Next consider ( M n ( Z ) ⊗ k ) S k ab for k not a multiple of n .The same argument shows that in this case σ k ( C ) = 0. But C n = 1 and σ k is multiplicative so 1 = σ k ( C n ) = 0.The case k = in seems to be difficult to attack with this method so weuse a different approach. Lemma 2.29.
Let F be an algebraically closed field, A a commutativealgebra over F and f : M n ( F ) → A be a multiplicative polynomial map ofdegree m . Then m = in and f ( a ) = det( a ) i A . Proof.
We have already shown that multiplicative maps can exist only fordegrees multiples of n so assume that m = in . For λ a scalar matrix, since f (1) = 1 A one must have f ( λ ) = λ m A . If a ∈ SL ( n, F ) then a is a productof commutators so that f ( a ) = 1 A .If a ∈ GL ( n, F ) we can write a = bλ, b ∈ SL ( n, F ) and λ a scalarso that f ( a ) = λ m A = det( a ) i A . Since GL ( n, F ) is Zariski dense in M n ( F ) and f is a polynomial it follows that for all matrices a we have f ( a ) = det( a ) i A . We claim that A := ( M n ( F ) ⊗ in ) S in ab = F. Let π be the projection of( M n ( F ) ⊗ in ) S in to A . If a ∈ M n ( F ) the map a π ( a ⊗ in ) is a multiplica-tive polynomial map so it is det( a ) i A and maps M n ( F ) to F A . Now theelements a ⊗ in span ( M n ( F ) ⊗ in ) S in and by construction π is surjective sothe claim follows.Now let us pass to the general case A := ( M n ( Z ) ⊗ in ) S in ab . We have A = ( M n ( Z ) ⊗ in ) S in /J where J is the ideal generated by commutators.The algebra A as abelian group if finitely generated. For any field F wehave the exact sequence0 −−−−→ J −−−−→ ( M n ( Z ) ⊗ in ) S in −−−−→ A −−−−→ F ⊗ J i −−−−→ F ⊗ ( M n ( Z ) ⊗ in ) S in π −−−−→ F ⊗ A −−−−→ F ⊗ ( M n ( Z ) ⊗ in ) S in = ( M n ( F ) ⊗ in ) S in and i ( F ⊗ J ) is the ideal of( M n ( F ) ⊗ in ) S in generated by commutators and π is surjective so by theprevious Lemma F ⊗ A = F .Since this is true for all F of all characteristics one must have A = Z .22n particular the first exact sequence splits and so for all commutativerings B one has0 −−−−→ B ⊗ J i −−−−→ ( M n ( B ) ⊗ in ) S in π −−−−→ B −−−−→ B = ( M n ( B ) ⊗ in ) S in ab . Azumaya algebras and invariants
For Azumaya algebras the Problemof Remark 2.9 has a positive answer:
Theorem 2.30. If R is a rank n Azumaya algebra, over its center Z themap D : Z = Γ n ( R ) ab → T n ( R ) G n is an isomorphism. Assume first that R = M n ( A ). A map M n ( A ) → M n ( B ) consists ofa morphism f : A → B and then an automorphism g : B ⊗ Z M n ( A ) → M n ( B ), this functor is thus classified by T n ( M n ( A )) = P n ⊗ A and we havethe universal map j : M n ( A ) → M n ( P n ⊗ A ) ≃ P n ⊗ M n ( A ). The map j isalso given by Formula (2): a XaX − ∈ M n ( P n ⊗ A ) , a ∈ M n ( A ) . (30)The condition for an element u ∈ T n ( M n ( A )) = P n ⊗ A to be invariant isfrom Formula (6):∆( u ) = 1 ⊗ u, ∆ : P n ⊗ A → P n ⊗ P n ⊗ A, comultiplicationBy the Hopf algebra properties if S is the antipode we have m ◦ S ⊗ ◦ ∆ = ǫ which in group terms just means f ( x − x ) = f (1)= ⇒ m ◦ S ⊗ ◦ ∆( u ) = m ◦ S ⊗ ⊗ u ) = m (1 ⊗ u ) = u = ǫ ( u ) ∈ A. In order to analyze the universal map R → M n ( T n ( R )) for an Azumayaalgebra R we use a fact from the Theory of polynomial identities.One knows [1] Theorem 10.2.19, that there is a multilinear non commu-tative polynomial ϕ ( x , . . . , x k ) with coefficients in Z which, when evaluatedin matrices M n ( A ) over any commutative ring A does not vanish and takesvalues in the center A . From this it follows that:23 emma 2.31. When we evaluate ϕ in any Azumaya algebras R of rank n over its center Z its values lie in Z and every element of Z is obtainedas a sum of evaluations of ϕ .Proof. In fact from the faithfully flat splitting W ⊗ Z R ≃ M n ( W ) it followsthat ϕ takes values in Z and it does not vanish on R otherwise it wouldvanish on M n ( W ). Since ϕ is multilinear the sums of evaluations of ϕ forman ideal J of Z and, if J = Z , then ϕ vanishes on the Azumaya algebras R/J R of rank n over its center Z/J a contradiction.
Lemma 2.32. If f : R → R is a ring homomorphism of two rank n Azumaya algebras with centers Z , Z then f ( Z ) ⊂ Z , R ≃ Z ⊗ Z R .Proof. Given a ∈ Z we have a is a sum of evaluations ϕ ( a , . . . , a k ) forsome elements a i ∈ R so f ( a ) is a sum of evaluations ϕ ( f ( a ) , . . . , f ( a k )) ∈ Z . We have that T n ( R ) classifies the functor hom R ( R, M n ( B )) and undersuch a map f : R → M n ( B ) we have f ( Z ) ⊂ B and B ⊗ Z R ≃ M n ( B )which is an isomorphism. We may, to begin with, assume that instead ofworking in the category of rings we work in that of Z algebras Z . Thenhom Z ( R, M n ( B )) is the set of isomorphisms g : B ⊗ Z R ≃ M n ( B ).This set, for a given B , may be empty but when it is non empty we saythat B splits R and on it the group P GL ( n, − ) acts in a simply transitiveway that is, by definition, we have that T n ( R ) is a torsor for P GL ( n, − ). Definition 2.33.
Let C be a category and G ( − ) , F ( − ) be respectively acovariant group valued and a set valued functor, together with a groupaction µ : G ( − ) × F ( − ) → F ( − ).We say that F is a torsor over G if for all X ∈ C given x, y ∈ F ( X )there is a unique element g ∈ G ( X ) with gy = x .If G , F are represented by two objects G , F and C has products we saythat F is a torsor over the group like object G (cf. [11]).Notice again that when F ( X ) = ∅ the condition is void.The previous condition can be conveniently reformulated as: the natural transformation of functors: G ( X ) × F ( X ) × δ −−−−→ G ( X ) × F ( X ) × F ( X ) µ × −−−−→ F ( X ) × F ( X ) (31) is an isomorphism (with δ the diagonal) .24ne knows that there is a faithfully flat extension j : Z → W so that W ⊗ Z R ≃ M n ( W ). One deduces for the universal object T n ( R ) that W ⊗ Z T n ( R ) ≃ W ⊗ P n and so that T n ( R ) is faithfully flat over Z .From Formula (31) the property of being a torsor can be stated in termsof the coaction of P n ⊗ Z over T n ( R ), as in Formula (5): η : T n ( R ) → ( P n ⊗ Z ) ⊗ Z T n ( R ) = P n ⊗ T n ( R ) . Since F and G are controvariant functors Formula (31) gives a dual formulafor their representing objects. This is the fact that the map 1 ⊗ m ◦ η ⊗ m is the multiplication m ( a ⊗ b ) = ab ) is an isomorphism:1 ⊗ m ◦ η ⊗ T n ( R ) ⊗ T n ( R ) η ⊗ −→ P n ⊗ T n ( R ) ⊗ T n ( R ) ⊗ m −→ P n ⊗ T n ( R ) . This formula at the level of points, in any commutative Z –algebra A , meansexactly that given two points ( x, y ) in hom( T n ( R ) , A ) there is a unique g ∈ G ( A ) with x = gy, or ( x, y ) = ( gy, y ).If u ∈ T n ( R ) we have1 ⊗ m ◦ η ⊗ u ⊗
1) = 1 ⊗ m ◦ η ( u ) ⊗ η ( u )1 ⊗ m ◦ η ⊗ ⊗ u ) = 1 ⊗ m ◦ ⊗ u = 1 ⊗ u. Since the map 1 ⊗ m ◦ η ⊗ η ( u ) = 1 ⊗ u is equivalent to u ⊗ ⊗ u which by faithfully flat descentis equivalent to u ∈ Z . This proves Theorem 2.30. Σ –algebras We have seen in Formula (19) the maps σ i ( f ) , A h X i + → S A . which arepolynomial laws homogeneous of degree i which commute with the action ofthe endomorphisms T ( X ). We can view the operators σ i as homogeneouspolynomial maps, with respect to S A , of degree i of A h X i + ⊗ S A to thecenter S A which satisfy the Amitsur identity (20). Thus we may set thefollowing definition: Definition 2.35.
1) A Σ algebra R is an algebra over a commutativering A equipped with polynomial laws σ i : R → R which satisfy:[ σ i ( a ) , b ] = 0 , σ i ( σ j ( a ) b ) = σ j ( a ) i σ i ( b ) , ∀ a, b ∈ B R, ∀ B ∈ C A . (32)25 i ( ab ) = σ i ( ba ) , σ i ( a j ) (22) = P i,j ( σ ( a ) , . . . , σ i · j ( a )) (33)and that also satisfy Amitsur’s Formula (20).2) The σ –algebra σ ( R ) of R is the algebra generated over A by theelements σ j ( a ) , j ∈ N , a ∈ R .It is a subalgebra of the center of R closed under the operations σ i . Definition 2.36.
An ideal I ⊂ R in a Σ–algebra is a σ –ideal if it is closedunder the maps σ i . Proposition 2.37. If I is a σ –ideal of R the maps σ i pass to the quotient R/I which is thus also a Σ –algebra.If I is a σ –ideal of R and B is a commutative A algebra, then B I is a σ –ideal of B R. Proof.
Given a ∈ R and b ∈ I we need to see that σ i ( a + b ) − σ i ( a ) ∈ I .This follows from Amitsur’s Formula.The second part also follows from Amitsur’s Formula. Remark . Since the σ i are polynomial laws the previous statementsextend to all B R .Then Σ–algebras also form a category, where hom Σ ( R, S ) denotes theset of homomorphisms f : R → S commuting with the operations σ i , i.e. f ( σ j ( r )) = σ j ( f ( r )) , ∀ j ∈ N , ∀ r ∈ R .The kernel of a σ –homomorphism is a σ –ideal and the usual homomor-phism theorem holds.This category has free algebras namely A h X i + ⊗ S A if we do not consideralgebras with 1 or A h X i ⊗ S A [ σ i (1)] by declaring the elements σ i (1) to beindependent variables. Definition 2.39.
A Σ–identity for a Σ–algebra R is an element of the freealgebra f ∈ A h X i ⊗ S A which vanishes under all evaluations of X in R .By abuse of notations we denote by S A the algebra S A [ σ i (1)] , i = 1 , . . . . As in the Theory of polynomial identities one has then the notions of T –ideal of A h X i ⊗ S A , of variety of Σ–algebras and of Σ or PI equivalence ofΣ–algebras. Remark .
1) In this language one defines a formal Cayley Hamiltonpolynomial in the free algebra by CH n ( x ) := x n + P ni =1 ( − i σ i ( x ) x n − i .Then an n – Cayley Hamilton A –algebra R can be also defined as a Σalgebra satisfying the following conditions, which we will refer to as26 efinition . three CH n conditions :1) σ i ( x ) = 0 , ∀ i > n , 2) CH n ( x ) = 0 , ∀ i > n, ∀ x ∈ B R , and B anycommutative A –algebra, and 3) σ i (1) = (cid:0) ni (cid:1) , ∀ i ≤ n . Remark . It then follows from the Theorem of Zubkov and Zieplies that σ n is a norm and that CH n ( x ) is the evaluation for t = x of σ n ( t − x ), see[7] and also [19].In particular all the maps σ i ( x ) are deduced from σ n ( x ) via the for-mulas σ n ( t − x ) = t n + P ni =1 ( − i σ i ( x ) t n − i and σ i ( x ) = 0 , ∀ i > n. revisited The first and second fundamental Theorem for matrix invariants for alge-bras may be viewed as the starting point of the Theory of Cayley—Hamiltonalgebras, in all characteristics. It is Theorem 2.22 which can be interpretedbest in the language of Σ–algebras. Let F = Z , or a field: Theorem 2.43.
The algebra F Σ ,n h X i of equivariant polynomial maps from X –tuples of n × n matrices, M n ( F ) X to n × n matrices M n ( F ) , is thefree Σ –algebra F h X i ⊗ S A modulo the T –ideal generated by the three CH n conditions of Definition 2.41 F Σ ,n h X i := F h X i ⊗ S A / h CH n ( x ) , σ i ( x ) = 0 , ∀ i > n, σ i (1) = (cid:18) ni (cid:19) . (34)To be concrete if X has ℓ elements, let A ℓ,n denote the polynomialfunctions on the space M n ( F ) ℓ (that is the algebra of polynomials over F in mn variables ξ i, ( j,h ) , i = 1 , . . . m ; j, h = 1 , . . . , n ).On this space, and hence on A ℓ,n , acts the group P GL ( n, − ) by conju-gation.The space of polynomial maps from M n ( F ) ℓ to M n ( F ) is M n ( A ℓ,n ) = M n ( F ) ⊗ A ℓ,n . This is a Σ–algebra in an obvious way, and on this space acts diagonally
P GL ( n, − ) commuting with the Σ–operators and the invariants F Σ ,n h x , . . . , x ℓ i = M n ( A ℓ,n ) P GL ( n, − ) = ( M n ( F ) ⊗ A ℓ,n ) P GL ( n, − ) ℓ variables in thevariety of Σ–algebras satisfying the three CH n conditions.For the σ –algebra of F Σ ,n h X i we have T n ( ℓ ) = A P GL ( n, − ) ℓ,n . Of course wemay let ℓ be also infinity (of any type) and have F Σ ,n h X i = M n ( A X,n ) P GL ( n, − ) = ( M n ( F ) ⊗ A X,n ) P GL ( n, − ) where A X,n is the polynomial ring on M n ( F ) X . Remark . If F is an infinite field one may take for P GL ( n, − ) theactual group P GL ( n, F ), otherwise one has two options. The first optionis to take for P GL ( n, − ) the group P GL ( n, G ) for G any infinite fieldcontaining F or take the categorical notion, valid also over Z , taking thecoaction η : A X,n ⊗ M n ( F ) → P n ⊗ A X,n ⊗ M n ( F ) (with P n the coordinatering of P GL ( n, − )) set( A X,n ⊗ M n ( F )) P GL ( n, − ) := { r ∈ ( A X,n ⊗ M n ( F )) | η ( r ) = 1 ⊗ r } . By the universal properties one sees that η ( r ) = 1 ⊗ r is equivalent to g ( r ) = 1 ⊗ r, ∀ g ∈ P GL ( n, B ) , ∀ B. The reader will understand at this point that the approach with traceand that with norm, in characteristic 0, are equivalent.For a proof of these Theorems in all characteristics or even Z –algebras,the Theorem of Zubkov, the reader may consult [8].One can then reformulate the definition of n –Cayley–Hamilton algebrain this language: Definition 2.45.
A Σ–algebra satisfying the three CH n conditions h CH n ( x ) , σ i ( x ) = 0 , ∀ i > n, σ i (1) = (cid:18) ni (cid:19) i (35)will be called an n –Cayley–Hamilton algebra or n –CH algebra .In other words an n –Cayley–Hamilton algebra R is a quotient, as Σ–algebra, of one free algebra F Σ ,n h X i .For n = 1 a 1–CH algebra is just a commutative algebra in which thenorm is the identity map or σ i ( a ) = 0 , ∀ i > σ ( a ) = a . Therefore theTheory of n –Cayley–Hamilton algebras may be viewed as a generalizationof Commutative algebra. 28 emark . For an n –Cayley–Hamilton algebra R the map σ n is a normand, from Remark 2.42 it follows that the σ algebra σ ( R ) coincides withits norm algebra . Remark . An n –Cayley–Hamilton algebra R satisfies all the polynomialidentities of M n ( Z ). Let R be any n –Cayley–Hamilton algebra over a commutative ring F .Choosing a set of generators X for R we may present R as a quotient ofa free algebra F Σ ,n h X i = ( M n ( F ) ⊗ A X,n ) P GL ( n, − ) modulo a Σ–ideal I . Now( M n ( F ) ⊗ A X,n ) I ( M n ( F ) ⊗ A X,n ) is an ideal of M n ( F ) ⊗ A X,n = M n ( A X,n )so there is an ideal J ⊂ A X,n , which is
P GL ( n, − ) stable with( M n ( F ) ⊗ A X,n ) I ( M n ( F ) ⊗ A X,n ) = M n ( J )from which one has a commutative diagram: Theorem 2.48.
We have a commutative diagram in which the first hor-izontal arrow i is an isomorphism. The second horizontal arrows are bothinjective and the vertical maps surjective: F Σ ,n h X i i −−−−→ ∼ = M n [ A X,n ] P GL ( n, − ) −−−−→ M n [ A X,n ] y y y R i R −−−−→ M n [ A X,n /J ] P GL ( n, − ) −−−−→ M n [ A X,n /J ] (36) If F ⊃ Q then i R is an isomorphism , [Strong embedding Theorem]. The fact that in characteristic 0, i R is an isomorphism depends uponthe fact that GL ( n ), in characteristic 0, is linearly reductive, and then theproof, see [16] or [1] Theorem 14.2.1, of this Theorem is based on the socalled Reynold’s identities.In general the nature of i R is not known, we do not know if parts ofthis statement are true. This is in my opinion the main open problem ofthe Theory. 29 Prime and simple Cayley Hamilton algebras
1. a simple
Σ algebra is one with no proper σ ideals,2. a prime Σ algebra is one in which if
I, J are two σ ideals with IJ = 0then either I = 0 or J = 0.3. Finally a semiprime Σ algebra is one in which if I is an ideal with I = 0 then I = 0.Notice that prime implies semiprime. Definition 3.3.
1) Given a Σ–algebra R the set K R := { x ∈ R | σ i ( xy ) = 0 , ∀ y ∈ R, ∀ i } (37)will be called the kernel of the Σ–algebra. R is called nondegenerate if K R = 0.2) The set e K R := { x ∈ R | σ i ( x ) is nilpotent , ∀ y ∈ R, ∀ i } (38)will be called the radical of the Σ–algebra. R is called regular if e K R = 0.By Amitsur’s Formula (20) both K R and e K R are σ –ideals of R .Moreover if B is a commutative A algebra B K R ⊂ K B R , B e K R ⊂ e K B R . If I is a σ –ideal in a Σ–algebra R we set K ( I ) ⊃ I (resp. ˜ K ( I )) to bethe ideal such that R/K ( I ) = K R/I (resp. R/ ˜ K ( I ) = e K R/I ).We call K ( I ) the radical kernel of I and ˜ K ( I ) the nil kernel of I , both σ –ideals. Lemma 3.4.
Let R be a n –CH algebra. An element r ∈ R is nilpotent ifand only if all the σ i ( r ) are nilpotent. roof. In one direction every element r ∈ R satisfies its characteristic poly-nomial, if σ i ( r ) is nilpotent for all i we have r n is a linear combinationof the commuting nilpotent elements σ i ( r ) r n − i hence the claim. Assumenow r nilpotent. The elements σ i ( r k ) , i · k ≤ N satisfy the relations of thecorresponding symmetric functions e i ( X k ) := e i ( x k , x k , . . . , x kN ).Now, for each k the polynomial ring Z [ x , x , . . . , x N ] is integral over thesubring Z [ e ( X k ) , e ( X k ) , . . . , e N ( X k )] so σ i ( r ) satisfies a monic polynomialof some degree ℓ whose coefficients are polynomials in the elements σ j ( r k )(and with 0 constant coefficient). If r k = 0 these coefficients are all 0 so σ i ( r ) ℓ = 0 . Proposition 3.5. K R is the maximal σ –ideal J where σ ( J ) = 0 .2. If R is an n –CH algebra we have e K R is the maximal ideal I with theproperty that B I is nil for all B .3. If R is an n –CH algebra R/ e K R is regular., i.e e K R/ e K R = 0 .4. If σ i ( a ) is nilpotent, then σ i ( a ) ∈ e K R .
5. If R is an n –CH algebra, over some commutative ring A , and I is anil ideal of R then B I is is a nil ideal of B R for every commutative B algebra.Proof.
1) The first part is clear. 2) Follows from the previous Lemma3.4.3) If the class of r ∈ R is in the radical ˜ S of S := R/ e K R we have thatfor each y ∈ B R , σ i ( ry ) is nilpotent, in B S hence σ i ( ry ) is nilpotent also in B R and so r ∈ e K R .4) As for the last statement, σ j ( σ i ( a ) y ) = σ i ( a ) j σ j ( y ) is nilpotent.5) This follows from the previous Lemma 3.4 and Amitsur’s Formula. Corollary 3.6.
Let R be a n –CH algebra with σ –algebra reduced (no nonzeronilpotent elements) then if r ∈ R is nilpotent we have r n = 0 . In particular we have
Corollary 3.7.
1) An n –CH algebra R is semiprime if and only if its σ –algebra is reduced and the radical e K R = 0 .2) An n –CH algebra R is prime if and only if its σ –algebra is a domainand e K R = 0 . ) An n –CH algebra R is simple if and only if its σ –algebra is a fieldand e K R = 0 .Proof.
1) Assume R semiprime. If the σ –algebra contains a non zero nilpo-tent element a then Ra is a nilpotent ideal a contradiction. Since R is aPI algebra, it is semiprime if and only if it does not contain a nonzero nilideal. So since e K R is nil and R is semiprime e K R = 0.Conversely if R has an ideal I = 0 with I = 0 then for each a ∈ B I wehave σ i ( a ) is nilpotent for all i , by Lemma 3.4 then I ⊂ e K R .2) As for the second statement let us show that the given conditionsimply R prime. In fact given two σ –ideals I, J with IJ = 0 since σ ( I ) ⊂ I, σ ( J ) ⊂ J we have σ ( I ) σ ( J ) = 0. Since these are ideals and σ ( R ) is adomain one of them must be 0. If σ ( I ) = 0 then I ⊂ K R = { } since R issemiprime and by the previous statement.Conversely if R is prime in particular it is semiprime so we must have e K R = 0. If we had two non zero elements a, b ∈ σ ( R ) with ab = 0 we wouldhave Ra · Rb = 0 and Ra, Rb are σ ideals, a contradiction.3) If R is simple it is prime so σ ( R ) is a domain. We need to show thatif a ∈ σ ( R ) then a is invertible, and the element b with ab = 1 is in σ ( R ).First the ideal aR is σ stable so it must be R and there is an element b with ab = ba = 1. Thus the field of fractions K of σ ( R ) is contained in R , butsince K is integral over σ ( R ), by the going up Theorem it coincides with σ ( R ).Conversely if σ ( R ) is a field and I is a nonzero proper σ ideal, for every a ∈ I we must have σ i ( a ) ∈ I = ⇒ σ i ( a ) = 0 , ∀ i . Then I ⊂ K R = { } . Proposition 3.8.
1) If R is a semiprime n Cayley-Hamilton algebra with a ∈ σ ( R ) not a zero divisor in σ ( R ) then a is not a zero divisor in R .2) If R is a prime Σ –algebra, σ ( R ) is a domain and R is torsion freerelative to σ ( R ) .Proof.
1) Let J := { r ∈ R | ar = 0 } , then J is an ideal and we claim it isnil hence by hypothesis 0. In fact taking one of the functions σ i we have0 = σ i ( ar ) = a i σ i ( r ) implies σ i ( r ) = 0 for all r ∈ J , since r satisfies its CHit must be r n = 0.2) If a ∈ σ ( R ) and J := { r ∈ R | ar = 0 } then both J and Ra are idealsclosed under the Σ operations and J Ra = 0. Since R is prime and Ra = 0it follows that J = 0. 32inally the local finiteness property: Proposition 3.9. An n –CH algebra R finitely generated over its σ –algebra σ ( R ) is a finite σ ( R ) module.Proof. The Cayley Hamilton identity implies that each element of R isintegral over σ ( R ) of degree ≤ n then this is a standard result in PI ringsconsequence of Shirshov’s Lemma, [1] Theorem 8.2.1.. In this section F denotes an infinite field, this hypothesis could be removedbut it simplifies the treatment. We want to study general CH algebras over F such that the values of the norm and hence of all the σ i are in F .First a simple fact. Let R = R ⊕ R be an F algebra and N : R → F a multiplicative polynomial map. Then setting e , e the two unit elementsof R , R we have N ( a, b ) = N (( a, e )( e , b )) = N ( a, e ) N ( e , b ).Set N ( a ) := N ( a, e ) , N ( b ) := N ( e , b ). Clearly N i : R i → F, i = 1 , N ( a, b ) = N ( a ) N ( b ). Proposition 3.10. If N is homogeneous of degree n there are two positiveintegers h , h > with h + h = n and N ( a ) is homogeneous of degree h while N ( b ) is homogeneous of degree h .Proof. Restrict N to F ⊕ F then N factors through a homomorphism of[( F ⊕ F ) ⊗ n ] S n → F .Now [( F ⊕ F ) ⊗ n ] S n is the direct sum of n + 1 copies of F , each withunit element the symmetrization e h e n − h of e ⊗ h ⊗ e ⊗ n − h .A homomorphism [( F ⊕ F ) ⊗ n ] S n → F thus factors though the projectionto one of this summands. The claim then follows from the remark that( αe + βe ) ⊗ n = P nh =0 α h β n − h e h e n − h . Corollary 3.11.
Under the previous hypotheses, N : R → F a multiplica-tive polynomial map of degree n . If R = ⊕ ki =1 R i then k ≤ n .Proof. N is a product of the norms N i each with some degree h i > n = P i h i . Corollary 3.12.
Under the previous hypotheses, R is an n –CH algebra ifand only if R i is an h i CH–algebra for i = 1 , . roof. χ ( a,b ) ( t ) = N (( te − a, te − b ) = N ( te − a ) N ( te − b ) = χ a ( t ) χ b ( t ) , so if R i is an h i CH–algebra for i = 1 , χ ( a,b ) (( a, b )) = ( χ a ( a ) , χ a ( b ))( χ b ( a ) , χ b ( b )) = 0 . (39)Conversely assume χ ( a,b ) (( a, b )) = 0. Given a ∈ R take α ∈ F so χ α ( t ) = ( t − α ) h and χ ( a,α ) (( a, α )) = χ a ( a )( a − α ) h = 0 , ∀ α ∈ F. The minimal polynomial f ( t ) of a thus divides χ a ( t )( t − α ) h , ∀ α ∈ F henceit divides χ a ( t ) so χ a ( a ) = 0 and similarly for b .A semisimple algebra S finite dimensional over a field F is isomorphicto the direct sum S = ⊕ i M k i ( D i ) of matrix algebras over division ringswhich are finite dimensional over F .We treated the theory in characteristic 0 in [20], so we assume that F has some positive characteristic p >
0. Let G i denote the center of D i .If F is separably closed then all the D i = G i are fields, purely insepara-ble over F (this follows from the fact that a division ring D of degree n hasa separable element of degree n over the center, see for instance Saltmen[27]). We ask in general which norms exist on S with values in F whichmake S an n –Cayley–Hamilton algebra.We start with a special case.Given two lists m := m , . . . , m k and a := a , . . . , a k of positive integerswith P j m j a j = n consider the algebra with norm NF ( m ; a ) := ⊕ ki =1 M m i ( F ) , ⊂ M n ( F ) , N ( r , . . . , r k ) = k Y i =1 det( r i ) a i (40)where the i th block is repeated a i . F ( m ; a ) is a subalgebra (of block diagonal matrices) of M n ( F ) and thenthe norm N equals the determinant, hence it is an n Cayley–Hamiltonalgebra, and, as σ – algebra, it is simple .Conversely we have the standard:34 roposition 3.13. If F is algebraically closed and S ⊂ M n ( F ) is a semisim-ple algebra then it is one of the algebras F ( m , . . . , m k ; a , . . . , a k ) .Proof. A semisimple algebra S over F is of the form S = ⊕ ki =1 M m i ( F ).An embedding of S in M n ( F ) is a faithful n –dimensional representa-tion of S . Now the representations of S are direct sums of the irreduciblerepresentations F m i of the blocks M m i ( F ), and a faithful n –dimensionalrepresentation of S is thus of the form ⊕ i ( F m i ) ⊕ a i , a i ∈ N , a i > , X i a i m i = n. For this representation the algebra S appears as block diagonal matrices,with an m i × m i block repeated a i times. The norm is the determinantdescribed by Formula (40).Formula (40) can be made axiomatic. Assume F is an infinite field. Theorem 3.14.
Suppose that F ( m ) := ⊕ qi =1 M m i ( F ) is equipped with aNorm N of degree n with values in F . Then there are positive integers a i with P i a i m i = n so that N ( r , . . . , r q ) = q Y i =1 det( r i ) a i . (41) Hence it is the n Cayley–Hamilton algebra F ( m ; a ) .Proof. From Corollary 3.12 we are reduced to the case R = M h ( F ). In thiscase the statement is a special case of Theorem 2.28. We have an even more abstract Theorem:
Theorem 3.15.
Let F be an algebraically closed field and S an n Cayley–Hamilton algebra with norm in F and radical e K S .Then S/ e K S is finite dimensional, simple, and isomorphic to one of thealgebras F ( m , . . . , m k ; a , . . . , a k ) as algebra with norm. roof. First remark that, since the values of σ i are all in F we have thatthe kernel equals the radical K S .Passing to S/K S we may thus assume that K S = 0. Let us first assumethat S is finite dimensional, then by Proposition 3.5 we have that S is asemisimple algebra so it is of the form S = ⊕ ki =1 M m i ( F ). Since it is an n Cayley–Hamilton algebra the statement follows from Theorem 3.14.Now let us show that it is finite dimensional. For any choice of a finiteset of elements A = { a , . . . , a k } ⊂ S let S A be the subalgebra generated bythese elements, since each σ takes values in F this is also a Σ–subalgebra.By a standard theorem of PI theory since S is algebraic of bounded degreeeach S A is finite dimensional. Then if J A is the radical of S A we have by theprevious part that dim S A /J A ≤ n . Let us choose A so that dim S A /J A ismaximal. We claim that S = S A and J A = 0.First let us show that J A ⊂ K S the Kernel of S . Let a ∈ J A and r ∈ S ; we need to show that σ i ( ra ) = 0 , ∀ i . If r ∈ S A this is the previousstatement, if r / ∈ S A then S A,r ) S A and we claim that J A,r ⊃ J A , in factotherwise dim S A,r /J A,r > dim S A /J A a contradiction.Then by the previous argument σ i ( ar ) = 0 so a ∈ K S but since S issimple K S = 0 and J A = 0. Next if S A = S we have again some S A,r ) S A and now J A,r = 0 a contradiction. If R is a simple PI algebra, over a field F , we have R = M k ( D ) with D a division ring finite dimensional over its center G ⊃ F (Theorem 11.2.1of [1]), let dim G D = h . If furthermore R is finite dimensional over F letdim F G = ℓ .In this last case the algebra R is endowed with a canonical Norm ho-mogeneous of degree khℓ which is a composition of two norms N R/F = N R/G ◦ N G/F
The norm N R/G can be defined as follows. We take a maximal subfield M ⊂ D separable over G then: M k ( D ) ⊗ G M = M k · h ( M ) . If a ∈ M k ( D ) define as Norm N R/G ( a ) := det( a ⊗
1) as matrix.It is a standard fact that N R/G ( a ) ∈ G . As for N G/F ( g ) , g ∈ G onetakes the determinant of the multiplication by g a ℓ × ℓ matrix over F .36ssume G is separable over F and ¯ F ⊃ G is the separable closure of G and F . We have the ℓ , F –embeddings of G in ¯ F , γ , . . . , γ ℓ given by GaloisTheory: G ⊗ F ¯ F = ¯ F ℓ , g ⊗ γ ( g ) , . . . , γ ℓ ( g )) = ⇒ N ( g ) = Y i γ i ( g ) ∈ F. (42)Notice that, in this case we also have a trace tr ( a ) ∈ F, ∀ a ∈ R , theseparability condition is given by the fact that the trace form tr ( ab ) is nondegenerate.In general let L be the separable closure of F in G , and a = [ L : F ] , p k =[ G : L ]. Let ¯ F be a separable closure of F we still have that L ⊗ F ¯ F = ¯ F ⊕ a , D ⊗ F ¯ F = D ⊗ L ( L ⊗ F ¯ F ) = ⊕ i D ⊗ L ¯ F where in the summands L embeds in ¯ F by the a different embeddings givenby Galois Theory as in Formula (42).Moreover G ⊗ L ¯ F is a field by Theorem 9 page 163 of [10]. Finally Lemma 3.17. D ⊗ L ¯ F = D ⊗ G ( G ⊗ L ¯ F ) ≃ M h ( G ⊗ L ¯ F ) and M k ( D ) ⊗ F ¯ F = M k ( D ) ⊗ G ( G ⊗ F ¯ F ) = M k ( D ) ⊗ G ( G ⊗ L ( L ⊗ F ¯ F ))= ⊕ i M hk ( G ⊗ L ¯ F ) . (43) Proof. D contains a maximal subfield M separable over G so generated bya single element a satisfying an irreducible separable polynomial f ( x ) = x h + P hi =1 α i x h − i , f ( a ) = 0 with coefficients α i in G . We have for somepower a p k that a p k satisfies x h + P hi =1 α p k i x h − i with coefficients α p k i in¯ F . Since M is purely inseparable over G [ a p k ] we have that a p k is also agenerator of M but being separable over ¯ F it is in ¯ F . Therefore G ⊗ L ¯ F isa splitting field for D .As for the norm N M k ( D ) /F one has to embed G ⊗ L ¯ F ⊂ M ℓ ( ¯ F ) and thenin the embedding M k ( D ) ⊗ F ¯ F ⊂ ⊕ i M hkp k ( ¯ F ) ⊂ M hkp k ℓ ( ¯ F ) , a ( a , . . . , a ℓ )we have N M k ( D ) /F ( a ) = Y i det( a i ) . roposition 3.18. Under this norm M k ( D ) is a k · h · p k · ℓ CH algebra.Proof.
The norm is induced by the determinant of the previous Formula.We ask now what is the general form of a norm and we will see that,contrary to what happens in characteristic 0 in general there are normswhich have degree strictly less than that of the canonical norm.We now do not even assume that D is finite dimensional over F .Let again L be the separable closure of F in G , and consider a norm N : R = M k ( D ) → F that is a multiplicative polynomial map homogeneousof some degree n . Theorem 3.19. [ L : F ] = ℓ < ∞ . There is a minimum integer k such that g p k ∈ L, ∀ g ∈ G . There is a b ∈ N such that n = khbp k ℓ . The norm N depends only upon b and will be denoted by N b and maps as N b : M k ( D ) N R/G −−−−→ G a a bpk −−−−−→ L N L/F −−−−→ F. (44)That is N b = N b . The proof is in several steps. Denote again by ¯ F a separable closure of F . Restricting the norm to L ⊗ F ¯ F we have, buCorollary 3.11, that [ L : F ] = ℓ is finite. The ℓ divides n by a combinationof Formula (42) and Theorem 3.14. Lemma 3.17 still holds.The norm N induces a norm N : M k ( D ) ⊗ F ¯ F → ¯ F which, by Propo-sition 3.10 is the product of the norms N i in the ℓ summands of Formula(43). If γ is an automorphism of ¯ F over F we have N ◦ ⊗ γ = γ ◦ N . Thisallows us to say that the norms N i induced according to Proposition 3.10in the a summand of Formula (43) have all the same degree m and n = ma .Set m = hk we have to analyze the norms N : M m ( G ⊗ L ¯ F ) → ¯ F whichby abuse of notation we still think of degree n .The first case is when m = 1.Changing notations let G ⊃ F be purely inseparable over F . Lemma 3.20.
The map π n : G ⊗ n → G, a ⊗ a ⊗ · · · ⊗ a n Q i a i is asurjective homomorphism with kernel the nil radical of G ⊗ n .Proof. By induction on n . The map π : G ⊗ G → G, π ( a ⊗ b ) = ab isa homomorphism with kernel the ideal generated by a ⊗ − ⊗ a whichis nilpotent sinc, for some h we have a p h ∈ F so ( a ⊗ − ⊗ a ) p h = a p h ⊗ − ⊗ a p h = 0 . Then by induction π n must factor through J ⊗ G with J the radical of G ⊗ n − . 38ut G ⊗ n /J ⊗ G = G ⊗ G and we are at the beginning of the induction. Lemma 3.21. If N : G → F is a multiplicative polynomial map of degree n then there is a minimal k so that a p k ∈ F, ∀ a ∈ G , n = bp k , b ∈ N and N ( a ) = a n . Proof.
Let a ∈ G be such that a p h ∈ F with h minimal, restrict N first to G = F [ a ]. A multiplicative polynomial map then factors though N : G u u ⊗ n −→ ( G ⊗ F G ⊗ · · · ⊗ F G ) S n ¯ N −→ F (45)with ¯ N a homomorphism which thus vanishes on the radical.Now the map π : G ⊗ n → G, a ⊗ a ⊗ · · · ⊗ a n Q i a i is a surjectivehomomorphism with kernel the radical of G ⊗ n . It follows that π restrictedto ( G ⊗ F G ⊗ · · · ⊗ F G ) S n induces an isomorphism of ( G ⊗ F G ⊗ · · · ⊗ F G ) S n modulo its radical with a field L with F ⊂ L ⊂ G . Thus if ¯ N exists, sinceit factors through ( G ⊗ F G ⊗ · · · ⊗ F G ) S n we must have L = F .Next ( G ⊗ F G ⊗ · · · ⊗ F G ) S n is generated by the elementary symmetricfunctions in the elements 1 ⊗ i ⊗ a ⊗ ⊗ n − i − so the image of π is generatedby the elements (cid:0) nj (cid:1) a i . In particular if they have to lie in F we must havethat a n ∈ F so n is a multiple of the minimal p h for which a p h ∈ F .For general G the statement follows by restricting to all subfields F [ a ]of previous type, assuming the existence of N we must in particular havethat for each a ∈ G we have [ F [ a ] , F ] = p d with p d dividing n . Thereforethere is a minimum k so that a p k ∈ F, ∀ a ∈ G and n is a multiple of p k .Finally N is the canonical norm G → ( G ⊗ F G ⊗ · · · ⊗ F G ) S n = F whichis a a n . Remark . If ˜ F is an algebraic closure of F , setting y = x − a ˜ F ⊗ F F [ a ] = ˜ F [ x ] / ( x p k − a p k ) ≃ ˜ F [ y ] /y p k . The norm N extends to N : ˜ F [ y ] /y p k a a ⊗ n −→ [( ˜ F [ y ] /y p k ) ⊗ n ] S n ¯ N −→ ˜ F . (46)There is a unique ¯ N in Formula (46) which vanishes on the radical.39 e pass to general m . Let N : M m ( G ) → F be a multiplicativepolynomial map, over F , of degree n . Consider the subspace of diagonalmatrices G m ⊂ M m ( G ), the norm N restricted to G m is a product of norms N i for the various summands, but the symmetric group S m ⊂ M m ( G )permutes the summands so the norms are all the same and by Lemma 3.21there is b so that N ( a , . . . , a m ) = Q i a bp k i = det(( a , . . . , a m )) bp k .Next consider elementary matrices e i,j ( α ) := 1 + αe i,j , i = j, α ∈ G .We have e i,j ( α ) e i,j ( β ) = e i,j ( α + β ) so N on this subgroup is a multi-plicative polynomial map from G , as additive F vector space, to the mul-tiplicative F . In a basis of G over F it is thus a polynomial of some degree k with f ( x + y , . . . , x h + y h ) = f ( x , . . . , x h ) f ( y , . . . , y h )this by degree implies that f = 1. Then by the usual Gaussian eliminationany matrix A is a product of a diagonal matrix times elementary matricesso that hence N ( A ) = det( A ) bp k for all A .Finally the norm N b of Formula 44 becomes under tensor product with¯ F the one of Theorem 3.19 so the Theorem follows.Let us summarize what we proved. Consider a general semisimple alge-bra over F R = ⊕ pi =1 M k i ( D i ) , dim G i D i = h i where G i is the center of thedivision algebra D i . Let L i ⊂ G i be the separable closure of F , dim F L = ℓ i and k i the minimum integer such that a p ki ∈ L i , ∀ a ∈ G i . Given positiveintegers b i , i = 1 , . . . , p we may define the norm N ( a , . . . , a p ) := p Y i =1 N b i ( a i ) , a i ∈ M k i ( D i ) , N b i ( a i ) Formula (44) . (47) Theorem 3.23.
The algebra R = ⊕ pi =1 M k i ( D i ) with the previous norm isan n CH algebra with n = P i k i h i b i p k i ℓ i .Conversely any norm on R is of the previous form.Proof. From Corollary 3.12 we may reduce to R = M k ( D ) and then, bysplitting to the case in which G is purely inseparable over F , R = M ℓ ( G )and N : M ℓ ( G ) → F, N ( a ) = det( a ) bp k . The characteristic polynomial isthus χ ( t ) = det( t − a ) bp k . Then clearly this vanishes for t = a by the usualCH Theorem for M ℓ ( G ). Corollary 3.24.
Let R = ⊕ pi =1 M k i ( D i ) be an n CH algebra over a field F as in Theorem 3.23. If for one i we have that the dimension of M k i ( D i ) over its center G i equals n then R = M k i ( D i ) , F = G i and the norm isthe reduced norm. Lemma 3.25.
Let S be a semiprime algebra with center an infinite field C and each element satisfies an algebraic equation of degree n over C . Then S is a finite dimensional central simple algebra.Proof. S satisfies a polynomial identity so it is enough to prove that S is aprime algebra (cf. [1] Theorem 11.2.6).Let P be a prime ideal of S , the prime algebra S/P is algebraic over C so its center is a field and being a PI ring it is a simple algebra, thusisomorphic to M k ( D ) with D a division algebra finite dimensional over itscenter. In particular P is a maximal ideal. By [1] Theorem 1.1.41 we have { } = T P is the intersection of all prime (maximal ideals). If there areonly finitely many maximal ideals P ∩ P ∩ . . . ∩ P k = { } then S = ⊕ i S/P i and its center is a field only if m = 1. But if we have m > n maximal idealswe still have S/ ∩ mi =1 P i = ⊕ mi =1 S/P i contains C m which contains elementswhich are not algebraic of degree n over C . Theorem 3.26. If S is a σ –simple n Cayley–Hamilton algebra then S isisomorphic to one of the algebras of Theorem 3.23.Proof. By Corollary 3.7 the algebra σ ( R ) = F is a field. Let C be the centerof R it is also an n Cayley–Hamilton algebra, commutative and with nonilpotent elements, we claim it is a finite direct sum of fields.In fact we claim that in C there are at most n orthogonal idempotents.In fact if e , . . . , e m are orthogonal idempotents we have that S = ⊕ i Se i ,by Corollary 3.10 we have n = P i h i , h i > e + · · · + e m is a decomposition into primitive orthogonalidempotents we have C = ⊕ i C i , C i := Ce i . Each C i has no nilpotentelements, no non trivial idempotents and it is algebraic over F then it is afield.We claim that S i := Se i is an h i simple CH algebra.In fact clearly an ideal of S i is an ideal of S so there are no nil ideals,the σ algebra is contained in F which is a field so, by the argument ofCorollary 3.7 is also a field and the conclusion of Corollary 3.7 applies.In positive characteristic the σ algebra of S i need not be F as thefollowing example shows. Let F be of positive characteristic p , consider on R = F ⊕ F the norm N ( a, b ) = a p b , the σ algebra of R is F but that of thefirst summand is F p which may be different from F .Now, changing notations we may assume that the center of S is a field C ⊃ F = σ ( S ) and conclude by Lemma 3.25.41ecall that, by Proposition 3.8 a σ –prime n Cayley–Hamilton algebra S is torsion free over σ ( R ) which is a domain. Thus we can embed S in S ⊗ σ ( S ) K . Corollary 3.27. If S is a σ –prime n Cayley–Hamilton algebra and K isthe field of fractions of σ ( S ) then S ⊗ σ ( S ) K ≃ ⊕ pi =1 M k i ( D i ) is a σ –simple n Cayley–Hamilton algebra isomorphic to one of the algebras of Theorem3.23 and containing S .There are finitely many minimal prime ideals P j = S ∩⊕ pi =1 , i = j M k i ( D i ) in S with intersection { } . In any associative algebra R one can define the spectrum of R as the set ofall its prime ideals, it is equipped with the Zariski topology .For commutative algebras the spectrum is a contravariant functor with f : A → B giving P f − ( P ). But in general a subalgebra of a primealgebra need not be prime and the functoriality fails.For Σ–algebras R we may define: Spec σ ( R ) := { P | P is a prime σ –ideal } . For an n Cayley-Hamilton algebra R we have, by Corollary 3.7 the map j : Spec σ ( R ) → Spec ( σ ( R )) , P P ∩ σ ( R ) and the remarkable fact: Proposition 3.29.
The map j : Spec σ ( R ) → Spec ( σ ( R )) , P P ∩ σ ( R ) is a homeomorphism, its inverse is p ˜ K ( p R ) . Proof.
First we have, for any Σ–algebra R , any I ⊂ σ ( R ) an ideal of σ ( R )that IR is a σ –ideal and IR ∩ σ ( R ) = I . In fact if r ∈ I ∩ σ ( R ) we have r = P i a i s i , a i ∈ I, s i ∈ R and, by Amitsur’s formula σ j ( r ) = σ j ( P i a i s i )is a polynomial in elements σ k ( u ) with u a monomial in the elements a i s i .So u = as, a ∈ I and σ k ( u ) = a k σ k ( s ) ∈ I. Let p ⊂ σ ( R ) be a prime σ –ideal. Since σ ( R/ p R ) = σ ( R ) / p is a domainwe have also ˜ K ( p R ) / p R ∩ σ ( R ) / p = { } hence σ ( R/ ˜ K ( p R )) = σ ( R ) / p .From Corollary 3.7 2) we have that the ideal ˜ K ( p R ) , is prime. In fact σ ( R/ ˜ K ( p R )) = σ ( R ) / p a domain and also the radical of R/ ˜ K ( p R ) is { } ,by 3) of Proposition 3.5. 42o the composition in one direction is the identity p = ˜ K ( p R ) ∩ σ ( R ).If P is a prime σ –ideal we need to show that P = ˜ K (( P ∩ σ ( R )) R ).We certainly have P ⊃ ˜ K (( P ∩ σ ( R )) R ) so it is enough to show that,if P ⊃ Q are two prime σ –ideals and P ∩ σ ( R ) = Q ∩ σ ( R ) then P = Q .In fact in R/Q , a prime σ –algebra, we have σ ( P/Q ) = 0 which implies
P/Q ⊂ K R/Q = 0.So for n Cayley-Hamilton algebras the spectrum is also a contravariantfunctor setting f : A → B, P ˜ K ( f − ( P )) . Some consequences
Assume that R is an n Cayley-Hamilton algebra.Let p be a prime σ –ideal of σ ( R ) and consider the local algebra σ ( R ) p and R p := R ⊗ σ ( R ) σ ( R ) p we have that Proposition 3.30. R p is local, as σ –ring, with maximal σ –ideal ˜ K ( R p p ) .Theorem 3.23 gives the possible residue σ –simple algebras R ( p ) := R p / ˜ K ( R p p ) . A special case is when R ( p ) is simple as algebra and of rank n overits center. In this case one can apply Artin’s characterization of Azumayaalgebras (cf. [1] Theorem 10.3.2) and deduce that Proposition 3.31. If R ( p ) is simple as algebra and of rank n over itscenter than R p is a rank n Azumaya algebra over its center σ ( R ) p . Let us analyze general prime ideals, not necessarily σ –ideals. Proposition 3.32.
Let S be an n Cayley–Hamilton algebra with σ –algebra A , P an algebra ideal of S which is prime and p = P ∩ A . Let F be the fieldof fractions of A/ p .Then S/P ⊗ A F = S/P ⊗ A/ p F is a simple algebra and P is one of theminimal primes of the prime σ –algebra S/ ˜ K ( p S ) (Corollary 3.27).Proof. We have P ⊃ ˜ K ( p S ) since ˜ K ( p S ) is nil modulo p S .Thus we have a surjective map S/ ˜ K ( p S ) → S/P which induces a sur-jective map S/ ˜ K ( p S ) ⊗ A F → S/P ⊗ A F with F the field of fractions of A/ p .Since S/P ⊗ A F = S/P ⊗ A/ p F is a prime algebra and, Corollary 3.27, S/ ˜ K ( p S ) ⊗ A F = ⊕ i S i is a direct sum of simple algebras we must have S/P ⊗ A F = S i for one of the summands and the claim follows.43his is a strong form of going up and lying over of commutative algebrain this general setting. Corollary 3.33.
Let S be an n Cayley–Hamilton algebra with σ –algebra A , M an algebra ideal of S which is maximal and m = M ∩ A . Then m isa maximal ideal of A .Proof. We have that
S/M is a simple algebra integral over A/ m hence itscenter, a field, is integral over A/ m . Thus A/ m is a field from the going uptheorem. Lemma 3.34.
Let S be an n Cayley–Hamilton algebra with σ –algebra A , M an algebra ideal of S so that S/M is simple of dimension n over itscenter and m = M ∩ A . Then M = m S .Proof. The algebra ¯ S := S/ m S is an n Cayley–Hamilton algebra with σ –algebra the field F = A/ m A by the previous Lemma.We have that ¯ S/ ˜ K ( ¯ S ) is a simple σ –algebra to which we can applyTheorem 3.23. From Corollary 3.24 t and the fact that one of the simplesummands of R := ¯ S/ ˜ K ( ¯ S ) is S/M = M k ( D ) simple of dimension n overits center we have that R = M k ( D ) with center F . Since S satisfies the PI’sof n × n matrices, by Artin’s characterization of Azumaya algebras we havethat S is Azumaya over its center A . By Theorem 2.28 the norm takes asvalues the center A which thus must equal F and so S = M k ( D ). Theorem 3.35.
Let S be an n Cayley–Hamilton algebra with σ –algebra A a local ring with maximal ideal m . Let M be an algebra ideal of S so that S/M is simple of dimension n over its center, then M = m S and S is arank n Azumaya algebra over A . T –ideals and relatively free algebras One purpose of this paper is to classify and then study prime and semiprime T –ideals in a free CH n –algebra. In order to explain the meaning and theresults of this program we recall first the classical theory of polynomialidentities of which the present paper is a natural development.Recall that, given an associative algebra R over an infinite field F , a polynomial identity of R is an element f ( x , . . . , x ℓ ) ∈ F h x , . . . , x ℓ , . . . i of the free algebra in countably many variables which vanishes under allevaluations in R , x i r i ∈ R of the variables x i .44he set I of polynomial identities of R is an ideal of F h x , . . . , x ℓ , . . . i with the special property of being closed under all substitutions of thevariables x i g i ∈ F h x , . . . , x ℓ , . . . i . Such an ideal is called a T –ideal.Conversely any T –ideal I is the ideal of polynomial identities of some alge-bra R , and we can take as R = F h x , . . . , x ℓ , . . . i /I .We want to investigate now the structure of prime T –ideals (in thesense of Σ–algebras) in the free n Cayley Hamilton algebra F Σ ,n h X i . Forsimplicity let us assume that all algebras are over an infinite field, thoughthis is not strictly necessary as the reader may show.If P is such an ideal then it is the T –ideal of Σ–identities of the prime Σ–algebra R := F Σ ,n h X i /P . These identities are also satisfied by the algebraof fractions of R so that finally a prime T –ideal of Σ–identities is the T –ideal of Σ–identities of a simple Σ–algebra S with σ ( S ) = L ⊃ F a field.Replace S , with ˇ S := S ⊗ L ˇ L where ˇ L is an algebraic closure of L . Althoughthis algebra is not necessarily simple the Σ–identities with coefficients in F of S and ˇ S , coincide.If ¯ L ⊂ ˇ L is the separable closure of L we have that ¯ S := S ⊗ L ¯ L = ⊕ i M m i ( G i ) with G i purely inseparable over ¯ L and furthermore there is aminimum k i with g p ki ∈ ¯ L, ∀ g ∈ G i .The structure of ˇ S := ¯ S ⊗ ¯ L ˇ L = ⊕ i M m i ( G i ⊗ ¯ L ˇ L ) can be deduced fromRemark 3.22. The commutative algebras G i ⊗ ¯ L ˇ L are obtained from thealgebraically closed field ˇ L by adding nilpotent elements a i , i ∈ I, of degrees p h i , h i ≤ k i , i.e. G i ⊗ ¯ L ˇ L = ˇ L [ a i ] , i ∈ I, a p hi i = 0. The norm N is a productof the norms N i : M m i ( ˇ L [ a i ]) → ˇ L [ a i ] is given by A det( A ) p ki . Here A ∈ M m i ( ˇ L [ a i ]) is a matrix with entries a i,j + u i,j , a i,j ∈ ˇ L and u p ki i,j = 0. Sothat det( A ) = det(( a i,j ))+ u, u p ki = 0 and finally det( A ) p ki = det(( a i,j )) p ki . This means that the norm N i is obtained from the norm det( A ) p ki of M m i ( ˇ L ) by extending the coefficients, thus as far as Σ–identities, thealgebra ˇ S := ⊕ i M m i ( ˇ L [ a i ]) is equivalent to ⊕ i M m i ( ˇ L ). Finally, since F isan infinite field, this is PI equivalent to ⊕ i M m i ( F ).The possible norms on ⊕ i M m i ( F ) are given by Theorem 3.14 and areof the form given by Formula (41), thus depend only on q integers m i , a i with P qi =1 m i a i = n .To complete the classification of prime T –ideals we have only to showthat two algebras associated to two different sequences of q integers m i , a i with P qi =1 m i a i = n are not PI equivalent. For this it is enough to see thatthe corresponding relatively free algebras contain the information on these45umbers. This part is identical to that developed in the previous paper [19]to which we refer. Theorem 3.37.
Prime T –ideals (in the sense of Σ –algebras) in the free n Cayley Hamilton algebra F Σ ,n h X i are classified by sequences of q integers m i , a i with P qi =1 m i a i = n . To such a sequence one associates the T –idealof identities of the algebra of Formula (40) F ( m ; a ) . References [1] E. Aljadeff, A. Giambruno, C. Procesi, A. Regev.
Rings with poly-nomial identities and finite dimensional representations of algebras ,A.M.S. Colloquium Publications, to appear. 1.1, 1.7, 2.26, 2.47.1, 3.1.1,3.16, 3.16, 3.28[2] S. A. Amitsur,
On the characteristic polynomial of a sum of matrices ,Linear and Multilinear Algebra (1979/80), no. 3, 177–182. 2.12[3] M. Artin, On Azumaya algebras and finite-dimensional representationsof rings , J. Algebra (1969), 532–563.[4] M. Auslander, O. Goldman, The Brauer Group of a CommutativeRing , Trans. A.M.S. (1960), 367–409. 2.26[5] G. Azumaya, On maximally central algebras , Nagoya Math. J. (1950), 119–150. 2.26[6] Chenevier, Ga¨etan, The p-adic analytic space of pseudocharacters ofa profinite group and pseudorepresentations over arbitrary rings.
Au-tomorphic forms and Galois representations. Vol. 1, 221–285, LondonMath. Soc. Lecture Note Ser., 414, Cambridge Univ. Press, Cambridge,2014. (document)[7] C. De Concini, C. Procesi,
A characteristic free approach to invarianttheory , Adv. Math. (1976), 330–354. 2.42[8] C. De Concini, C. Procesi, The invariant theory of matrices
A.M.S.University Lecture Series v. , 151 pp. (2017). (document), 1.4, 2.3.1,2.3.1, 2.8, 2.8, 2.13, 2.12, 2.12, 2.12, 2.21, 2.42.1[9] S. Donkin, Invariants of several matrices , Inventiones Mathematicae, , 1992, n. 2, 389–401. (document), 2.8, 2.214610] N. Jacobson,
Lectures in abstract algebra, Volume III–Theory of fieldsand Galois theory , Van Nostrand 1964. 3.16[11] Milne, James S. (1980), ´Etale cohomology , Princeton MathematicalSeries, 33, Princeton University Press. 2.33[12] Nyssen, Louise,
Pseudo-repr´esentations.
Math. Ann. 306 (1996), no.2, 257–283.[13] C. Procesi,
Finite dimensional representations of algebras , Israel J.Math. (1974), 169–182. (document)[14] C. Procesi, The invariant theory of n × n matrices , Adv. Math. (1976), 306–381. 1.4, 1.4, 2.8[15] C. Procesi, Trace identities and standard diagrams , Ring theory. Lec-ture notes in Pure and Appl. Mathematics M. Dekker (1979), pp.191–218.[16] C. Procesi,
A formal inverse to the Cayley Hamilton theorem , J. Al-gebra (1987), 63–74. (document), 2.47.1[17] C. Procesi,
Deformations of Representations , Lecture Notes in Pureand Appl. Math., (1998), 247–276, Dekker. (document)[18] C. Procesi,
Lie Groups, An approach through invariants and represen-tations , Springer Universitext, 2007 pp. xxiv+596,[19] C. Procesi,
Cayley Hamilton algebras , to apear, 2020 2.7, 2.42, 3.36[20] C. Procesi, T –ideals of Cayley Hamilton algebras , 2020,http://arxiv.org/abs/2008.02222 (document), 3.9.1[21] Yu. P. Razmyslov, Identities with trace in full matrix algebras over afield of characteristic zero , (Russian) Izv. Akad. Nauk SSSR Ser. Mat. (1974), 723–756. 1.4[22] Yu. P. Razmyslov, Identities of algebras and their representations , Pro-ceedings of the International Conference on Algebra, Part 2 Novosi-birsk, (1989), 173–192, Contemp. Math., , Part 2, Amer. Math.Soc., Providence, RI, (1992). 4723] Yu. P. Razmyslov,
Trace identities of matrix algebras via a field ofcharacteristic zero , Math. USSR Izvestia (translation). (1974), 727–760. 2.8[24] C. Reutenauer, M.-P. Sch¨utzenberger, A formula for the determinantof a sum of matrices , Lett. Math. Phys. (1987), no. 4, 299–302.2.12[25] N. Roby, Lois polynomes et lois formelles en th´eorie des modules , An-nales Scientifiques de l’´Ecole Normale Sup´erieure. Troisi`eme S´erie, (1963), 213–348. 2.1[26] N. Roby, Lois polynˆomes multiplicatives universelles , Comptes RendusHebdomadaires des S´eances de l’Acad´emie des Sciences. S´eries A et B, n. 19 (1980), A869–A871. 2.1, 2.3.1[27] Saltman, David J.
Lectures on division algebras.
CBMS Regional Con-ference Series in Mathematics, 94. Published by American Mathemat-ical Society, Providence, RI; on behalf of Conference Board of theMathematical Sciences, Washington, DC, 1999. viii+120 pp. 3.9.1[28] F. Vaccarino,
Generalised symmetric functions and invariants of ma-trices , Math. Z. (2008), 509–526. 2.8[29] F. Vaccarino,
Homogeneous multiplicative polynomial laws are deter-minants , J. Pure Appl. Algebra (2009), no. 7, 1283–1289.[30] Ziplies, Dieter,
A characterization of the norm of an Azumaya algebraof constant rank through the divided powers algebra,
Beitr¨age zur Al-gebra und Geometrie, Contributions to algebra and geometry, pp. 53- 70, (1971). 2.26[31] D. Ziplies,
Generators for the divided powers algebra of an algebra andtrace identities , Beitr¨age zur Algebra und Geometrie (1987), 9–27.2.8[32] D. Ziplies, Abelianizing the divided powers algebra of an algebra , J.Algebra (1989), n. 2, 261–274. 2.3.1[33] A. N. Zubkov,
On a generalization of the Razmyslov-Procesi theorem ,Algebra i Logika, Sibirski˘ı Fond Algebry i Logiki. Algebra i Logika,35