APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

A Decomposition Theorem for CK1 of Central Simple Algebras

Let A be a central simple algebra over its center F. Define CK₁ A = Coker(K₁ F → K₁ A). We prove that if A and B are F-central simple algebras of coprime degrees, then CK₁(A? _F B) = CK₁ A × CK₁ B.     

Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p ₁ ? p ₂ ? ··· ? p _n be a chain of nonmaximal prime ideals of T such that T _{p n} is a regular local ring. We construct a chain of excellent local domains A _n  ? A _n?1 ? ··· ? A ₁ such that for each 1 ≤ i ≤ n, the completion of A _i is T, the generic formal fiber of A _i is local with maximal ideal p _i , and if I is a nonzero ideal of A _i then A _i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht _T Q, then there is a chain of excellent local domains B ₀ ? B ₁ ? ··· ? B _h  ? T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B _i ) = i, the completion of B _i is isomorphic to T[[X ₁, X ₂,…, X _i ]] where the X _j 's are indeterminants, and the formal fiber of Q ∩ B _i is local.     

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A?B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N _v  = {f ∈ D[X] | c(f)^?1 = D}. In this article, we study when the Nagata ring D[X] _N (more generally, D[X] _{N v} ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] _{N v} is a GPVD if and only if D is a t-GPVD and D[X] _{N v} has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] _{N v} is a GPVD. As a corollary, we have that D[X] _N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] _{N v} is a GPVD.     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

The eigenvalues of the sum of matrices with prescribed invariant polynomials *

Let A and B be n×n matrices over a field F, and c ₁,…,c_n ∈F. We give a sufficient condition for the existence of matrices A' and B' similar to A and B, respectively, such that A' + B' has eigenvalues c ₁,…,c_n .     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.

     

Lie Triple Derivable Mappings on Rings

Let ? be a ring containing a nontrivial idempotent. In this article, under a mild condition on ?, we prove that if δ is a Lie triple derivable mapping from ? into ?, then there exists a Z _{A, B} (depending on A and B) in its centre 𝒵(?) such that δ(A + B) = δ(A) + δ(B) + Z _{A, B} . In particular, let ? be a prime ring of characteristic not 2 containing a nontrivial idempotent. It is shown that, under some mild conditions on ?, if δ is a Lie triple derivable mapping from ? into ?, then δ = D + τ, where D is an additive derivation from ? into its central closure T and τ is a mapping from ? into its extended centroid 𝒞 such that τ(A + B) = τ(A) + τ(B) + Z _{A, B} and τ([[A, B], C]) = 0 for all A, B, C ∈ ?.     

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ? = ?_{?, ?1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 ? regular.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings

Let (A, B) be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and T: A ∪ B → A ∪ B be a continuous and asymptotically relatively nonexpansive map. We prove that there exists x ∈ A ∪ B such that ‖x ? Tx‖ = dist(A, B) whenever T(A) ? B, T(B) ? A. Also, we establish that if T(A) ? A and T(B) ? B, then there exist x ∈ A and y ∈ B such that Tx = x, Ty = y and ‖x ? y‖ = dist(A, B). We prove the aforementioned results when the pair (A, B) has the rectangle property and property UC. In the case of A = B, we obtain, as a particular case of our results, the basic fixed point theorem for asymptotically nonexpansive maps by Goebel and Kirk.     

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ?: T → K: = T/M the natural surjection, and R the pullback ?^?1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x ₁]]…[x _n ]] to be catenarian, where each [x _i ]] is fixed as either [x _i ] or [[x _i ]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x ₁]]…[x _n ]]) → Spec(k[x ₁]]…[x _n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x ₁]]…[x _n ]] is catenarian.     

On the Equations De.ning Curves in a Polynomial Algebra

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I ² is free A[T]/I-module of rank n and (A[T]/I) is torsion in K ₀(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers.     

A determinantal version of the frobenius-könig theorem

Let F be a field and let {d ₁,…,d_k } be a set of independent indeterminates over F. Let A(d ₁,…,d_k ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d ₁,…,d_k }. We assume that no d ¹ appears twice in A(d ₁,…,d_k ). We show that if det A(d ₁,…,d_k ) = 0 then A(d ₁,…,d_k ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ? p ? 1: for some positive integer p.     

Closed Polynomials in Polynomial Rings over Unique Factorization Domains

Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[f], where f ∈ R[x ₁,…, x _n ]?R. For a polynomial f ∈ R[x ₁,…, x _n ]?R, we prove that R[f] is a maximal element of M(R, n) if and only if it is integrally closed in R[x ₁,…, x _n ] and Q(R)[f] ∩ R[x ₁,…, x _n ] = R[f]. Moreover, we prove that, in the case where the characteristic of R equals zero, R[f] is a maximal element of M(R, n) if and only if there exists an R-derivation on R[x ₁,…, x _n ] whose kernel equals R[f].     

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B _n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB _n contains an ideal K _n of dimension (2 ⁿ  ? 1)², and we construct an explicit isomorphism of K _n with the matrix algebra M _{2 ⁿ ?1}(F).     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

An Explicit Computation of a Family of Trivializing Étale Covers

We explicitly compute étale covers of the smooth Fermat curves Y _p+1 = Proj k[u, v, w]/(u ^p+1 + v ^p+1 ? w ^p+1) which trivialize the vector bundles Syz(u ², v ², w ²)(3), where k is a field of characteristic p ≥ 3.