ABSTRACT Analyzing square-central elements in central simple algebras of degree 4, we show that e... more ABSTRACT Analyzing square-central elements in central simple algebras of degree 4, we show that every two elementary abelian Galois maximal subfields are connected by a chain of nontrivially-intersecting pairs. Similar results are proved for non-central quaternion subalgebras, and for central quaternion subalgebras when they exist. Along these lines we classify the maximal square-central subspaces. We also show that every two standard quadruples of generators of a biquaternion algebra are connected by a chain of basic steps, in each of which at most two generators are being changed.
International Journal of Algebra and Computation, 2008
This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where ... more This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.
ABSTRACT We describe the multilinear identities of the superalgebra M 2, 1(G) of matrices over th... more ABSTRACT We describe the multilinear identities of the superalgebra M 2, 1(G) of matrices over the Grassmann algebra, in the minimal possible degree, which is 9.
ABSTRACT The Zariski closure of an arbitrary representable (not necessarily associative) algebra ... more ABSTRACT The Zariski closure of an arbitrary representable (not necessarily associative) algebra is studied in the general context of universal algebra, with an application being that the codimension sequence is exponentially bounded.
In a series of papers, we used full quivers as tools in describing PI-varieties of algebras and p... more In a series of papers, we used full quivers as tools in describing PI-varieties of algebras and providing a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. In this paper, utilizing ideas from that work, we give a full exposition of Belov's theorem that relatively free affine PI-algebras over an arbitrary field are representable. (Kemer proved the theorem over an infinite field.)
A set of quadratic $n$-fold Pfister forms is linked if it has a common $(n-1)$-fold factor. We st... more A set of quadratic $n$-fold Pfister forms is linked if it has a common $(n-1)$-fold factor. We study this notion over a field $F$ of characteristic $2$, where one has to distinguish between left- and right-linkage. Every linked set must be ``tight'', namely generate a group of forms in $I_q^nF/I_q^{n+1}F$. Following Sivatzki, who studied triples of quaternion algebras, we associate to any tight set an invariant in $I_q^{n+1}F$, which is zero whenever the set is left-linked. In fact, a left-linked set generates a group of forms in $I_q^nF$; we call such a set ``strongly tight''. We show that a right-linked set is strongly tight if and only if it is pairwise left-linked. For any $n$, we construct a set of $n+1$ quadratic $n$-fold Pfister forms which is strongly tight, and in particular has zero invariant, but does not have any common $1$-fold Pfister factor. For $n = 2$ this answers a problem of Sivatzki on the negative.
ABSTRACT We study the generalized Clifford algebras associated to homogeneous binary forms of pri... more ABSTRACT We study the generalized Clifford algebras associated to homogeneous binary forms of prime degree p, focusing on exponentiation forms of p-central spaces in division algebra.For a two-dimensional p-central space, we make the simplifying assumption that one basis element is a sum of two eigenvectors with respect to conjugation by the other. If the product of the eigenvalues is 1 then the Clifford algebra is a symbol Azumaya algebra of degree p, generalizing the theory developed for p=3p=3. Furthermore, when p=5p=5 and the product is not 1, we show that any quotient division algebra of the Clifford algebra is a cyclic algebra or a tensor product of two cyclic algebras, and every product of two cyclic algebras can be obtained as a quotient. Explicit presentation is given to the Clifford algebra when the form is diagonal.
ABSTRACT Analyzing square-central elements in central simple algebras of degree 4, we show that e... more ABSTRACT Analyzing square-central elements in central simple algebras of degree 4, we show that every two elementary abelian Galois maximal subfields are connected by a chain of nontrivially-intersecting pairs. Similar results are proved for non-central quaternion subalgebras, and for central quaternion subalgebras when they exist. Along these lines we classify the maximal square-central subspaces. We also show that every two standard quadruples of generators of a biquaternion algebra are connected by a chain of basic steps, in each of which at most two generators are being changed.
International Journal of Algebra and Computation, 2008
This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where ... more This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.
ABSTRACT We describe the multilinear identities of the superalgebra M 2, 1(G) of matrices over th... more ABSTRACT We describe the multilinear identities of the superalgebra M 2, 1(G) of matrices over the Grassmann algebra, in the minimal possible degree, which is 9.
ABSTRACT The Zariski closure of an arbitrary representable (not necessarily associative) algebra ... more ABSTRACT The Zariski closure of an arbitrary representable (not necessarily associative) algebra is studied in the general context of universal algebra, with an application being that the codimension sequence is exponentially bounded.
In a series of papers, we used full quivers as tools in describing PI-varieties of algebras and p... more In a series of papers, we used full quivers as tools in describing PI-varieties of algebras and providing a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. In this paper, utilizing ideas from that work, we give a full exposition of Belov's theorem that relatively free affine PI-algebras over an arbitrary field are representable. (Kemer proved the theorem over an infinite field.)
A set of quadratic $n$-fold Pfister forms is linked if it has a common $(n-1)$-fold factor. We st... more A set of quadratic $n$-fold Pfister forms is linked if it has a common $(n-1)$-fold factor. We study this notion over a field $F$ of characteristic $2$, where one has to distinguish between left- and right-linkage. Every linked set must be ``tight'', namely generate a group of forms in $I_q^nF/I_q^{n+1}F$. Following Sivatzki, who studied triples of quaternion algebras, we associate to any tight set an invariant in $I_q^{n+1}F$, which is zero whenever the set is left-linked. In fact, a left-linked set generates a group of forms in $I_q^nF$; we call such a set ``strongly tight''. We show that a right-linked set is strongly tight if and only if it is pairwise left-linked. For any $n$, we construct a set of $n+1$ quadratic $n$-fold Pfister forms which is strongly tight, and in particular has zero invariant, but does not have any common $1$-fold Pfister factor. For $n = 2$ this answers a problem of Sivatzki on the negative.
ABSTRACT We study the generalized Clifford algebras associated to homogeneous binary forms of pri... more ABSTRACT We study the generalized Clifford algebras associated to homogeneous binary forms of prime degree p, focusing on exponentiation forms of p-central spaces in division algebra.For a two-dimensional p-central space, we make the simplifying assumption that one basis element is a sum of two eigenvectors with respect to conjugation by the other. If the product of the eigenvalues is 1 then the Clifford algebra is a symbol Azumaya algebra of degree p, generalizing the theory developed for p=3p=3. Furthermore, when p=5p=5 and the product is not 1, we show that any quotient division algebra of the Clifford algebra is a cyclic algebra or a tensor product of two cyclic algebras, and every product of two cyclic algebras can be obtained as a quotient. Explicit presentation is given to the Clifford algebra when the form is diagonal.
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Papers by U. Vishne