Let A be a central simple algebra with involution sigma of first or second kind. Let v be a valuation on the sigma-fixed part F of Z(A). A sigma-special v-gauge g on A is a kind of value function on A extending v on F, such that g(sigma(x) x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a sigma-special v-gauge g if and only if sigma is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a sigma-special v-gauge g if and only if sigma remains anisotropic after scalar extension from F to the Henselization of F re v; when this occurs, g is the unique sigma-invariant v-gauge on A.
We extend the classical links between valuations and orderings on fields to Tignol-Wadsworth gauges and positive cones on finite-dimensional simple algebras with involution. We also study the compatibility of gauges and positive cones, and prove lifting results in the style of the Baer-Krull theorem for fields.
Associative algebras with involution over a field of zero characteristic are considered. It is proved that in this case for any finitely generated associative algebra with involution there exists a finite dimensional algebra with involution which satisfies exactly the same identities with involution.
We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings. If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_L$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_L$ generated by $U(L)$. Let $k$ be a field of characteristic zero and $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group a…
We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra $M_2(F)$, endowed with an orthogonal involution. Using the theory of gauges, developed by Tignol-Wadsworth, we construct examples of algebras isomorphic to a tensor product of quaternion algebras with $k$ split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with $k$ factors $M_2…
We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra $Q$. We classify these algebras in degree~4 and give an example of such a division algebra with orthogonal involution of degree~8 that does not contain $Q$ with its canonical involution, even though it contains $Q$ and is totally decomposable into a tensor product of quaternion algebras with involution.
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution, is shown to have an affirmative answer.
We consider associative algebras with involution graded by a finite abelian group G over a field of characteristic zero. Suppose that the involution is compatible with the grading. We represent conditions permitting PI-representability of such algebras. Particularly, it is proved that a finitely generated (Z/qZ)-graded associative PI-algebra with involution satisfies exactly the same graded identities with involution as some finite dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4. This is an analogue of the theorem of A.Kemer for ordinary identities, and an extension …
We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians and some other related invariants is studied. Also, in arbitrary characteristic, a criterion is obtained for an orthogonal involution on a biquaternion algebra to be metabolic.
We consider associative algebras with involution over a field of characteristic zero. We proved that any algebra with involution satisfies the same identities with involution as the Grassmann envelope of some finite dimensional $Z_4$-graded algebra with graded involution. As a consequence we obtain the positive solution of the Specht problem for identities with involution: any associative algebra with involution over a field of characteristic zero has a finite basis of identities with involution. These results are analogs of theorems of A.R.Kemer for ordinary identities.
A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogonal type.
The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discussed.
Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$ satisfying $(A_g)^*\subseteq A_{g^{-1}}$, for all $g\in G$. We describe such involutions for the full $n\times n$ matrix algebra over $F$ and for the algebra of $n\times n$ upper triangular matrices.
We introduce positive cones on algebras with involution. These allow us to prove analogues of Artin's solution to Hilbert's 17th problem, the Artin-Schreier theorem characterizing formally real fields, and to define signatures with respect to positive cones. We consider the space of positive cones of an algebra with involution and investigate its topological properties, showing in particular that it is a spectral space. As an application we solve the problem of the existence of positive involutions.
In this paper we describe graded automorphisms and antiautomorphisms of finite order on matrix algebras endowed with a group gradings by a finite abelian group over an arbitrary algebraically closed field of charcteristic different from 2.
A classification of (countable) direct limits of finite dimensional involution simple associative algebras over an algebraically closed field of arbitrary characteristic is obtained. This also classifies the corresponding dimension groups. The set of invariants consists of two supernatural numbers and two real parameters.
We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.
Let F be a field of characteristic different from $2$, and let $UT_2(F)$ be the algebra of $2\times 2$ upper triangular matrices over $F$. For every involution of the first kind on $UT_2(F)$, we describe the set of all $*$-central polynomials for this algebra.
An involution $#$ on an associative ring $R$ is \textit{formally real} if a sum of nonzero elements of the form $r^# r$ where $r \in R$ is nonzero. Suppose that $R$ is a central simple algebra (i.e. $R=M_n(D)$ for some integer $n$ and central division algebra $D$) and $#$ is an involution on $R$ of the form $r^# = a^{-1} r^\ast a$, where $\ast$ is some transpose involution on $R$ and $a$ is an invertible matrix such that $a^\ast=\pm a$. In section 1 we characterize formal reality of $#$ in terms of $a$ and $\ast|_D$. In later sections we apply this result to the study of formal reality of invo…
We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution.
We present new additive results for the core inverse in a Banach algebra with involution. We obtain necessary and sufficient conditions under which the sum of two core invertible elements in a Banach algebra with involution is core invertible. Then we apply our results to block complex matrices and obtain certain conditions under which a block complex is core invertible. These generalize many known results, e.g.,~\cite[Theorem 4.3]{XCZ}, ~\cite[Theorem 2.5]{XS}.
We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra $I(X,\mathbb{K})$, where $X$ is a finite poset (star, Y and Rhombuses) and $\mathbb{K}$ is a finite field of characteristic different from 2. Using the techniques describing here we show an algorithm to calculate the number of involutions on any finite poset.
We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of $*$-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional $*$-PI-exponent. We also construct a family of infinite-dimensional algebras $C_α$ such that ${\rm exp}^*(C_α)$ does not exist.
In earlier work we developed the theory of signatures of hermitian forms over algebras with involution with respect to orderings on the base field of the algebra and obtained in particular that the total signature of a hermitian form is a continuous function from the space of orderings of that field to $\mathbb{Z}$. In this note we give another presentation of signatures and also introduce and study the stability index of algebras with involution.
We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.
Let $k$ be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free $PGL_n$-varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of the semidirect product $P_{n,τ}:=PGL_n\rtimes<τ>$, where $τ$ is the automorphism of $\PGLn$ given by $τ(h)=(h^{-1})^transpose$. Basic properties of an involution are described in terms of the action of $P_{n,τ}$ on the associated variety, and in particular in…
Let $FI(X,K)$ be the finitary incidence algebra of a non-connected partially ordered set $X$ over a field $K$ of characteristic different from $2$. For the case where every multiplicative automorphism of $FI(X,K)$ is inner, we present necessary and sufficient conditions for two involutions on $FI(X,K)$ to be equivalent.
We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.
Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra.
In characteristic two, it is shown that a central simple algebra of degree equal to a power of two with anisotropic orthogonal involution is totally decomposable, if it becomes either anisotropic or metabolic over all extensions of the ground field. A similar result is obtained for the case where this algebra with involution is Brauer-equivalent to a quaternion algebra and it becomes adjoint to a bilinear Pfister form over all splitting fields of the algebra.
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.
In 1976 Procesi and Schacher developed an Artin-Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.
The `transcendental methods' in the algebraic theory of quadratic forms are based on two major results, proved in the 60's by Cassels and Pfister, and known as the representation and the subform theorems. A generalization of the representation theorem was proven by Jean-Pierre Tignol in 1996, in the setting of central simple algebras with involution. This paper studies the subform question for orthogonal involutions. A generic characterization of direct summands is given; an analogue of the subform theorem is proven for division algebras and algebras of index at most 2.
An involution of a real commutative algebra $A$ is a real-linear homomorphism $f : A \rightarrow A$ such that $f^2 = \mathrm{Id}$. We show that there are six involutions of the algebra of bicomplex numbers, contrary to the actual number of four stated in the literature. We also characterize $n$-involutions satisfying the additional property $f^n = \mathrm{Id}$ for some integer $n \geq 2$. We show there are eight $n$-involutions and they occur only for $n = 2$ and $n= 4$. We use our result to give a new characterization of the invertible elements of the algebra of bicomplex numbers.
Let $\mathbb{F}$ be a field of characteristic $p$, and let $UT_n(\mathbb{F})$ be the algebra of $n \times n$ upper triangular matrices over $\mathbb{F}$ with an involution of the first kind. In this paper we describe: the set of all $*$-central polynomials for $UT_n(\mathbb{F})$ when $n\geq 3$ and $p\neq 2$ ; the set of all $*$-polynomial identities for $UT_3(\mathbb{F})$ when $\mathbb{F}$ is infinite and $p>2$.
Let $1\leq m \leq n$ be integers with $m|n$ and $\cat{Alg}_{n,m}$ the class of central simple algebras of degree $n$ and exponent dividing $m$. In this paper, we find new, improved upper bounds for the essential dimension and 2-dimension of $\cat{Alg}_{n,2}$. In particular, we show that $\ed_{2}(\cat{Alg}_{16,2})=24$ over a field $F$ of characteristic different from 2.
Every finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each element a of A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero.
We exploit various inclusions of algebraic groups to give a new construction of groups of type E8, determine the Killing forms of the resulting E8's, and define an invariant of central simple algebras of degree 16 with orthogonal involution "in I^3", equivalently, groups of type D8 with a half-spin representation defined over the base field. The determination of the Killing form is done by restricting the adjoint representation to various twisted forms of PGL2 and requires very little computation. An appendix by Kirill Zainoulline contains a type of "index reduction" result for groups of type …
Following [Botur, M., Chajda, I., Halaš, R.: Are basic algebras residuated structures?, Soft Comput. 14 (2010), 251-255] we discuss the connections between left-residuated partially ordered groupoids and the so-called basic algebras, which are a non-commutative and non-associative generalization of MV-algebras and orthomodular lattices.
Let $R$ be an algebra over a commutative ring $k$. Suppose that $R$ is endowed with a descending filtration indexed on an ordered group $(G,<)$ such that the restriction to $k$ is positive. We show that the existence of free algebras on a certain set of generators in the induced graded ring $grad(R)$ implies the existence of free group algebras in $R$. Our best results are obtained for division rings endowed with a valuation.
Let $K$ be a field and $X$ a partially ordered set (poset). Let $FI(X,K)$ and $I(X,K)$ be the finitary incidence algebra and the incidence space of $X$ over $K$, respectively, and let $D(X,K)=FI(X,K)(+)I(X,K)$ be the idealization of the $FI(X,K)$-bimodule $I(X,K)$. In the first part of this paper, we show that $D(X,K)$ has an anti-automorphism (involution) if and only if $X$ has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on $D(X,K)$. In the second part, we obtain the classification of involutions on $D(X,K)$ to the case when …
