On the *-Polynomial Identities of a Class of *-Minimal Algebras

Let F be an infinite field. We consider certain block-triangular algebras with involution U _n , with n ∈ ?, having minimal *-exponent. We describe their *-polynomial identities, and in case char.F = 0, their structure as a T _*-ideal under the action of general linear groups. These goals are achieved by means of Y-proper polynomials. We also compute explicitly the irreducible modules occurring in the decomposition of B _Y (U ₃) and their multiplicities.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

The graded Gelfand--Kirillov dimension of verbally prime algebras

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M _n (F), M _n (E) and M _a,b (E) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand--Kirillov dimensions.     

Central polynomials for matrices over finite fields

Let c(x ₁,?…?,?x _d ) be a multihomogeneous central polynomial for the n?×?n matrix algebra M _n (K) over an infinite field K of positive characteristic p. We show that there exists a multihomogeneous polynomial c ₀(x ₁,?…?,?x _d ) of the same degree and with coefficients in the prime field 𝔽 _p which is central for the algebra M _n (F) for any (possibly finite) field F of characteristic p. The proof is elementary and uses standard combinatorial techniques only.     

Identities of Group Algebras

We consider polynomial identities of group algebras over a field F of characteristic zero. We prove that any PI group algebra satisfies the same identities as a matrix algebra M _n (F ), where n is the maximal degree of finite dimensional representations of the group over algebraic extensions of F.     

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

A problem on conjugacy of matrices

Let F be an infinite field and n?12. Then the number of conjugacy classes of the upper triangular nilpotent matrices in M_n(F) under action by the subgroup of GL_n(F) consisting of all the upper triangular matrices is infinite.     

The elementary equivalence of free algebras in cantor varieties

We study properties of free algebras in the Cantor varieties C_m,n. A free algebra of rank r in C_m,n is denoted F_{C m,n}(r). We argue that the following hold: (1) any two C_m,n-free algebras F_{C m,n}(r) and F_{C m,n}(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, r≥m, and s≥m, are elementary equivalent; (2) any two C_m,n-free algebras F_{C m,n}(r) and F_{C m,n}(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, are universally equivalent, that is, share one ∀-theory; (3) an elementary theory Th(F_{C m,n}(r)) for an arbitrary C_m,n-free algebra of (finite or infinite) rank r, treated in a signature Ω, is decidable; (4) an elementary theory Th(K) for an arbitrary nonempty class of free algebras in C_m,n, treated in a signature Ω, is decidable. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 228–248, March–April, 1999.     

Involutions in triangular groups

We describe involutions, i.e. elements of order 2, in the groups T _n (K) – of upper triangular matrices of dimension n (n?∈??), and T _∞(K) – of upper triangular infinite matrices, where K is a field of characteristic different from 2. Using the obtained result, we give a formula for the number of all involutions in T _n (K) in the case when K is a finite field.     

*-Polynomial Identities of Matrices with the Symplectic Involution: The Low Degrees

Abstract

The *-polynomial identities of minimal degree of M _n (F) are determined for n = 2, 4, * the symplectic involution.     

Commuting Involution

We are given a semiprime unital ring A with * such that x*x = xx* for all elements x of A. We will show that both elements x + x* and xx* are central elements. In the case in which A is a quaternion algebra over a field F in the sense given by Albert, we show that * is unique and coincides with the canonical involution. We also provide specific constructions of quaternion division algebras A with canonical involution over a field F of one of the following types: (i) F is a function field in two variables over a ground field of unspecified characteristic; (ii) F is a function field over the Galois field GF(2ⁿ); and (iii) F is a function field over the Galois field GF(pⁿ) where p is an odd prime number and n is a natural number.     

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))?Γ(M _m (E)), then n = m and |F|=|E|.     

Graded central polynomials for the matrix algebra of order two

Let K be an infinite integral domain, and let A = M ₂(K) be the matrix algebra of order two over K. The algebra A can be given a natural -grading by assuming that the diagonal matrices are the 0-component while the off-diagonal ones form the 1-component. In this paper we study the graded identities and the graded central polynomials of A. We exhibit finite bases for these graded identities and central polynomials. It turns out that the behavior of the graded identities and central polynomials in the case under consideration is much like that in the case when K is an infinite field of characteristic 0 or p > 2. Our proofs are characteristic-free so they work when K is an infinite field, char K = 2. Thus we describe finite bases of the graded identities and graded central polynomials for M ₂(K) in this case as well. A. Krasilnikov has been partially supported by CNPq and FINATEC.     

Additive Maps on Matrix Algebras Preserving Invertibility or Singularity

We characterize the additive singularity preserving almost surjective maps on Mn （F）, the algebra of all n×n matrices over a field F with char F=0. We also describe additive invertibility preserving surjective maps on Mn （F） and give examples showing that all the assunlptions in these two theorems are indispensable.     

Polynomial Identities of RA and RA2 Loop Algebras

Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z _F over F. For RA2 loops of type M(Dih(A), ^?1,g ₀), we prove that the loop algebra is in the variety generated by the algebra  ₃ which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, ^?1,g ₀), where G is a non-Abelian group of exponent 4 having exactly 2 squares.     

A presentation of the group Sl * (2, A), A a simple artinian ring with involution

Let (A,*) be an involutive ring. Then the groups Sl _*(2, A), are a non commutative version of the special linear groups Sl(2, F) defined over a field F. In particular, if A = M(n, F) and * is transposition, then Sl _*(2, M _n (F)) = Sp(2n, F). The above groups were defined by Pantoja and Soto-Andrade, and a set of generators for the group SSl _*(2, A) (which is either Sl _*(2, A) or a index 2 subgroup of Sl _*(2, A)) was given in the case when A is an artinian ring. In this paper, we prove that the mentioned generators provide a presentation of the mentioned groups in the case of simple artinian rings.Partially supported by FONDECYT project 1030907 and Pontificia Universidad Católica de Valparaíso     

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom _KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl ₂(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl ₂(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl ₂(K).     

On Commuting Graphs of Finite Matrix Rings

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M _n (F)), for a finite field F and n ≥ 2, then R ≈ M _n (F). Here we prove this conjecture when n = 2.     

Inverse-preserving Linear Maps Between Spaces of Matrices over Fields

Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.