A subalgebra of the full matrix algebra Mn(K),K a field, satisfying the identity x1,y1x2,y2⋯xq,yq=0 is called a Dq subalgebra of Mn(K). In the paper we deal with the structure, conjugation and ...isomorphism problems of maximal Dq subalgebras of Mn(K).
We show that a maximal Dq subalgebra A of Mn(K) is conjugated with a block triangular subalgebra of Mn(K) with maximal commutative diagonal blocks. By analysis of conjugations, the sizes of the obtained diagonal blocks are uniquely determined. It reduces the problem of conjugation of maximal Dq subalgebras of Mn(K) to the analogous problem in the class of commutative subalgebras of Mn(K). Further examining conjugations, in case A is contained in the upper triangular matrix algebra Un(K), we prove that A is already in a block triangular form.
We consider the isomorphism problem in a certain class of maximal Dq subalgebras of Mn(K) which contain all Dq subalgebras of Mn(K) with maximum dimension. In case K is algebraically closed, we invoke Jacobson's characterization of maximal commutative subalgebras of Mn(K) with maximum (K-)dimension to show that isomorphic subalgebras in this class are already conjugated. To illustrate it, we invoke results from 19 and find all isomorphism (equivalently conjugation) classes of Dq subalgebras of Mn(K) with maximum possible dimension, in case K is algebraically closed.
Let F be a field of characteristic zero and M2,1(F) the algebra of 3×3 matrices over F endowed with non-trivial Z2-grading. The transpose involution t on M2,1(F) preserves the homogeneous components ...of the grading and so, we consider (M2,1(F),t) as a superalgebra with graded involution. We study the (Z2,⁎)-identities of this algebra and make explicit the decomposition of the space of multilinear (Z2,⁎)-identities into the sum of irreducibles under the action of the group (Z2×Z2)≀Sn in order to determine all the irreducible characters appearing with non-zero multiplicity in the decomposition of the ⁎-graded cocharacter of (M2,1(F),t). Along the way, using the representation theory of the general linear group, we determine all the (Z2,⁎)-identities of (M2,1(F),t) up to degree 3.
Let UJ2 be the Jordan algebra of 2×2 upper triangular matrices. This paper is devoted to continue the description given by recent works about the gradings and graded polynomial identities on UJ2(K) ...when K is an infinite field of characteristic 2. Due to the definition of Jordan algebras in terms of the commutative and Jordan identities being unsuitable in characteristic 2, we decided to study the gradings of the non-associative commutative algebra of 2×2 upper triangular matrices UT2=(UT2(K),∘) with the product x∘y=xy+yx. More precisely, fixed K a field of characteristic 2, we classify the gradings of (UT2(K),∘) and also, given an arbitrary grading, we calculate the generators of the ideals of graded identities and give a positive answer to the Specht property for the variety of commutative algebras generated by (UT2(K),∘) in each grading when K is infinite.
For an arbitrary q≥2, we find an upper bound for the dimension of a subalgebra of the full matrix algebra Mn(K) over an arbitrary field K satisfying the identityx1,y1,z1⋅x2,y2,z2⋅⋯⋅xq,yq,zq=0, and we ...show that this upper bound is sharp by presenting an example in block triangular form of a subalgebra of Mn(K) with dimension equal to the obtained upper bound. We apply this result to Lie solvable algebras of index 2, i.e., algebras satisfying the identity x1,y1,x2,y2=0. To be precise, for n≤4, we find the sharp upper bound for the dimension of a Lie solvable subalgebra of Mn(K) of index 2, and for n>4, we obtain the relatively tight (at least for small values of n>4) interval2+⌊3n28⌋,2+⌊5n212⌋ for the maximum dimension of a Lie solvable subalgebra of Mn(K) of index 2, the exact value of which is not known.
We exhibit a class C of finite dimensional algebras with superinvolution over an algebraically closed field of characteristic zero, with the remarkable property that each member of C generates a ...minimal variety of algebras with superinvolution. This sums up to the fact that any affine minimal variety of algebras with superinvolution is generated by a suitable member of C, thus providing a complete characterization of the affine minimal varieties of algebras with superinvolution.
Consider a prime ring
R
which is noncommutative with the maximal left ring of quotients of
R
denoted by
Q
m
l
(
R
)
and the extended centroid
C
. An additive map
G
:
R
→
Q
m
l
(
R
)
satisfying
G
(
g
...2
)
+
η
g
G
(
g
)
∈
C
for all
g
∈
R
, where
η
∈
C
is called a weak Jordan right η-centralizer. In this paper, we establish the characterization of weak Jordan right η-centralizers in prime rings. As an application, we describe X-generalized skew derivations which behave like weak Jordan right η-centralizers.
To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities ...satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞cnsup(A)n is an integer that can be explicitly computed.
Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then(1)C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group.
As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that limn→∞logncnsup(A)expsup(A)n exists and is a half integer.
We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks ...of a finite dimensional algebra with involution *. To any star-algebra A is attached a numerical sequence c_n^*(A), n\ge 1, called the sequence of *-codimensions of A. Its asymptotic is an invariant giving a measure of the *-polynomial identities satisfied by A. It is well known that for a PI-algebra such a sequence is exponentially bounded and \exp ^*(A)=\lim _{n\to \infty }\sqrt n{c_n^*(A)} can be explicitly computed. Here we prove that if A is a star-fundamental algebra, <TD NOWRAP ALIGN="CENTER">\displaystyle C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n, <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> where C_1>0,C_2, t are constants and t is explicitly computed as a linear function of the dimension of the skew semisimple part of A and the nilpotency index of the Jacobson radical of A. We also prove that any finite dimensional star-algebra has the same *-identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in J. Algebra 383 (2013), pp. 144-167 we get that if A is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, \lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n} exists and is an integer or half an integer.
Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that
for ...all
where
are fixed positive integers. Then, one of the following conditions holds:
there exist
such that
and
for all
m = s, n = t and there exist
and
such that g(x) = xa and
for all
C is a finite field,
the
matrix ring over C for some integer
and there exist
such that g(x) = xa and h(x) = ux for all
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