Let D be a Noetherian infinite integral domain, denote by M2(D) and by sl2(D) the 2×2 matrix algebra and the Lie algebra of the traceless matrices in M2(D), respectively. In this paper we study the ...weak polynomial identities for the natural grading by the cyclic group Z2 of order 2 on M2(D) and on sl2(D). We describe a finite basis of the graded polynomial identities for the pair (M2(D),sl2(D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M2(D),sl2(D)), is finitely generated. The polynomial identities for M2(D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra sl2(D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M2(D),gl2(D)) were also described, by Krasilnikov and the second author.
In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets.
Asymptotic codimensions of Mk(E) Berele, Allan; Regev, Amitai
Advances in mathematics (New York. 1965),
03/2020, Letnik:
363
Journal Article
Recenzirano
We show that the codimension sequence of the algebra of k×k matrices over the Grassmann algebra, cn(Mk(E)), is asymptotic to αn1−k22(2k2)n, where α is an undetermined constant.
Graded A-identities for M 1,1 (E) Augusto Naves, Fernando; Luiz Talpo, Humberto
Linear & multilinear algebra,
12/19/2022, Letnik:
70, Številka:
20
Journal Article
Recenzirano
Let F be a field of characteristic 0, E be the unitary infinite dimensional Grassmann algebra over F and consider the algebra
with its natural
-grading. We describe the graded A-identities for
and we ...compute its graded A-codimensions.
Let A and B be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose A and B are graded by a semigroup S so that the graded identical relations of A ...are the same as those of B. Then A is isomorphic to B as an S-graded algebra.
Let so3(K) be the Lie algebra of 3×3 skew-symmetric matrices over a field K of characteristic 0. The ideal I(M3(K),so3(K)) of the weak polynomial identities of the pair (M3(K),so3(K)) consists of the ...elements f(x1,…,xn) of the free associative algebra K〈X〉 with the property that f(a1,…,an)=0 in the algebra M3(K) of all 3×3 matrices for all a1,…,an∈so3(K). The generators of I(M3(K),so3(K)) were found by Razmyslov in the 1980s. In this paper the cocharacter sequence of I(M3(K),so3(K)) is computed. In other words, the GLp(K)-module structure of the algebra generated by p generic skew-symmetric matrices is determined. Moreover, the same is done for the closely related algebra of SO3(K)-equivariant polynomial maps from the space of p-tuples of 3×3 skew-symmetric matrices into M3(K) (endowed with the conjugation action). In the special case p=3 the latter algebra is a module over a 6-variable polynomial subring in the algebra of SO3(K)-invariants of triples of 3×3 skew-symmetric matrices, and a free resolution of this module is found. The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.
Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory ...and the sequence of ⁎-graded codimensions {cngri(A)}n≥1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T2⁎-equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums.
Let R be a unitary associative algebra over a field. We call an algebra A a generalized R-algebra when A is endowed with an R-module action with the property that, for each r∈R, there exists finitely ...many elements r+=(r1+,r2+)∈R2 and r−=(r1−,r2−)∈R2 such that, for all a1,a2∈A,r⋅(a1a2)=∑r+(r1+⋅a1)(r2+⋅a2)+∑r−(r2−⋅a2)(r1−⋅a1). Suppose an associative generalized R-algebra A satisfies an identical relation of the formx1⋯xd−∑1≠σ∈Sd∑r(r1⋅xσ(1))⋯(rd⋅xσ(d))≡0, where Sd denotes the symmetric group of degree d and the inner sum runs over finitely many r=(r1,…,rd)∈Rd. We prove: if the algebra of endomorphisms on A defined by the action of R is m-dimensional, then A satisfies a classical polynomial identity of degree bounded by an explicit function of d and m only. We also prove the analogous result holds when A is a Lie algebra, thus extending a collection of results in associative and Lie PI-theory.
Let K be a field (finite or infinite) of char(K)≠2 and let UT2(K) be the 2×2 upper triangular matrix algebra over K. If ⋅ is the usual product on UT2(K) then with the new product a∘b=(1/2)(a⋅b+b⋅a) ...we have that UT2(K) is a Jordan algebra, denoted by UJ2=UJ2(K). In this paper, we describe the set I of all polynomial identities of UJ2 and a linear basis for the corresponding relatively free algebra. Moreover, if K is infinite we prove that I has the Specht property. In other words I, and every T-ideal containing I, is finitely generated as a T-ideal.
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