The graded identities of upper triangular matrices of size two

Valenti, A.

doi:10.1016/s0022-4049(01)00169-4

Cited by 38 publications

(26 citation statements)

References 10 publications

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“…In [20] we classified the G-gradings on U T 2 . It turns out that if G is any (non-necessarily abelian) group, then any G-grading on U T 2 is elementary and can be induced by a pair (1, g), for some g ∈ G. Let e ij be the usual matrix units.…”

Section: Classifying Varieties Of Almost Polynomial Growthmentioning

confidence: 99%

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Group graded algebras and almost polynomial growth

Valenti¹

2011

Journal of Algebra

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Let F be a field of characteristic 0, G a finite abelian group and A a\ud G-graded algebra. We prove that A generates a variety of G-graded\ud algebras of almost polynomial growth if and only if A has the same\ud graded identities as one of the following algebras:\ud (1) FCp , the group algebra of a cyclic group of order p, where p\ud is a prime number and p | |G|;\ud (2) UTG\ud 2 (F ), the algebra of 2×2 upper triangular matrices over F\ud endowed with an elementary G-grading;\ud (3) E, the infinite dimensional Grassmann algebra with trivial Ggrading;\ud (4) in case 2 | |G|, EZ2 , the Grassmann algebra with canonical Z2-\ud grading

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Section: Classifying Varieties Of Almost Polynomial Growthmentioning

confidence: 99%

“…In [20] we computed the Z 2 -cocharacter of U T Z 2 2 and showed that U T Z 2 2 generates a variety of Z 2 -graded algebras of almost polynomial growth. We point out that, by looking at the decomposition into homogeneous spaces, it is readily seen that [20] actually shows the following:…”

Section: Classifying Varieties Of Almost Polynomial Growthmentioning

confidence: 99%

Group graded algebras and almost polynomial growth

Valenti¹

2011

Journal of Algebra

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“…By evaluating y 1 = 1 F , y 2 = a, and y 3 = b we get f (n−2,1,1) = ab −ba = 0 which is a contradiction. Now it is immediate to see that [y 1 , y 2 ][y 3 , y 4 ] ≡ 0 is an identity of A 1 and A 2 and the proof is complete [32]. 2…”

Section: Classifying Varieties Of Slow Growthmentioning

confidence: 92%

“…If g = 1 G the result is obvious since [y 1 , y 2 ] ≡ 0, z g 1 z g 2 ≡ 0 and tĝ 1 ≡ 0 are identities for A. Therefore A ∈ var(UT g 2 ) [32]. Suppose now g = 1 G .…”

Section: Classifying Varieties Of Slow Growthmentioning

confidence: 96%

Graded algebras with polynomial growth of their codimensions

Koshlukov

Mattina

2015

Journal of Algebra

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Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with invo-lution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group. §1. Introduction Let F be a field of characteristic zero and F X = F x 1 , x 2 ,. .. the free algebra of countable rank over F. If A is a PI-algebra over F , that is, an algebra satisfying a polynomial identity, we let Id(A) be the T-ideal of F X of identities of A. It is well known that Id(A) is completely determined by the multilinear polynomials it contains; if V n = Span{x σ(1) · · · x σ(n) | σ ∈ S n } is the space of multilinear poly-nomials in x 1 ,. .. , x n , then the sequence c n (A) = dim F Vn Vn∩Id(A) , n = 1, 2,. .. , is called the sequence of codimensions of A and it is an important numerical invariant of Id(A). It was proved by Regev in [R] that for any PI-algebra A, c n (A) is exponentially bounded, i.e., there exist constants a, α > 0 such that c n (A) ≤ aα n for all n. In this paper we study algebras A whose codimension sequence is polynomially bounded i.e., such that for all n, c n (A) ≤ an t for some constants a, t. Kemer in [K1] gave a characterization of such T-ideals in the language of the representation theory of S n. It also follows from [K2] that if c n (A) is polynomially bounded, then Id(A) = Id(B) for a suitable finite dimensional algebra B. For any finite dimensional algebra A over an algebraically closed field we shall prove that A has polynomial growth of the codimensions if and only if A = A 0 ⊕A 1 ⊕ · · · ⊕ A m where A 0 , A 1 ,. .. , A m are F-algebras such that 1) for i = 1,. .. , m, A i = B i + J i , where B i ∼ = F and J i is a nilpotent ideal of A i , 2) A 0 , J 1 ,. .. , J m are nilpotent right ideals of A and 3) A i A k = 0 for all i, k ∈ {1,. .. , m}, i = k and B i A 0 = 0.

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“…When n = 2 and K is of characteristic 0, the above is the only non-trivial Z 2 -grading allowed. A detailed description of the Z 2 -graded identities of UT 2 (K) and their numerical invariants were given in [18].…”

Section: Preliminariesmentioning

confidence: 99%

Gradings on the algebra of upper triangular matrices and their graded identities

2004

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Graded polynomial identities play an important role in the structure theory of PI algebras.Many properties of the ideals of identities are described in the language of graded identities and graded algebras. In this paper we study the elementary gradings on the algebra UT_n(K) of n × n upper triangular matrices over an infinite field. We describe these gradings by means of the graded identities that they satisfy. Namely we prove that there exist |G|^{n−1} nonisomorphic elementary gradings on UT_n(K) by the finite group G, and show that nonisomorphic gradings produce different graded identities. Furthermore we describe generators for the ideals of graded identities for a given (but arbitrary) elementary grading on UT_n(K), and produce linear bases of the corresponding relatively free graded algebra

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The graded identities of upper triangular matrices of size two

Cited by 38 publications

References 10 publications

Group graded algebras and almost polynomial growth

Group graded algebras and almost polynomial growth

Graded algebras with polynomial growth of their codimensions

Gradings on the algebra of upper triangular matrices and their graded identities

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