A new proof of the Amitsur-Levitski identity

Rosset, Shmuel

doi:10.1007/bf02756797

Cited by 44 publications

(17 citation statements)

References 2 publications

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“…As it is shown in [20, p. 104 Proof This statement is a direct implication of the Amitsur-Levitzki theorem [21]. (C(3, 1)).…”

Section: Corollary 46mentioning

confidence: 65%

Regev’s Conjecture and Codimensions of P.I. Algebras

Gordienko

2008

Acta Appl Math

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We study the periodicity of the proper cocharacters and show that Regev's conjecture holds in unitary algebras of P.I. exponent 2. Also we discuss the asymptotic behaviour of the codimensions and cocharacters of Clifford algebras and deal with other important examples of P.I. algebras.Keywords Polynomial identities · Codimensions · Cocharacters · Regev's conjecture · P.I. algebras · P.I. exponent · Commutator · Proper polynomial · Associative algebras · Clifford algebras · Grassmann envelope Mathematics Subject Classification (2000) Primary 16R10 · Secondary 05C90 · 15A66 · 16R40 · 16W55 · 20C30In the 80's conjectures about the asymptotic behaviour of the codimensions of associative algebras were made by S.A. Amitsur and A. Regev.Let A be an associative algebra and c n (A) be its codimension sequence.Amitsur's conjecture was proved in 1999 by M.V. Zaicev and A. Giambruno [1, Chap. 6, Theorem 5.2, p. 143]. Regev's conjecture was proved by V.S. Drensky for algebras of polynomial growth [2], by A. Regev for finite dimensional semisimple algebras [3], by M.V. Zaicev and A. Giambruno for algebras of block-triangular matrices [4]. Regev's conjecture in its generality still remains an open problem. 34 A.S. GordienkoV.S. Drensky has shown [2] that the multiplicity sequence of the irreducible cocharacters with fixed lower rows of every polynomial growth algebra is periodic. In Sect. 1 we show that this sequence is, from some place, a constant. Therefore, we obtain a new proof of Regev's conjecture for algebras of P.I. exponent 1. Furthermore, we study the periodicity of the proper cocharacters and show that Regev's conjecture holds in unitary algebras of P.I. exponent 2.Despite the intense activity in this field, only few examples are known in which the codimensions, cocharacters, and bases for the polynomial identities are precisely computed. The cocharacters of the algebra of 2 × 2 matrices were found by V.S. Drensky and E. Formanek [5,6]. The codimensions of this algebra were precisely computed by C. Procesi [7]. The codimensions and cocharacters of the Grassmann algebra were calculated by D. Krakowski and A. Regev [8]. The basis for the polynomial identities of the algebras UT n (F ) of upper triangular matrices was found by Yu.N. Malcev [9]. In Sect. 2 we compute the codimension, cocharacter, colength sequences, and a basis for the polynomial identities of the algebra A 1 = x a c 0 y b 0 0 x over a field of characteristic 0. In Sect. 3 we discuss the asymptotic behaviour of the codimensions and cocharacters of Clifford algebras. The multilinear identities of minimal degree for Clifford algebras of rank 1 are found. In Sect. 4 we present the Grassmann envelope of a finite dimensional superalgebra with the T -ideal of identities generated by [x 1 , x 2 , x 3 , x 4 ]. It provides us a version of its codimensions calculation.Here we give an improved account of the results obtained in [10][11][12][13].

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“…As it is shown in [20, p. 104 Proof This statement is a direct implication of the Amitsur-Levitzki theorem [21]. (C(3, 1)).…”

Section: Corollary 46mentioning

confidence: 65%

Regev’s Conjecture and Codimensions of P.I. Algebras

Gordienko

2008

Acta Appl Math

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“…The hard part of this result, namely that M d (k) satisfies S 2d = 0, is known as the Amitsur-Levitzki Theorem [39]. All known proofs are either messy (e.g., by graph theory [135]) or tricky (e.g., using exterior algebras [128]). The reader is invited to attempt to find a new proof!…”

Section: Var(c) = H S P(c)mentioning

confidence: 99%

An Invitation to General Algebra and Universal Constructions

Bergman

2015

Universitext

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“…Other proofs of the Amitsur-Levitzki theorem were later obtained: in 1963, R. G. Swan [24] reduced the original problem to a graph theory problem; in 1974, L. H. Rowen [21], used a direct method in his proof; in 1976, S. Rosset [20] gave a fast proof based on the Hamilton-Cayley theorem. Finally in 1981 [11], Kostant closed the subject once and for all by providing a very nice interpretation of the theorem in the context of representation theory and generalizing it using his separation of variables theorem [12].…”

Section: Introductionmentioning

confidence: 99%

THE AMITSUR–LEVITZKI THEOREM FOR THE ORTHOSYMPLECTIC LIE SUPERALGEBRA 𝔬𝔰𝔭(1, 2n)

Pinczon¹,

Ushirobira²

2006

J. Algebra Appl.

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Communicated by Ian M. MussonBased on Kostant's cohomological interpretation of the Amitsur-Levitzki theorem, we prove a super version of this theorem for the Lie superalgebras osp(1, 2n). We conjecture that no other classical Lie superalgebra can satisfy an Amitsur-Levitzki type super identity. We show several (super) identities for the standard super polynomials. Finally, a combinatorial conjecture on the standard skew supersymmetric polynomials is stated.

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A new proof of the Amitsur-Levitski identity

Cited by 44 publications

References 2 publications

Regev’s Conjecture and Codimensions of P.I. Algebras

Regev’s Conjecture and Codimensions of P.I. Algebras

An Invitation to General Algebra and Universal Constructions

THE AMITSUR–LEVITZKI THEOREM FOR THE ORTHOSYMPLECTIC LIE SUPERALGEBRA 𝔬𝔰𝔭(1, 2n)

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