The minimal degree standard identity on M n E 2 and M n E 3
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THE MINIMAL DEGREE STANDARD IDENTITY ON Mn E 2 AND Mn E 3
BY
´zs∗ Barbara Anna Bala Department of Computer Science and Information Theory Budapest University of Technology and Economics M˝ uegyetem rkp. 3, 1111 Budapest, Hungary e-mail: [email protected] AND
´ros∗∗ Szabolcs M´ esza R´enyi Institute Re´ altanoda utca 13-15, 1053 Budapest, Hungary e-mail: [email protected]
ABSTRACT
We prove an Amitsur–Levitzki-type theorem for Grassmann algebras, stating that the minimal degree of a standard identity that is a polynomial identity of the ring of n × n matrices over the m-generated Grassmann algebra is at least 2 m + 4n − 4 for all n, m ≥ 2 and this bound is sharp for 2 m = 2, 3 and any n ≥ 2. The arguments are purely combinatorial, based on computing sums of signs corresponding to Eulerian trails in directed graphs.
∗ The research reported in this paper has been supported by the National Research,
Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program). The research reported in this paper was supported by the BMEArtificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC). ∗∗ This research was partially supported by National Research, Development and Innovation Office, NKFIH K 119934. Received April 28, 2019 and in revised form June 5, 2019
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´ ´ AROS ´ B. A. BALAZS AND SZ. MESZ
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Isr. J. Math.
1. Introduction In [8, 9] R. G. Swan gave a graph-theoretic proof of the Amitsur–Levitzki theorem which states that the standard identity of degree 2n holds for the ring of n × n matrices over a commutative ring. We generalize these methods and extend the Amitsur–Levitzki theorem to the case where the commutative ring is replaced by a finite-dimensional Grassmann algebra. Let R be a commutative unital ring of characteristic zero, m, n ≥ 1 and denote by Mn E m the ring of n × n matrices over the m-generated Grassmann algebra def
E m = Rvi | 1 ≤ i ≤ m/(vi2 , vi vj + vj vi | 1 ≤ i, j ≤ m). We say that the standard identity of degree k is a polynomial identity of Mn E m if and only if for any x1 , . . . , xk ∈ Mn E m def
sk (x1 , . . . , xk ) =
sgn(π)xπ(1) xπ(2) · · · xπ(k) = 0
π∈Sk
where Sk denotes the symmetric group on the set {1, . . . , k}. The algebra Mn E m satisfies the standard identity of degree 2m n2 +1, by basic linear algebra. This upper bound is far from being optimal, and the problem of finding the exact bound was raised in [7]. For n = 1 it is easy to see that the answer is 2 m 2 + 2, while for m = 1 it is 2n by the Amitsur–Levitzki theorem (see [1]). By [2, Thm. 5.5], Mn E m satisfies the standard identity of degree mn2 + 1 if m is odd. This bound is still not the best possible: In [4] Frenkel proved that Mn E m satisfies the standard m n2 +1 identity of degree 2n( m 2 + 1) and also that of degree 2( 2 + 2 ), but not the identity of degree 2( m 2 + n) − 1. In particular, for n = 2 the bound m 2( 2 + 2) is optimal and for m = 2 and 3 the minimal degree is between 2n + 2 and 4n. Frenkel also asked whether the standard identity of degree 2 m 2 +
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