Amitsur-Levitzki Theorem Research Articles

A graph-theoretic approach to a conjecture of Dixon and Pressman

Let Matn(F) be the algebra of n × n matrices over a field F. For A1,…, Ak ∈ Matn(F), we define the linear transformation $$L\left( {{A_1}, \ldots ,{A_k}} \right):{\rm{Ma}}{{\rm{t}}_n}\left( F \right) \to {\rm{Ma}}{{\rm{t}}_n}\left( F \right)$$ by \(L\left( {{A_1}, \ldots ,{A_k}} \right)\left( {{A_{k + 1}}} \right) = \sum\nolimits_{\sigma \in {\rm{Sy}}{{\rm{m}}_{k + 1}}} {{\mathop{\rm sgn}} \left( \sigma \right){A_{\sigma \left( 1 \right)}}{A_{\sigma \left( 2 \right)}} \ldots {A_{\sigma \left( {k + 1} \right)}}} \). The Amitsur—Levitzki theorem asserts that L(A1,…, Ak) is identically 0 for every A1,…, Ak, as long as k ⩾ 2n − 1. Dixon and Pressman conjectured that if F = ℝ is the field of real numbers and k is an even integer between 2 and 2n − 2, then the kernel of L(A1,…, Ak) is of dimension k for A1,…, Ak ∈ Matn(ℝ) in general position. We prove this conjecture using graph-theoretic techniques.

A variant of Rosset's approach to the Amitsur-Levitzki theorem and some $\mathbb{Z}_{2}$-graded identities of $\mathrm{M}_{n}(E)$

In the spirit of Rosset's proof of the Amitsur-Levitzki theorem, we show how the standard identiy (for matrices over a commutative base ring) and the addition of external Grassmann variables can be used to derive a certain $\mathbb{Z}_{2}$-graded polynomial identity of $\mathrm{M}_{n}(E)$.

Minimal degree of standard identities of matrix algebras with symplectic graded involution

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.

Generalized commutators and a problem related to the Amitsur–Levitzki theorem

The generalized commutator of a list of k real matrices is defined as a multilinear skew-symmetric function and the linear operator on the vector space is defined by . The Amitsur–Levitzki theorem shows that when . We investigate the kernel of T and prove that for all integers k and n such that we have where if k is even; if k is odd and n is even; and if k and n are both odd. We conjecture that this result is best possible and that for almost all when k and n are in this range. This conjecture is supported by some computational evidence but so far remains open.

A graph theoretical equivalence for Amitsur–Levitzki theorem

Let n be a positive integer. A remarkable theorem of Amitsur–Levitzki in PI algebras states that if are any matrices with entries in any commutative ring, then , where is the standard polynomial over 2n variables . Now, let n and m be positive integers and let G be a multigraph with n vertices and m edges. Also consider a fixed labelling of edges, then any Eulerian trail in G (if exists), as a successive label of edges, corresponds to an element of with a specified parity sign. So, we may call an Eulerian trail as an odd or even Eulerian trail. R.G. Swan proved that when , then the number of even and odd Eulerian trails between any two vertices G (may be identical) are equal. We show that this latter theorem of Swan is a graph theoretical equivalence for Amitsur–Levitzki Theorem. Meanwhile, we present the Euler and Hamilton matrix of a graph with the entries denoting, respectively the number of possible Eulerian trails and Hamiltonian cycles of the graph.

Path decompositions of digraphs and their applications to Weyl algebra

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the n-th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem and helps to study skew-symmetric polynomials on certain subspaces of Weyl algebra. For instance, path decompositions can be used to study minimal polynomial identities on Weyl algebra, similarly as Eulerian tours applicable for Amitsur–Levitzki theorem. We introduce the G-Stirling functions which enumerate decompositions by sources (and sinks) of paths.

On the theorem of Amitsur-Levitzki

We present a proof of the Amitsur-Levitzki theorem which is a basis for a general theory of equivariant skew-symmetric maps on matrices.

ON APPLICATIONS OF PI-ALGEBRAS IN THE ANALYSIS OF QUANTUM CHANNELS

In this paper, we discuss some constructive procedures which can be used in characterizations of linear transformations which preserve the set of states of a fixed quantum system. Our methods are based on analyzing an explicit form of a linear positive map in its Kraus representation. In particular, we discuss the so-called partial commutativity of operators and its applications to investigation of decoherence-free subspaces. These subspaces can also be considered as a special class of quantum error correcting codes. Using the concept of standard polynomials and Amitsur–Levitzki theorem and other ideas from the so-called polynomial identity algebras (PI-algebras) we discuss some effective algorithms for analyzing properties of quantum channels.

An Application of Eulerian Graph to PI on <i>Mn</i>(<i>C</i>)

We obtain a new class of polynomial identities on the ring of n × n matrices over any commutative ring with 1 by using the Swan’s graph theoretic method [1] in the proof of Amitsur-Levitzki theorem. Let be an Eulerian graph with k vertices and d edges. Further let be an integer and assume that . We prore that is an PI on Mn(C). Standard and Chang [2] -Giambruno-Sehgal [3] polynomial identities are the spectial examples of our conclusions.

An analog of the Amitsur-Levitzki theorem for matrix superalgebras

We construct a polynomial identity of degree 2(nk + n + k) — min{n, k} for the matrix superalgebra M n,k over a field of characteristic zero. The conjecture is formulated that M n,k lacks any identities of lower degree.

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

Based 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.

On subalgebras of n× n matrices not satisfying identities of degree 2 n−2

The Amitsur–Levitzki theorem asserts that M n ( F) satisfies a polynomial identity of degree 2 n. (Here, F is a field and M n ( F) is the algebra of n× n matrices over F.) It is easy to give examples of subalgebras of M n ( F) that do satisfy an identity of lower degree and subalgebras of M n ( F) that satisfy no polynomial identity of degree ⩽2 n−2. In this paper we prove that the subalgebras of n× n matrices satisfying no nonzero polynomial of degree less than 2 n are, up to F-algebra isomorphisms, the class of full block upper triangular matrix algebras.

Minimal identities for right-symmetric algebras

An algebra A with multiplication A × A → A , ( a, b ) → a ∘ b , is called right-symmetric, if a ∘ ( b ∘ c ) − ( a ∘ b ) ∘ c = a ∘ ( c ∘ b ) − ( a ∘ c ) ∘ b , for any a , b , c ε A . The multiplication of right-symmetric Witt algebras W n = { d ∂ i : u ε U , U = K[ x 1 ±1 , …, x ±1 n ], or = K[ x 1 , …, x n ], i = 1, …, n }, p = 0, or W ( m ) = { u∂ i : u ε U , U = O n .( m )}, p > 0, are given by u∂ i ∘ u∂ j = ν∂ j ( u ) ∂ i . An analogue of the Amitsur-Levitzki theorem for right-symmetric Witt algebras is established. Right-symmetric Witt algebras of rank n satisfy the standard right-symmetric identity of degree 2 n + 1 : Σ σε Sym 2 n sign( σ ) a σ (1) ∘ ( a σ (2) ∘ … ∘ ( a σ (2 n ) ∘ a 2 n +1 ) …) = 0. The minimal degree for left polynomial identities of W n r sym , W n +r sym , p = 0, is 2 n + 1. All left polynomial identities of right-symmetric Witt algebras of minimal degree follow from the left standard right-symmetric identity S 2 n r sym = 0, if p ≠ 2.

Eulerian Polynomial Identities and Algebras Satisfying a Standard Identity

Using the trivial observation that one can get polynomial identities on R from the ones of Mk(R) we derive from the Amitsur-Levitzki theorem a subset of the identities on n × n matrices, obtained recently by Szigeti, Tuza, and Révész starting from directed Eulerian graphs, which generate the same T-ideal of the free algebra. After that we show that by this method we get a generating set of the T-ideal of identities on the 2 × 2 matrix ring over a field of characteristic 0. We reformulate a problem on algebras satisfying a standard identity in terms of Eulerian identities and use this equivalence in both directions. We apply a result of Braun to Eulerian identities on 3 × 3 matrices, and we give a simpler example which answers the question investigated by Braun. Finally we give an upper estimation on the minimal degree of the standard identity which is satisfied by the matrix algebra over an algebra satisfying some standard identity.

An algorithm for determining the simplicity of a modular group representation

Let G be a finite group an F and arbitrary splitting field for G. Using the Amitsur-Levitzki theorem on the polynomial identities satisfied by a matrix ring over a commutative ring an algorithm is given for finding the minimal dimension t for which there is a simple t-dimensional FG-submodule U in a faithful FG-module M of dimension n <∞. As a corollary a new criterion is obtained for checking the simplicity of a finite-dimensional group representation.

Polynomial Identities of Incidence Algebras

In this paper we determine the polynomial identities satisfied by incidence algebras. One of our results is logically equivalent to the Amitsur-Levitzki Theorem on the polynomial identities satisfied by ${K_n}$, the algebra of of $n \times n$ matrices over a field $K$.

Polynomial identities of incidence algebras

In this paper we determine the polynomial identities satisfied by incidence algebras. One of our results is logically equivalent to the Amitsur-Levitzki Theorem on the polynomial identities satisfied by K n {K_n} , the algebra of of n × n n \times n matrices over a field K K .