Search results for " Trace algebras"

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Trace identities and almost polynomial growth

2021

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2 \times 2$ matrices generated by $e_{11}+e_{22}$ and $e_{12}$. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynomial or exponential.

PolynomialPure mathematicsTrace (linear algebra)Trace algebrasField (mathematics)01 natural sciencesPolynomial identitiesMatrix (mathematics)16R10 16R30 16R50Polynomial identitieCodimensions growth Polynomial identities Trace algebras0103 physical sciencesDiagonal matrixFOS: Mathematics0101 mathematicsCommutative propertyMathematicsCodimensions growth; Polynomial identities; Trace algebrasAlgebra and Number TheoryCodimensions growth010102 general mathematicsTrace algebraMathematics - Rings and AlgebrasExponential functionSettore MAT/02 - AlgebraRings and Algebras (math.RA)010307 mathematical physicsVariety (universal algebra)
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Defining relations of minimal degree of the trace algebra of 3 X 3 matrices

2008

The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d 2. Minimal sets of generators of Cnd are known for n = 2 and n = 3 for any d as well as for n = 4 and n = 5 and d = 2. The defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Starting with the generating set of C3d given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set of defining relations of C3d is equal to 7 for any d 3. We have determined all relations of minimal degree. For d = 3 we have also found the defining relations of degree 8. The proofs are based on methods of representati…

generic matrices matrix invariants trace algebras defining relationstrace algebra
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