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Differential identities of matrix algebras

Jose Brox, Carla Rizzo

Abstract

We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $M_k(F)$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$.

Differential identities of matrix algebras

Abstract

We study the differential identities of the algebra of matrices over a field of characteristic zero when its full Lie algebra of derivations, , acts on it. We determine a set of 2 generators of the ideal of differential identities of for . Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with -action generated by has almost polynomial growth for all .
Paper Structure (19 sections, 31 theorems, 145 equations, 1 figure)

This paper contains 19 sections, 31 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General setting
  3. Preliminaries
  4. The variety of $L$-algebras
  5. The variety of $(L,U)$-algebras
  6. Identities and growth
  7. Computing $L$-data through $U$-data
  8. Matrix setting
  9. Derivations of $M_k(F)$
  10. Explicit description of $U^{\mathop{\mathrm{op}}\nolimits}$
  11. Computations involving $U^{\mathop{\mathrm{op}}\nolimits}$
  12. Explicit description of $U(L)^{\mathop{\mathrm{op}}\nolimits}$
  13. $U$-polynomials and $U$-identities
  14. Differential identities of $M_k(F)$
  15. $U$-identities of $M_k(F)$
  16. ...and 4 more sections

Key Result

Theorem 2.1.4

Let $L$ be a finite-dimensional split-semisimple Lie algebra over a field of characteristic $0$ and $\rho$ be a finite-dimensional representation of $L$. Then the enveloping associative algebra $[\rho(L)]$ of $\rho$ is split semisimple. Moreover, if $\rho$ is irreducible of dimension $d$, then $[\rh

Figures (1)

  • Figure 1: Relationships between the different free algebras

Theorems & Definitions (62)

  • Theorem 2.1.4: Full matrix algebras as enveloping algebras
  • proof
  • Lemma 2.1.6
  • proof
  • Lemma 2.2.2: Primitive element
  • proof
  • Lemma 2.5.1: Relating $L$-identities to $U$-identities
  • Remark 2.5.2: $\mathop{\mathrm{Id^U\langle X\rangle}}\nolimits$ is not a $T_L$-ideal
  • Proposition 2.5.3: Structure of $\mathop{\mathrm{Id^U\langle X\rangle}}\nolimits$
  • proof
  • ...and 52 more