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<p>Let <i>K</i> be a field of characteristic zero. In this paper we study the asymptotic behavior of the sequences of codimensions for the polynomial identities of pairs (associative algebra, vector space), called for short AS pairs, and denoted by (<i>A, S</i>). Here <i>A</i> is an associative <i>K</i>-algebra generated by a vector subspace <i>S</i>. More precisely, we show that if <i>A</i> is a ℤ<sub>2</sub>-graded algebra of finite dimension and <i>S</i> is a homogeneous vector subspace of <i>A</i>, then the Grassmann envelope of the pair (<i>A, S</i>) has codimension sequence that is exponentially bounded. Furthermore, we prove that if <i>S</i> is a simple Lie or Jordan algebra, then the PI exponent of the Grassmann envelope of (<i>A, S</i>) exists and is a positive integer.</p>

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Codimension growth and existence of PI exponent of pairs (associative algebra, vector space)

  • 期刊论文

  • 发表日期：

    2025年9月29日

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  • David Levi da Silva Macêdo,
  • Claudemir Fideles,
  • Plamen Koshlukov

摘要

Let K be a field of characteristic zero. In this paper we study the asymptotic behavior of the sequences of codimensions for the polynomial identities of pairs (associative algebra, vector space), called for short AS pairs, and denoted by (A, S). Here A is an associative K-algebra generated by a vector subspace S. More precisely, we show that if A is a ℤ2-graded algebra of finite dimension and S is a homogeneous vector subspace of A, then the Grassmann envelope of the pair (A, S) has codimension sequence that is exponentially bounded. Furthermore, we prove that if S is a simple Lie or Jordan algebra, then the PI exponent of the Grassmann envelope of (A, S) exists and is a positive integer.