Identities of graded algebras and codimension growth

2004/01/16 by Yu. Bahturin, Yu. A. Bahturin, M. Zaicev +1 · 6 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Rings, Modules, and Algebras

paper · pdf · doi:10.1090/s0002-9947-04-03426-9

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals circled-plus upper A Subscript g"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mo> ⊕ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">A=⊕ _{g∈ G}A_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript e"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to that of the whole of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finite dimensional and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript e"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has polynomial growth.

Cited by

• On infinite dimensional algebras with regular gradings
• Graded polynomial identities and exponential growth
• Z₂-graded Gelfand-Kirillov dimension of the Grassmann algebra
• On the classification of quantum symmetries
• On the codimension growth of G-graded algebras
• One proof of the original Kemer's theorems (concerning the text of C.\n Procesi "What happened to PI-theory", arxiv.org/abs/1403.5673)

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