The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra

2009/03/17 by V. V. Bavula, Bavula, V. V.
Mathematics · #16D25 #16D60 #16E10 #16G99 #Algebraic Geometry (math.AG) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AG #math.RA #msc:16D25 #msc:16D60 #msc:16E10 #msc:16G99

paper · pdf · doi:10.48550/arxiv.0903.3049

41 pages

arxiv created 2009/06/15 · arxiv updated 2009/12/01

Abstract

The algebra \mS_n in the title is obtained from a polynomial algebra P_n in n variables by adding commuting, \em left (but not two-sided) inverses of the canonical generators of P_n. Ignoring non-Noetherian property, the algebra \mS_n belongs to a family of algebras like the Weyl algebra A_n and the polynomial algebra P_2n. The group of automorphisms G_n of the algebra \mS_n is found: G_n=S_n\ltimes \mTⁿ\ltimes \Inn (\mS_n) ⊇ S_n\ltimes \mTⁿ\ltimes \underbrace\GL_∞ (K)\ltimes... \ltimes \GL_∞ (K)_2ⁿ-1 \rm times=:G_n' where S_n is the symmetric group, \mTⁿ is the n-dimensional torus, \Inn (\mS_n) is the group of inner automorphisms of \mS_n (which is huge), and \GL_∞ (K) is the group of invertible infinite dimensional matrices. This result may help in understanding of the structure of the groups of automorphisms of the Weyl algebra A_n and the polynomial algebra P_2n. An analog of the \em Jacobian homomorphism: \Aut_K-\rm alg(P_2n)\ra K^*, so-called, the \em global determinant is introduced for the group G_n' (notice that the algebra \mS_n is \em noncommutative and neither left nor right Noetherian).

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