English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper, we discuss the finite generation problem for the invariant subalgebras of finite abelian groups which act linearly on the 2 x 2 generic matrix algebra, and we obtain some conditions on the groups to ensure that their invariant subalgebras are finitely generated. Introduction. Let R(m) = K{X, Y} be a generic matrix algebra generated by two m x m generic matrices over a field K. A theorem of Fisher and Montgomery [2] shows that if G = (g) is a finite cyclic group such that char K t ICl, then RG the subalgebra of invariants, is not finitely generated whenever g is not scalar and mI CGI-[VI[G]+1. When m = 2 and G is a finite subgroup of SL(2, K), Formanek and Schofield [3] have proved that if char K t IGI, then RG is always finitely generated. In this paper we consider the case when m = 2 and G is a finite abelian subgroup of GL(2, K) such that char K t IGl. In this case we may assume that K is algebraically closed. So G acts diagonally on R = R(2) and RG is generated by some monomials in the generators of R. We obtain necessary and sufficient conditions on G to ensure that RG is finitely generated. They are as follows: RG is finitely generated if and only if G contains no pseudoreflections or, equivalently, G = (g) is cyclic and the eigenvalues of g in some extension field of K have the same order. In some sense, this provides a converse to Theorem 1 of Fisher and Montgomery [2]. 1. For cyclic groups. LEMMA 1.1 (FORMANEK AND SCHOFIELD). Let G be any finite subgroup of GL(2, K) such that char K t IGI and let [R, R] = J be the commutator ideal of R. Then (i) for each m > 1, Jm = J(XY YX)m-1, and (ii) G acts linearly on R/J2 and (R/J2)G is a finitely generated K-algebra. PROOF. See [3, Lemmas 6 and 7]. LEMMA 1.2. Let u E R be homogeneous and Z = XY YX. Then u = vZ + aYrXs + E bi,j,,,lY'XjZYKXl Received by the editors February 25, 1987 and, in revised form, August 19, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A38.

Invariants of finite abelian groups acting on the algebra of two 2×2 generic matrices