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Algebras associated with LSGOP. (English) Zbl 0652.20012

An LSGOP (lower substractive G orbit poset) for a finite group G is a rooted poset equipped with a G-action satisfying the following conditions: (1) the G-action on P is transitive; and (2) if \(\rho\) is the root of P, then x.\(\rho\leq y.\rho \leq z.\rho\) \(\Leftrightarrow\) \((x^{- 1}y.\rho \leq x^{-1}z.\rho\) and x.\(\rho\leq z.\rho)\) for all x, y, z in G.
For the main part of the paper, the stabiliser of the root \(\rho\) is taken to be \(\{\) \(1\}\) and P is identified with G with the root \(\rho\) being identified with 1 of G. Thus G is a group with a partial order P and least element 1.
If k is a field, the k-algebra \(k[G]_ P\) associated to with P is defined to be the k-vector space on symbols \(u_ g\) (g\(\in G)\) with a multiplication \(u_ g.u_ h=e_ P(g,h)u_{gh}\) (g,h\(\in G)\), where \(e_ P(g,h)\) is 1 (\(\in k)\) if \(g\leq gh\) and is 0 otherwise. Condition (2) is then equivalent to \(k[G]_ P\) being associative, i.e. the “2- cosickle” condition \(e_ P(g,h)e_ P(gh,k)=e_ P(h,k)e_ P(g,hk)\) holds.
Such algebras and LSGOP’s are defined and motivated in a paper by D. Haile, R. Larson and M. Sweedler [Am. J. Math. 105, 689-814 (1983; Zbl 0523.13005)]. Section 2 shows that if P and Q are trees on G, then \(k[G]_ P\sim k[G]_ Q\) as k-algebras if and only if there is a poset isomorphism \(\sigma\) : \(P\to Q\) such that \(\sigma (g)\sigma (h)=\sigma (gh)\) holds whenever \(e_ P(g,h)=1.\)
In section 3, various invariants of k-algebras \(k[G]_ P\) are obtained when P is not necessarily a tree. These include the numbers of elements at various levels above the root and also down from maximal elements (ages). An invariant diagram is constructed from \(k[G]_ P\) together with sources, sinks and isolated points.
These have been sufficient to distinguish the different algebras \(k[G]_ P\) for all groups of order \(\leq 5\), by a case-by-case check. Section 4 gives a breakup of the radical of \(k[G]_ P\) as a direct sum of indecomposable right (or left) ideals, when P is a tree.
Reviewer: S.B.Conlon

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
06F15 Ordered groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
20J05 Homological methods in group theory
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)

Citations:

Zbl 0523.13005
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References:

[1] Conlon, S. B., Twisted group algebras and their representations, J. Austral. Math. Soc., 4, 152-173 (1964) · Zbl 0134.26305
[2] Haile, D.; Larson, R.; Sweedler, M., A New invariant for \(C\) over \(R\): Almost invertible cohomology theory and the classification of idempotent cohomology class and algebras by partially ordered sets with a Galois group action, Amer. J. of Math., 105, 689-814 (1983) · Zbl 0523.13005
[3] Jacobson, N., Theory of Rings (1943), Amer. Math. Soc: Amer. Math. Soc Providence, RI
[4] Shaw, R., (Linear Algebra and Group Representations, Vol. II (1983), Academic Press: Academic Press London)
[5] Shiravanagi, K., On LSGOP’s and Algebras Associated with Them, (Master’s thesis (1984), University of Tokyo), [In Japanese]
[6] Sweedler, M., Groups of simple algebras, Publ. Math. IHES, 44, 79-189 (1975) · Zbl 0314.16008
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