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Computation of the Grothendieck group of the algebra \(\mathbb{C}(PSL(2,k))\), where \(k\) is a countable algebraically closed field. (English. Russian original) Zbl 0746.16004

Leningr. Math. J. 2, No. 6, 1251-1259 (1991); translation from Algebra Anal. 2, No. 6, 98-106 (1990).
Let \(F_ p\) be a prime finite field of characteristic \(p\), and \({\overline F}_ p\) its algebraic closure. Consider the countable group \(\Gamma=PSL(2,{\overline F}_ p))\) and its complex group algebra \({\mathbb{C}}(\Gamma)\). We give a complete description of the Grothendieck group of finitely generated projective modules over this algebra and the cone of faithful projective modules.

MSC:

16E20 Grothendieck groups, \(K\)-theory, etc.
20G05 Representation theory for linear algebraic groups
19A31 \(K_0\) of group rings and orders
16D40 Free, projective, and flat modules and ideals in associative algebras
16S34 Group rings
20C32 Representations of infinite symmetric groups
20G40 Linear algebraic groups over finite fields
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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