Structure of Hyperbolic Unitary Groups II: Classification of E-Normal Subgroups

Abstract

This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group U 2n pR, Λq which are normalized by the elementary subgroup EU 2n pR, Λq, under the condition that R is a quasi-finite ring with involution, i.e a direct limit of module finite rings with involution, and n ě 3. 2010 Mathematics Subject Classification: 20G35, 20H25 Keywords: unitary groups, E-normal subgroups, sandwich classification This implies EU 2n ppR, Λq, pRxs 0 R, Γpxs 0 qqq Ď H¨U 2n ppR, Λq, pI, Γqq, since both H and U 2n ppR, Λq, pI, Γqq are normalized by EU 2n pR, Λq. Hence EU 2n ppR, Λq, pRxs 0 R, Γpxs 0 qqq "rEU 2n pR, Λq, EU 2n ppR, Λq, pRxs 0 R, Γpxs 0 qqqs ĎrEU 2n pR, Λq, H¨U 2n ppR, Λq, pI, Γqqs ĎrEU 2n pR, Λq, Hsp H rEU 2n pR, Λq, U 2n ppR, Λq, pI, Γqqsq "rEU 2n pR, Λq, Hsp H EU 2n ppR, Λq, pI, Γqqq ĎH by Lemma 3.16. But this contradicts the maximality of pI, Γq since clearly Rxs 0 R Ę I or Γpxs 0 q Ę Γ. Thus H Ď CU 2n ppR, Λq, pI, Γqq. uniqueness Assume that EU 2n ppR, Λq, pI, Γqq Ď H Ď CU 2n ppR, Λq, pI, Γqq and EU 2n ppR, Λq, pI 1 , Γ 1 qq Ď H Ď CU 2n ppR, Λq, pI 1 , Γ 1 qq.It follows that EU 2n ppR, Λq, pI, Γqq "rEU 2n pR, Λq, EU 2n ppR, Λq, pI, Γqqs ĎrEU 2n pR, Λq, CU 2n ppR, Λq, pI 1 , Γ 1 qqs "EU 2n ppR, Λq, pI 1 , Γ 1 qq.It is easy to deduce that I Ď I 1 and Γ Ď Γ 1 . By symmetry it follows that I " I 1 and Γ " Γ 1 . ð:Suppose that EU 2n ppR, Λq, pI, Γqq Ď H Ď CU 2n ppR, Λq, pI, Γqq.Then rEU 2n pR, Λq, Hs Ď rEU 2n pR, Λq, CU 2n ppR, Λq, pI, Γqqs " EU 2n ppR, Λq, pI, Γqq Ď H and hence H is normalized by EU 2n pR, Λq.Definition 4.12 Let R be a ring, r Þ Ñr an involution on R and λ P centerpRq. Then we call R quasi-finite if it is a direct limit of subrings R i (i P Φ where Φ is some index set) which are almost commutative, involution invariant and contain λ (recall that a ring is called almost commutative if it is module finite over its center).

The E-normal structure of odd dimensional unitary groups

“…(a) It is easy to show thatλ = λ −1 . (b) If λ ∈ Center(R) then one gets the usual concept of an involution which is for instance used in [28]. (c) The map¯:…”

“…Let Ω be a relative odd form parameter for I. [2], [28] or [30]). + Z • R. Ω(Z) is called the relative odd form parameter defined by Z.…”

“…The proof can be repaired when 2 is invertible in R. In 2012, H. You [35] proved that (1.2) holds if R is commutative and n ≥ 4 and one year later H. You and X. Zhou [36] added the case that R is commutative and n = 3. Recently the second author [28] proved that (1.2) holds if R is semilocal or almost commmutative and n ≥ 3.…”

Sandwich classification for O 2n+1(R) and U 2n+1(R,Δ) revisited

“…For generalized hyperbolic unitary groups the SCT was announced by A. Bak and N. Vavilov in [3], but the proof has never been published. It appeared later in [10] (which is essentially the author's thesis). Recently A. Stepanov [12] proved the SCT for all Chevalley groups using the universal localization method.…”

The E-normal structure of Petrov’s odd unitary groups over commutative rings

“…The proof can be repaired when 2 is invertible in R. In 2012, H. You [23] proved that (3) holds if R is commutative and n ě 4 and one year later H. You and X. Zhou [24] added the case that n " 3. In 2017, the author [17] proved that (3) holds if R is almost commmutative and n ě 3. In 2005, V. A. Petrov [16] defined odd unitary groups UpVq.…”