Abstract Root Subgroups and Quadratic Action

Timmesfeld, F. G.

doi:10.1006/aima.1998.1779

Cited by 46 publications

(54 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Section: Eo(n R A) ≤ H ≤ Cgo(n R A)supporting

confidence: 73%

“…In the survey [93], one can find a detailed exposition of these results. Similar results should follow also from the general theorems of Franz Timmesfeld on quadratic action subgroups [85].The possibility of generalizing theorems of Dye-King-Li type to arbitrary commutative rings by using decomposition of unipotents was suggested in [92,83]. However, before…”

mentioning

confidence: 54%

“…• Consider the field case R = K. Since E(E 6 , R) contains a one-parameter unipotent subgroup X α that acts on K 27 quadratically, such a result could be deduced from the results of [85]. For such fields, algebraically closed or finite, this would follow also from the results of Seitz and Testerman [80], or the results of Aschbacher and Kondratiev [38,39,26,68].…”

Section: Problem 3 Describementioning

confidence: 99%

See 2 more Smart Citations

Overgroups of $\mathrm{EO}(n,R)$

Vavilov

Petrov

2008

St. Petersburg Math. J.

View full text Add to dashboard Cite

Abstract. Let R be a commutative ring with 1, n a natural number, and let l = [n/2]. Suppose that 2 ∈ R * and l ≥ 3. We describe the subgroups of the general linear group GL(n, R) that contain the elementary orthogonal group EO(n, R). The main result of the paper says that, for every intermediate subgroup H, there exists a largest ideal A R such that EEO(n, R, A) = EO(n, R)E(n, R, A) H. Another important result is an explicit calculation of the normalizer of the group EEO(n, R, A). If R = K is a field, similar results were obtained earlier by Dye, King, Shang Zhi Li, and Bashkirov. For overgroups of the even split elementary orthogonal group EO(2l, R) and the elementary symplectic group E p (2l, R), analogous results appeared in previous papers by the authors (Zapiski Nauchn. Semin. POMI, 2000, v. 272; Algebra i Analiz, 2003, v. 15, no. 3).The present work, together with [19], constitutes a final and fully developed version of the paper "Overgroups of classical groups over commutative rings", which was predicted in §11 of the survey [92]. Together with our papers [18,19] it completely describes the subgroups of the general linear group G = GL(n, R) over a commutative ring R, containing the elementary subgroup of a split classical group in the vector representation. In [78], the second-named author generalized these results to the even unitary groups. Namely, in [18] we described the overgroups of EO(2l, R) under the assumption that l ≥ 3, 2 ∈ R * , and in [19] we described the overgroups of Ep(2l, R) under the assumption that either l ≥ 3, or l = 2 and the ring R does not have the residue field F 2 . In the present paper, we consider the last remaining case; namely, we describe the overgroups of EO(n, R), where n = 2l + 1. As in the papers [18,19], it turns out that when n ≥ 7 and 2 ∈ R * , for every such subgroup H there exists a unique ideal A in R such that H lies between the group EEO(n, R, A) = EO(n, R)E (n, R, A) and its normalizer in GL(n, R). Since the even-dimensional case is very similar to the odd-dimensional one, and is in fact slightly easier, in the present paper we also give another proof, and a refinement of the results of [18]. The principal result of the present paper can be stated as follows. Theorem 1. Let R be a commutative ring. Suppose that 2 ∈ R* and n ≥ 7. Then for every subgroup H in G = GL(n, R) containing the elementary orthogonal group EO(n, R) there exists a unique ideal A R such that An important pendant to Theorem 1 is the following result, in which we explicitly calculate the normalizer of EEO(n, R, A). Namely, consider the reduction homomorphism ρ A : GL(n, R) −→ GL(n, R/A) and denote by CGO(n, R, A) the full preimage of the group GO(n, R) with respect to ρ A . The condition for a matrix to belong to CGO(n, R, A) is described by obvious quadratic congruences on its entries. Now we are in a position to state the second major result of the present paper. EEO(n, R, A) ≤ H ≤ N G (EEO(n, R, A)). Theorem 2. Under the assumptions of Theorem 1, for any ideal A R we have N G (EEO(n,...

show abstract

Section: Eo(n R A) ≤ H ≤ Cgo(n R A)supporting

confidence: 73%

mentioning

confidence: 54%

Section: Problem 3 Describementioning

confidence: 99%

See 1 more Smart Citation

Overgroups of $\mathrm{EO}(n,R)$

Vavilov

Petrov

2008

St. Petersburg Math. J.

View full text Add to dashboard Cite

show abstract

Section: Eep(2l R A) ≤ H ≤ Cgsp(2l R A)mentioning

confidence: 89%

“…It seems that the most powerful specific results in this direction were obtained by E. L. Bashkirov in [3]- [7]; in fact, he described the subgroups of GL(n, K) that contain a classical group not over the field K itself, but over its subfield L ≤ K such that the extension K/L is algebraic. These results must follow also from Timmesfeld's general theorems on quadratic action subgroups [76].The possibility to generalize these results to arbitrary commutative rings by using decomposition of unipotents was mentioned in [84,73], but detailed proofs have never been published. In fact, it was even claimed there that by the same method one could describe the subgroups of GL(n, R) normalized by the elementary classical group.…”

mentioning

confidence: 99%

Overgroups of elementary symplectic groups

Vavilov

Petrov

2004

St. Petersburg Math. J.

View full text Add to dashboard Cite

Abstract. Let R be a commutative ring, and let l ≥ 2; for l = 2 it is assumed additionally that R has no residue fields of two elements. The subgroups of the general linear group GL(n, R) that contain the elementary symplectic group Ep(2l, R) are described. In the case where R = K is a field, similar results were obtained earlier by Dye, King, and Shang Zhi Li.In the present paper we consider a description of the subgroups in the general linear group G = GL(2l, R) over a commutative ring R that contain the elementary symplectic group Ep(2l, R). It turns out that for every such group H there exists a unique ideal A in R such that H lies between the group EEp(2l, R, A) = Ep(2l, R) E(2l, R, A) and its normalizer in GL(2l, R). More specifically, our main objective in the present paper is a proof of the following result. Theorem 1. Let R be a commutative ring. Suppose that either l ≥ 3, or l = 2 and the ring R has no residue fields of two elements. Then for every subgroup H in G = GL(2l, R) that contains the elementary symplectic group Ep(2l, R), there exists a unique ideal A R such thatEEp(2l, R, A) ≤ H ≤ N G (EEp(2l, R, A)).An important supplement to Theorem 1 is the following result, in which we explicitly calculate the normalizer of EEp(2l, R, A). Namely, consider the reduction homomorphism ρ A : GL(2l, R) −→ GL(2l, R/A) and denote by CGSp(2l, R, A) the complete preimage of the group GSp(2l, R) with respect to ρ A . Then the condition for a matrix to belong to CGSp(2l, R, A) is described by obvious quadratic congruences on its entries. Now we are in a position to state the second major result of the present paper. Theorem 2. Under the assumptions of Theorem 1, for any ideal A R we have N G (EEp(2l, R, A)) = CGSp(2l, R, A).Thus, combining Theorems 1 and 2, we see that for any subgroup H with Ep(2l, R) ≤ H ≤ GL(2l, R) there exists a unique ideal such that EEp(2l, R, A) ≤ H ≤ CGSp(2l, R, A).2000 Mathematics Subject Classification. Primary 20G35. The present paper has been written in the framework of the RFBR projects nos. 01-01-00924 and 00-01-00441, and INTAS 00-566. The theorem on decomposition of unipotents mentioned in §13 is a part of first author's joint work with A. Bak and was carried out at the University of Bielefeld with the support of AvH-Stiftung, . At the final stage, the work of the authors was supported by express grants of the Russian Ministry of Higher Education 'Geometry of root subgroups' PD02-1.1-371 and 'Overgroups of semisimple groups' E02-1.0-61. In the case where R = K is a field, there are only two possibilities for A, namely, A = 0 or A = K. We have EEp(2, K, 0) = Ep(2l, K) = Sp(2l, K), CGSp(2l, K, 0) = GSp(2l, K).On the other hand, EEp(2, K, K) = E(2l, K) = SL(2l, K), CGSp(2l, K, K) = GL(2l, K).In this special case, Theorem 1 asserts that any overgroup of Sp(2l, K) in GL(2l, K) either is contained in GSp(2l, K), or contains SL(2l, K), so that this theorem boils down to a theorem of R. Dye [48]. Subsequently, R. Dye, O. King, and Shang Zhi Li generalized this result to overgroups of ot...

show abstract

Lie Algebras Generated by Extremal Elements

Cohen¹,

Steinbach²,

Ushirobira³

et al. 2001

Journal of Algebra

View full text Add to dashboard Cite

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals) over a field of characteristic distinct from 2. There is an associative bilinear form on such a Lie algebra; we study its connections with the Killing form. Any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal numbers of extremal generators for the Lie algebras of type A n n ≥ 1 , B n n ≥ 3 , C n n ≥ 2 , D n n ≥ 4 , E n n = 6 7 8 , F 4 and G 2 are shown to be n + 1, n + 1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.

show abstract

Abstract Root Subgroups and Quadratic Action

Cited by 46 publications

References 26 publications

Overgroups of $\mathrm{EO}(n,R)$

Overgroups of $\mathrm{EO}(n,R)$

Overgroups of elementary symplectic groups

Lie Algebras Generated by Extremal Elements

Contact Info

Product

Resources

About