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Lie algebra so(7), type B13
Semisimple complex Lie subalgebras
Up to linear equivalence, there are total 16 semisimple subalgebras (including the full subalgebra). The subalgebras are ordered by rank, Dynkin indices of simple constituents and dimensions of simple constituents.
The upper index indicates the Dynkin index, the lower index indicates the rank of the subalgebra.
Computation time in seconds: 24.75.
10867091 total arithmetic operations performed = 10475268 additions and 391823 multiplications.
The base field over which the subalgebras were realized is: Q
Number of root subalgebras other than the Cartan and full subalgebra: 8
Number of sl(2)'s: 6
Subalgebra A11 ↪ B13
1 out of 16
Subalgebra type: A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: A21+A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A11
Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0), (0, 0, 1)
Contained up to conjugation as a direct summand of: 2A11 , A21+A11 , A31+A11 , A21+2A11 .
Elements Cartan subalgebra scaled to act by two by components: A11: (1, 2, 2): 2
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−9
Positive simple generators: g9
Cartan symmetric matrix: (2)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2)
Decomposition of ambient Lie algebra: V2ω1⊕6Vω1⊕6V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ψ1+2ψ2⊕V4ψ1⊕Vω1+2ψ1⊕V2ψ2⊕V2ω1⊕Vω1−2ψ1+2ψ2⊕Vω1+2ψ1−2ψ2⊕2V0⊕Vω1−2ψ1⊕V−2ψ2⊕Vω1−2ψ1−2ψ2⊕V−4ψ1
Made total 276 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A21 ↪ B13
2 out of 16
Subalgebra type: A21 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: 2A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A21
Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0), (0, 0, 1)
Contained up to conjugation as a direct summand of: A21+A11 , 2A21 , A21+2A11 .
Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−6
Positive simple generators: g6
Cartan symmetric matrix: (1)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4)
Decomposition of ambient Lie algebra: 5V2ω1⊕6V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ω1+2ψ1⊕V2ψ2⊕V2ω1−2ψ1+2ψ2⊕V2ω1⊕V4ψ1−2ψ2⊕V2ω1+2ψ1−2ψ2⊕2V0⊕V2ω1−2ψ1⊕V−4ψ1+2ψ2⊕V−2ψ2
Made total 8302747 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A31 ↪ B13
3 out of 16
Subalgebra type: A31 (click on type for detailed printout).
Centralizer: A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A21+A11
Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0)
Contained up to conjugation as a direct summand of: A31+A11 .
Elements Cartan subalgebra scaled to act by two by components: A31: (2, 3, 4): 6
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−6+g−7
Positive simple generators: g7+g6
Cartan symmetric matrix: (2/3)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6)
Decomposition of ambient Lie algebra: 2V3ω1⊕2V2ω1⊕2Vω1⊕3V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V3ω1+2ψ⊕V4ψ⊕Vω1+2ψ⊕2V2ω1⊕V3ω1−2ψ⊕V0⊕Vω1−2ψ⊕V−4ψ
Made total 2102 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A41 ↪ B13
4 out of 16
Subalgebra type: A41 (click on type for detailed printout).
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: A41: (2, 4, 4): 8
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−2+g−8
Positive simple generators: 2g8+2g2
Cartan symmetric matrix: (1/2)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (8)
Decomposition of ambient Lie algebra: V4ω1⊕5V2ω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ω1+4ψ⊕V2ω1+2ψ⊕V4ω1⊕V2ω1⊕V0⊕V2ω1−2ψ⊕V2ω1−4ψ
Made total 730989 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A101 ↪ B13
5 out of 16
Subalgebra type: A101 (click on type for detailed printout).
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1)
Elements Cartan subalgebra scaled to act by two by components: A101: (4, 6, 6): 20
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−1+g−5
Positive simple generators: 3g5+4g1
Cartan symmetric matrix: (1/5)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (20)
Decomposition of ambient Lie algebra: V6ω1⊕2V4ω1⊕V2ω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+2ψ⊕V6ω1⊕V2ω1⊕V4ω1−2ψ⊕V0
Made total 1004970 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A281 ↪ B13
6 out of 16
Subalgebra type: A281 (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Elements Cartan subalgebra scaled to act by two by components: A281: (6, 10, 12): 56
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−1+g−2+g−3
Positive simple generators: 6g3+10g2+6g1
Cartan symmetric matrix: (1/14)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (56)
Decomposition of ambient Lie algebra: V10ω1⊕V6ω1⊕V2ω1
Made total 7185 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A11 ↪ B13
7 out of 16
Subalgebra type: 2A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A11 .
Centralizer: A21 .
The semisimple part of the centralizer of the semisimple part of my centralizer: 2A11
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1)
Contained up to conjugation as a direct summand of: A21+2A11 .
Elements Cartan subalgebra scaled to act by two by components: A11: (1, 2, 2): 2, A11: (1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−9, g−1
Positive simple generators: g9, g1
Cartan symmetric matrix: (2002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2002)
Decomposition of ambient Lie algebra: V2ω2⊕3Vω1+ω2⊕V2ω1⊕3V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+ω2+2ψ⊕V2ψ⊕V2ω2⊕Vω1+ω2⊕V2ω1⊕V0⊕Vω1+ω2−2ψ⊕V−2ψ
Made total 359 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A21+A11 ↪ B13
8 out of 16
Subalgebra type: A21+A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A21 .
Centralizer: A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A21+A11
Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0)
Contained up to conjugation as a direct summand of: A21+2A11 .
Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 2): 4, A11: (0, 1, 2): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−6, g−7
Positive simple generators: g6, g7
Cartan symmetric matrix: (1002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4002)
Decomposition of ambient Lie algebra: 2V2ω1+ω2⊕V2ω2⊕V2ω1⊕3V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ω1+ω2+2ψ⊕V4ψ⊕V2ω2⊕V2ω1⊕V2ω1+ω2−2ψ⊕V0⊕V−4ψ
Made total 1950 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A21 ↪ B13
9 out of 16
Subalgebra type: 2A21 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A21 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 2): 4, A21: (0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−4+g−8, −g−2+g−7
Positive simple generators: g8+g4, g7−g2
Cartan symmetric matrix: (1001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4004)
Decomposition of ambient Lie algebra: V2ω1+2ω2⊕2V2ω2⊕2V2ω1
Made total 50361 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A31+A11 ↪ B13
10 out of 16
Subalgebra type: A31+A11 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A31 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Elements Cartan subalgebra scaled to act by two by components: A31: (2, 3, 4): 6, A11: (0, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−6+g−7, g−2
Positive simple generators: g7+g6, g2
Cartan symmetric matrix: (2/3002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6002)
Decomposition of ambient Lie algebra: V3ω1+ω2⊕V2ω2⊕Vω1+ω2⊕2V2ω1
Made total 454 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A12 ↪ B13
11 out of 16
Subalgebra type: A12 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A11 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: A12: (1, 2, 2): 2, (0, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: g−9, g2
Positive simple generators: g9, g−2
Cartan symmetric matrix: (2−1−12)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−1−12)
Decomposition of ambient Lie algebra: Vω1+ω2⊕2Vω2⊕2Vω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω2+4ψ⊕Vω1+2ψ⊕Vω1+ω2⊕V0⊕Vω2−2ψ⊕Vω1−4ψ
Made total 359 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B12 ↪ B13
12 out of 16
Subalgebra type: B12 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A11 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1)
Elements Cartan subalgebra scaled to act by two by components: B12: (1, 2, 2): 2, (0, -2, -2): 4
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: g−9, g5
Positive simple generators: g9, g−5
Cartan symmetric matrix: (2−1−11)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−2−24)
Decomposition of ambient Lie algebra: V2ω2⊕2Vω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ψ⊕V2ω2⊕V0⊕Vω1−2ψ
Made total 225983 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra G12 ↪ B13
13 out of 16
Subalgebra type: G12 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A31 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Elements Cartan subalgebra scaled to act by two by components: G12: (2, 3, 4): 6, (-1, -1, -2): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: g−6+g−7, g8
Positive simple generators: g7+g6, g−8
Cartan symmetric matrix: (2/3−1−12)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6−3−32)
Decomposition of ambient Lie algebra: Vω2⊕Vω1
Made total 450 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A21+2A11 ↪ B13
14 out of 16
Subalgebra type: A21+2A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A21+A11 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 2): 4, A11: (0, 1, 2): 2, A11: (0, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: g−6, g−7, g−2
Positive simple generators: g6, g7, g2
Cartan symmetric matrix: (100020002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (400020002)
Decomposition of ambient Lie algebra: V2ω1+ω2+ω3⊕V2ω3⊕V2ω2⊕V2ω1
Made total 452 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A13 ↪ B13
15 out of 16
Subalgebra type: A13 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A12 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Elements Cartan subalgebra scaled to act by two by components: A13: (1, 2, 2): 2, (0, -1, 0): 2, (-1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: g−9, g2, g1
Positive simple generators: g9, g−2, g−1
Cartan symmetric matrix: (2−10−12−10−12)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−10−12−10−12)
Decomposition of ambient Lie algebra: Vω1+ω3⊕Vω2
Made total 442 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B13 ↪ B13
16 out of 16
Subalgebra type: B13 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A12 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Elements Cartan subalgebra scaled to act by two by components: B13: (1, 2, 2): 2, (0, -1, 0): 2, (0, 0, -2): 4
Dimension of subalgebra generated by predefined or computed generators: 21.
Negative simple generators: g−9, g2, g3
Positive simple generators: g9, g−2, g−3
Cartan symmetric matrix: (2−10−12−10−11)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−10−12−20−24)
Decomposition of ambient Lie algebra: Vω2
Made total 442 arithmetic operations while solving the Serre relations polynomial system.
Of the 6 h element conjugacy classes 2 had their Weyl group orbits computed and buffered. The h elements and their computed orbit sizes follow. h element | orbit size |
(6, 10, 12) | size not computed |
(4, 6, 6) | size not computed |
(2, 4, 4) | size not computed |
(2, 3, 4) | size not computed |
(2, 2, 2) | 6 |
(1, 2, 2) | 12 |
Number of sl(2) subalgebras: 6.
Let h be in the Cartan subalgebra. Let α1,...,αn be simple roots with respect to h. Then the h-characteristic, as defined by E. Dynkin, is the n-tuple (α1(h),...,αn(h)).
The actual realization of h. The coordinates of h are given with respect to the fixed original simple basis. Note that the h-characteristic is computed using a possibly different simple basis, more precisely, with respect to any h-positive simple basis.
A regular semisimple subalgebra might contain an sl(2) such that the sl(2) has no centralizer in the regular semisimple subalgebra, but the regular semisimple subalgebra might fail to be minimal containing. This happens when another minimal containing regular semisimple subalgebra of equal rank nests as a root subalgebra in the containing SA. See Dynkin, Semisimple Lie subalgebras of semisimple Lie algebras, remark before Theorem 10.4.
The sl(2) submodules of the ambient Lie algebra are parametrized by their highest weight with respect to the Cartan element h of sl(2). In turn, the highest weight is a positive integer multiple of the fundamental highest weight ψ. Vlψ is l+1-dimensional.
Type + realization link | h-Characteristic | Realization of h | sl(2)-module decomposition of the ambient Lie algebra ψ= the fundamental sl(2)-weight. | Centralizer dimension | Type of semisimple part of centralizer, if known | The square of the length of the weight dual to h. | Dynkin index | Minimal containing regular semisimple SAs | Containing regular semisimple SAs in which the sl(2) has no centralizer |
A281 | (2, 2, 2) | (6, 10, 12) | V10ψ+V6ψ+V2ψ
| 0 | 0 | 56 | 28 | B^{1}_3; | B^{1}_3; |
A101 | (2, 2, 0) | (4, 6, 6) | V6ψ+2V4ψ+V2ψ+V0
| 1 | 0 | 20 | 10 | A^{1}_3; B^{1}_2; | A^{1}_3; B^{1}_2; |
A41 | (0, 2, 0) | (2, 4, 4) | V4ψ+5V2ψ+V0
| 1 | 0 | 8 | 4 | A^{2}_1+2A^{1}_1; A^{1}_2; | A^{2}_1+2A^{1}_1; A^{1}_2; |
A31 | (1, 0, 1) | (2, 3, 4) | 2V3ψ+2V2ψ+2Vψ+3V0
| 3 | A11 | 6 | 3 | A^{2}_1+A^{1}_1; | A^{2}_1+A^{1}_1; |
A21 | (2, 0, 0) | (2, 2, 2) | 5V2ψ+6V0
| 6 | 2A11 | 4 | 2 | 2A^{1}_1; A^{2}_1; | 2A^{1}_1; A^{2}_1; |
A11 | (0, 1, 0) | (1, 2, 2) | V2ψ+6Vψ+6V0
| 6 | A21+A11 | 2 | 1 | A^{1}_1; | A^{1}_1; |
Length longest root ambient algebra squared/4= 1/2
Given a root subsystem P, and a root subsubsystem P_0, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant e(P, P_0) (which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an sl(2) is an S-subalgebra in the root subalgebra generated by P, such that it has characteristic 2 for all simple roots of P lying in P_0, then e(P, P_0)= 0. It turns out by direct computation that, in the current case of B^{1}_3, e(P,P_0)= 0 implies that an S-sl(2) subalgebra of the root subalgebra generated by P with characteristic with 2's in the simple roots of P_0 always exists. Note that Theorem 10.3 is stated in one direction only.
h-characteristic: (2, 2, 2)
Length of the weight dual to h: 56
Simple basis ambient algebra w.r.t defining h: 3 vectors:
(1, 0, 0), (0, 1, 0), (0, 0, 1)
Containing regular semisimple subalgebra number 1: B^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: V10ψ+V6ψ+V2ψ
Below is one possible realization of the sl(2) subalgebra.
h=12h3+10h2+6h1
e=6/5g3+5g2+6g1
The polynomial system that corresponds to finding the h, e, f triple:
x1x4−6 x2x5−10 2x3x6−12
h-characteristic: (2, 2, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 3 vectors:
(1, 0, 0), (0, 1, 0), (0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{1}_3
Containing regular semisimple subalgebra number 2: B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: V6ψ+2V4ψ+V2ψ+V0
Below is one possible realization of the sl(2) subalgebra.
h=6h3+6h2+4h1
e=3g7+3/5g2+2g1
The polynomial system that corresponds to finding the h, e, f triple:
x2x5−4 x3x6+x1x4−6 2x1x4−6
h-characteristic: (0, 2, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 3 vectors:
(1, 0, 0), (0, 1, 0), (0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_1+2A^{1}_1
Containing regular semisimple subalgebra number 2: A^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: V4ψ+5V2ψ+V0
Below is one possible realization of the sl(2) subalgebra.
h=4h3+4h2+2h1
e=1/2g7+g6+1/5g2
The polynomial system that corresponds to finding the h, e, f triple:
2x1x4−2 x3x6+x2x5+2x1x4−4 2x2x5+2x1x4−4
h-characteristic: (1, 0, 1)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 3 vectors:
(1, 0, 0), (0, 1, 0), (0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: 2V3ψ+2V2ψ+2Vψ+3V0
Below is one possible realization of the sl(2) subalgebra.
h=4h3+3h2+2h1
e=1/2g7+g6
The polynomial system that corresponds to finding the h, e, f triple:
2x1x3−2 x2x4+2x1x3−3 2x2x4+2x1x3−4
h-characteristic: (2, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 3 vectors:
(1, 0, 0), (0, 1, 0), (0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2A^{1}_1
Containing regular semisimple subalgebra number 2: A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: 5V2ψ+6V0
Below is one possible realization of the sl(2) subalgebra.
h=2h3+2h2+2h1
e=g9+1/2g1
The polynomial system that corresponds to finding the h, e, f triple:
x2x4+x1x3−2 2x1x3−2 2x1x3−2
h-characteristic: (0, 1, 0)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 3 vectors:
(1, 0, 0), (0, 1, 0), (0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: V2ψ+6Vψ+6V0
Below is one possible realization of the sl(2) subalgebra.
h=2h3+2h2+h1
e=g9
The polynomial system that corresponds to finding the h, e, f triple:
x1x2−1 2x1x2−2 2x1x2−2
Calculator input for loading subalgebras directly without recomputation. Subalgebras found so far: 16
Orbit sizes: A^28_1: n/a; A^10_1: n/a; A^4_1: n/a; A^3_1: n/a; A^2_1: 6; A^1_1: 12;
Current subalgebra chain length: 0
SetOutputFile("subalgebras_B^{1}_3");
LoadSemisimpleSubalgebras {}(AmbientDynkinType=B^{1}{}\left(3\right);CurrentChain=\left(\right);NumExploredTypes=\left(\right);NumExploredHs=\left(\right);Subalgebras=\left((DynkinType=A^{1}{}\left(1\right);ElementsCartan=(122)
;generators=\left(g{}\left(-9\right), g{}\left(9\right)\right)), (DynkinType=A^{2}{}\left(1\right);ElementsCartan=(222)
;generators=\left(g{}\left(-6\right), g{}\left(6\right)\right)), (DynkinType=A^{3}{}\left(1\right);ElementsCartan=(234)
;generators=\left(g{}\left(-6\right)+g{}\left(-7\right), g{}\left(6\right)+g{}\left(7\right)\right)), (DynkinType=A^{4}{}\left(1\right);ElementsCartan=(244)
;generators=\left(g{}\left(-2\right)+g{}\left(-8\right), 2 g{}\left(2\right)+2 g{}\left(8\right)\right)), (DynkinType=A^{10}{}\left(1\right);ElementsCartan=(466)
;generators=\left(g{}\left(-1\right)+g{}\left(-5\right), 4 g{}\left(1\right)+3 g{}\left(5\right)\right)), (DynkinType=A^{28}{}\left(1\right);ElementsCartan=(61012)
;generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-3\right), 6 g{}\left(1\right)+10 g{}\left(2\right)+6 g{}\left(3\right)\right)), (DynkinType=2 A^{1}{}\left(1\right);ElementsCartan=(122100)
;generators=\left(g{}\left(-9\right), g{}\left(9\right), g{}\left(-1\right), g{}\left(1\right)\right)), (DynkinType=A^{2}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=(222012)
;generators=\left(g{}\left(-6\right), g{}\left(6\right), g{}\left(-7\right), g{}\left(7\right)\right)), (DynkinType=2 A^{2}{}\left(1\right);ElementsCartan=(222022)
;generators=\left(g{}\left(-8\right)+g{}\left(-4\right), g{}\left(8\right)+g{}\left(4\right), -g{}\left(-2\right)+g{}\left(-7\right), -g{}\left(2\right)+g{}\left(7\right)\right)), (DynkinType=A^{3}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=(234010)
;generators=\left(g{}\left(-6\right)+g{}\left(-7\right), g{}\left(6\right)+g{}\left(7\right), g{}\left(-2\right), g{}\left(2\right)\right)), (DynkinType=A^{1}{}\left(2\right);ElementsCartan=(1220−10)
;generators=\left(g{}\left(-9\right), g{}\left(9\right), g{}\left(2\right), g{}\left(-2\right)\right)), (DynkinType=B^{1}{}\left(2\right);ElementsCartan=(1220−2−2)
;generators=\left(g{}\left(-9\right), g{}\left(9\right), g{}\left(5\right), g{}\left(-5\right)\right)), (DynkinType=G^{1}{}\left(2\right);ElementsCartan=(234−1−1−2)
;generators=\left(g{}\left(-6\right)+g{}\left(-7\right), g{}\left(6\right)+g{}\left(7\right), g{}\left(8\right), g{}\left(-8\right)\right)), (DynkinType=A^{2}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=(222012010)
;generators=\left(g{}\left(-6\right), g{}\left(6\right), g{}\left(-7\right), g{}\left(7\right), g{}\left(-2\right), g{}\left(2\right)\right)), (DynkinType=A^{1}{}\left(3\right);ElementsCartan=(1220−10−100)
;generators=\left(g{}\left(-9\right), g{}\left(9\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(1\right), g{}\left(-1\right)\right)), (DynkinType=B^{1}{}\left(3\right);ElementsCartan=(1220−1000−2)
;generators=\left(g{}\left(-9\right), g{}\left(9\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(3\right), g{}\left(-3\right)\right))\right))