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Lie algebra so(5), type B12
Semisimple complex Lie subalgebras

so(5), type B12
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.

Page generated by the calculator project.
Up to linear equivalence, there are total 5 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: 1.484.
173923 total arithmetic operations performed = 146696 additions and 27227 multiplications.
The base field over which the subalgebras were realized is: Q
Number of root subalgebras other than the Cartan and full subalgebra: 3
Number of sl(2)'s: 3
Subalgebra A11B12
1 out of 5
Subalgebra type: A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A11
Basis of Cartan of centralizer: 1 vectors: (1, 0)
Contained up to conjugation as a direct summand of: 2A11 .

Elements Cartan subalgebra scaled to act by two by components: A11: (1, 2): 2
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g4
Positive simple generators: g4
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: V2ω12Vω13V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ψVω1+2ψV2ω1V0Vω12ψV4ψ
Made total 274 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A21B12
2 out of 5
Subalgebra type: A21 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: B12
Basis of Cartan of centralizer: 1 vectors: (0, 1)

Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g3
Positive simple generators: g3
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: 3V2ω1V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ω1+2ψV2ω1V0V2ω12ψ
Made total 129363 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A101B12
3 out of 5
Subalgebra type: A101 (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B12

Elements Cartan subalgebra scaled to act by two by components: A101: (4, 6): 20
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g1+g2
Positive simple generators: 3g2+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: V6ω1V2ω1
Made total 1774 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A11B12
4 out of 5
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: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B12

Elements Cartan subalgebra scaled to act by two by components: A11: (1, 2): 2, A11: (1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g4, g1
Positive simple generators: g4, 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: V2ω2Vω1+ω2V2ω1
Made total 357 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B12B12
5 out of 5
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: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B12

Elements Cartan subalgebra scaled to act by two by components: B12: (1, 2): 2, (0, -2): 4
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: g4, g2
Positive simple generators: g4, g2
Cartan symmetric matrix: (2111)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2224)
Decomposition of ambient Lie algebra: V2ω2
Made total 357 arithmetic operations while solving the Serre relations polynomial system.

Of the 3 h element conjugacy classes 2 had their Weyl group orbits computed and buffered. The h elements and their computed orbit sizes follow.
h elementorbit size
(4, 6)size not computed
(2, 2)4
(1, 2)4

Number of sl(2) subalgebras: 3.
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 linkh-CharacteristicRealization of hsl(2)-module decomposition of the ambient Lie algebra
ψ= the fundamental sl(2)-weight.
Centralizer dimensionType of semisimple part of centralizer, if knownThe square of the length of the weight dual to h.Dynkin index Minimal containing regular semisimple SAsContaining regular semisimple SAs in which the sl(2) has no centralizer
A101(2, 2)(4, 6)V6ψ+V2ψ
0 02010B^{1}_2; B^{1}_2;
A21(2, 0)(2, 2)3V2ψ+V0
1 0422A^{1}_1; A^{2}_1; 2A^{1}_1; A^{2}_1;
A11(0, 1)(1, 2)V2ψ+2Vψ+3V0
3 A1121A^{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}_2, 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)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 2 vectors: (1, 0), (0, 1)
Containing regular semisimple subalgebra number 1: B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: V6ψ+V2ψ
Below is one possible realization of the sl(2) subalgebra.
h=6h2+4h1
e=3/2g2+4g1
The polynomial system that corresponds to finding the h, e, f triple:
x1x34 2x2x46 


h-characteristic: (2, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 2 vectors: (1, 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: 3V2ψ+V0
Below is one possible realization of the sl(2) subalgebra.
h=2h2+2h1
e=g4+1/2g1
The polynomial system that corresponds to finding the h, e, f triple:
x2x4+x1x32 2x1x32 


h-characteristic: (0, 1)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 2 vectors: (1, 0), (0, 1)
Containing regular semisimple subalgebra number 1: A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: V2ψ+2Vψ+3V0
Below is one possible realization of the sl(2) subalgebra.
h=2h2+h1
e=g4
The polynomial system that corresponds to finding the h, e, f triple:
x1x21 2x1x22 


Calculator input for loading subalgebras directly without recomputation. Subalgebras found so far: 5
Orbit sizes: A^10_1: n/a; A^2_1: 4; A^1_1: 4;
Current subalgebra chain length: 0


SetOutputFile("subalgebras_B^{1}_2");
LoadSemisimpleSubalgebras {}(AmbientDynkinType=B^{1}{}\left(2\right);CurrentChain=\left(\right);NumExploredTypes=\left(\right);NumExploredHs=\left(\right);Subalgebras=\left((DynkinType=A^{1}{}\left(1\right);ElementsCartan=(12)
;generators=\left(g{}\left(-4\right), g{}\left(4\right)\right)), (DynkinType=A^{2}{}\left(1\right);ElementsCartan=(22)
;generators=\left(g{}\left(-3\right), g{}\left(3\right)\right)), (DynkinType=A^{10}{}\left(1\right);ElementsCartan=(46)
;generators=\left(g{}\left(-1\right)+g{}\left(-2\right), 4 g{}\left(1\right)+3 g{}\left(2\right)\right)), (DynkinType=2 A^{1}{}\left(1\right);ElementsCartan=(1210)
;generators=\left(g{}\left(-4\right), g{}\left(4\right), g{}\left(-1\right), g{}\left(1\right)\right)), (DynkinType=B^{1}{}\left(2\right);ElementsCartan=(1202)
;generators=\left(g{}\left(-4\right), g{}\left(4\right), g{}\left(2\right), g{}\left(-2\right)\right))\right))