# Lie Algebras Graded by the Root Systems BC\(_r\), \(r\geq 2\)

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*Bruce Allison; Georgia Benkart; Yun Gao*

#### Table of Contents

# Table of Contents

## Lie Algebras Graded by the Root Systems BC$_{r}$, $r2$

- Table of Contents vii8 free
- I. Introduction 112 free
- II. The g-module decomposition of a BC[sub(r)]-graded Lie algebra, r ≥ 3 (excluding type D[sub(3)]) 1829
- General analysis 1829
- Multiplication in a BC[sub(r)]-graded Lie algebra, r ≥ 3 2233
- Properties of the algebras a and b 2435
- Lie algebras graded by C[sub(r)], r ≥ 3 2738
- Properties of the space C 2940
- Bilinear maps on C 3041
- The structure theorem for BC[sub(r)]-graded Lie algebras, r ≥ 3 (not of type D[sub(3)]) 3243
- The coordinate algebra b 3344

- III. Models for BC[sub(r)]-graded Lie algebras, r ≥ 3 (excluding type D[sub(3)]) 3647
- IV. The g-module decomposition of a BCr-graded Lie algebra with grading subalgebra of type B[sub(2)], C[sub(2)], D[sub(2)] or D[sub(3)] 5263
- The homomorphisms ϖ and τ 5364
- The homomorphism ϑ 5465
- The decomposition in the B[sub(2)], C[sub(2)], and D[sub(3)] cases 5566
- The coordinate algebra in the B[sub(2)], C[sub(2)], and D[sub(3)] cases 5667
- Calculation of the inner derivations in the B[sub(2)], C[sub(2)], and D[sub(3)] cases 5869
- BC[sub(r)]-coordinate algebras in the B[sub(2)], C[sub(2)], and D[sub(3)] cases 6778
- The decomposition in the D[sub(2)] case 6879
- The coordinate algebra in the D[sub(2)] case 7081
- Calculation of the inner derivations in the D[sub(2)] case 7283
- BC[sub(2)]-coordinate algebras in the D[sub(2)] case 7687
- BC[sub(r)]-coordinate algebras in general 7687

- V. Central extensions, derivations and invariant forms 7788
- Skew-dihedral homology 7990
- The universal central extension of a BC[sub(r)]-graded algebra, r ≥ 2 8394
- Derivations of BC[sub(r)]-graded Lie algebras, r ≥ 2 8798
- Invariant forms of BC[sub(r)]-graded Lie algebras, r ≥ 2 90101
- Appendix of proofs for types B[sub(2)], C[sub(2)], D[sub(2)] and D[sub(3)] 94105

- VI. Models of BC[sub(r)]-graded Lie algebras with grading subalgebra of type B[sub(2)], C[sub(2)], D[sub(2) or D[sub(3)] 100111
- Structurable algebras and the Kantor construction 101112
- Peirce decompositions in structurable algebras 103114
- Models of B[sub(r)]-graded Lie algebras with grading subalgebra of type B[sub(2)], D[sub(2)] or D[sub(3)] 106117
- The coordinate algebra in the B[sub(2)] case 112123
- Examples in the B[sub(2)] case 116127
- The coordinate algebra in the D[sub(2)] case 118129
- Examples in the D[sub(2)] case 123134
- The coordinate algebra in the D[sub(3)] case 124135
- Examples in the D[sub(3)] case 127138
- J-ternary algebras and the Lie algebra construction L(J,X) 128139
- Peirce decompositions in J-ternary algebras 128139
- Models of BC[sub(2)]-graded Lie algebras with grading subalgebra of type C[sub(2)] 130141
- The coordinate algebra in the C[sub(2)] case 132143
- Examples in the C[sub(2)] case 135146

- VII. Appendix: Peirce decompositions in structurable algebras 138149
- References 156167