Lie superalgebras.(English)Zbl 0366.17012
Lie superalgebras (author’s notion) are defined as \(\mathbb Z_2\)-graded generalizations of Lie algebras on a vector space \(A_0 +A_1\) such that graded skew symmetry and a graded version of Jacobi-identity
\[ [a, [b,c]] = [[a,b],c] + (-1)^{ik} [b,[a,c]] \]
for \(a\) in \(A_i\), \(b\) in \(A_k\) holds. Sometimes they are called “graded Lie algebras” which is misleading since they are not Lie algebras with a compatible graduation. Mathematicians studied them first some twenty years ago. Recently they gained interest by physicists, especially in the classification of particles with different statistics. This work now is a nearly complete algebraic theory of finite-dimensional superalgebras, giving general constructions as well as classifications of the complex semisimple ones.
In detail: It is shown that any superalgebra \(G\) has a unique maximal solvable ideal (radical) \(R\) such that \(G/R\) is semisimple, i.e. contains no solvable ideals. However, \(G\) in general is not a semidirect sum of \(G\) and \(G/R\). Lie’s theorem that a finite-dimensional irreducible representation of a solvable algebra is one-dimensional no longer is true here. A classification of these representations is given in section 5, and in addition a necessary and sufficient condition for such a representation to be one-dimensional. Furthermore the decomposition of semisimple algebras in a direct sum of simple ones is no longer possible here, but a description of semisimple algebras in terms of simple ones is derived.
Chapters 2–4 contain the main part of the work, the principal difficulty of the classification of the complex semisimple algebras lying in the fact that the Killing form may be degenerate. This result is given in two kinds of simple algebras, the classical ones: besides ordinary simple Lie algebras four series of the form \(A(m,n)\), \(B(m,n)\), \(C(n)\), \(D(m,n)\), \(m- n\ne 1\), and two exceptional algebras \(F(4)\), \(G(3)\), and the second kind with vanishing Killing form, two series \(A(n,n)\), \(D(n+1,n)\), two strange series \(P(n)\), \(Q(n)\), and a one-parameter family of 17-dimensional exceptional algebras \(D(1,2,\alpha)\). Also a classification of simple complex 2-graded superalgebras is given, where besides the above ones there are four series \(W(n)\), \(S(n)\), \(\tilde S(n)\) and \(H(n)\) constructed in terms of Grassmann algebras and Hamiltonian vector fields. Recently B. Kostant has given a rigorous globalization of superalgebras in terms of supermanifolds.
\[ [a, [b,c]] = [[a,b],c] + (-1)^{ik} [b,[a,c]] \]
for \(a\) in \(A_i\), \(b\) in \(A_k\) holds. Sometimes they are called “graded Lie algebras” which is misleading since they are not Lie algebras with a compatible graduation. Mathematicians studied them first some twenty years ago. Recently they gained interest by physicists, especially in the classification of particles with different statistics. This work now is a nearly complete algebraic theory of finite-dimensional superalgebras, giving general constructions as well as classifications of the complex semisimple ones.
In detail: It is shown that any superalgebra \(G\) has a unique maximal solvable ideal (radical) \(R\) such that \(G/R\) is semisimple, i.e. contains no solvable ideals. However, \(G\) in general is not a semidirect sum of \(G\) and \(G/R\). Lie’s theorem that a finite-dimensional irreducible representation of a solvable algebra is one-dimensional no longer is true here. A classification of these representations is given in section 5, and in addition a necessary and sufficient condition for such a representation to be one-dimensional. Furthermore the decomposition of semisimple algebras in a direct sum of simple ones is no longer possible here, but a description of semisimple algebras in terms of simple ones is derived.
Chapters 2–4 contain the main part of the work, the principal difficulty of the classification of the complex semisimple algebras lying in the fact that the Killing form may be degenerate. This result is given in two kinds of simple algebras, the classical ones: besides ordinary simple Lie algebras four series of the form \(A(m,n)\), \(B(m,n)\), \(C(n)\), \(D(m,n)\), \(m- n\ne 1\), and two exceptional algebras \(F(4)\), \(G(3)\), and the second kind with vanishing Killing form, two series \(A(n,n)\), \(D(n+1,n)\), two strange series \(P(n)\), \(Q(n)\), and a one-parameter family of 17-dimensional exceptional algebras \(D(1,2,\alpha)\). Also a classification of simple complex 2-graded superalgebras is given, where besides the above ones there are four series \(W(n)\), \(S(n)\), \(\tilde S(n)\) and \(H(n)\) constructed in terms of Grassmann algebras and Hamiltonian vector fields. Recently B. Kostant has given a rigorous globalization of superalgebras in terms of supermanifolds.
Reviewer: Hans Tilgner (Berlin)
MSC:
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B65 | Infinite-dimensional Lie (super)algebras |
| 17B70 | Graded Lie (super)algebras |
Keywords:
\(\mathbb Z_2\)-graded generalizations of Lie algebras;Lie superalgebras;graded skew symmetry;graded version of Jacobi-identityCite
Full Text:DOI
References:
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