G. M. Benkart, J. M. Osborn, and D. J. Britten :
“Flexible Lie-admissible algebras with the solvable radical of \( A^- \) abelian and Lie algebras with nondegenerate forms ,”
Hadronic J.
4 : 2
(1980–1981 ),
pp. 274–326 .
MR
613337 Zbl
0456.17002 article
Abstract
People
BibTeX
If \( A \) is a flexible Lie-admissible algebra, then \( A \) under the product \( (xy) = xy - yx \) is a Lie algebra, denoted by \( A^- \) . This paper investigates finite-dimensional, simple, flexible, Lie-admissible algebra \( A \) over an algebraically closed field of characteristic zero, for which the solvable radical \( R \) of \( A^- \) is Abelian. The technique employed is to view \( A \) as a module for a semisimple Lie algebra of derivations, and then to use representation theory to gain information about products in \( A \) . In the final section of the paper we construct examples of simple flexible Lie-admissible algebras from Lie algebras with nondegenerate associative symmetric bilinear forms. These examples illustrate the great diversity of algebras which can occur when the assumption that \( R \) is Abelian is dropped.
@article {key613337m,
AUTHOR = {Benkart, Georgia M. and Osborn, J. Marshall
and Britten, Daniel J.},
TITLE = {Flexible {L}ie-admissible algebras with
the solvable radical of \$A^-\$ abelian
and {L}ie algebras with nondegenerate
forms},
JOURNAL = {Hadronic J.},
FJOURNAL = {Hadronic Journal},
VOLUME = {4},
NUMBER = {2},
YEAR = {1980--1981},
PAGES = {274--326},
NOTE = {\textit{Proceedings of the third workshop
on {L}ie-admissible formulations} (4--9
August 1980, Boston). MR:613337. Zbl:0456.17002.},
ISSN = {0162-5519},
}
G. Benkart, J. M. Osborn, and D. Britten :
“On applications of isotopy to real division algebras ,”
pp. 497–529
in
Proceedings of the third workshop on Lie-admissible formulations
(4–9 August 1980, Boston ),
published as Hadronic J.
4 : 2 .
Hadronic Press (Nonantum, MA ),
1980–1981 .
MR
613342 Zbl
0451.17002 incollection
Abstract
People
BibTeX
In this paper we illustrate how the notion of isotopy can be used to solve various problems concerning finite-dimensional division algebras over the real numbers. In particular, we show that the 8-dimensional division algebras which have the same derivation algebra as the octonions, and hence which most resemble the octonions, are not in general isotopes of the octonions. Secondly, using a result of Hopf, we argue that every commutative division algebra is the reals or is isomorphic to a special kind of isotope of the complex numbers. Finally, by considering a certain class of algebras, we show how isotopy is a useful tool for determining necessary and sufficient conditions on the multiplication constants in order to have a division algebra.
@article {key613342m,
AUTHOR = {Benkart, Georgia and Osborn, J. Marshall
and Britten, Daniel},
TITLE = {On applications of isotopy to real division
algebras},
JOURNAL = {Hadronic J.},
FJOURNAL = {Hadronic Journal},
VOLUME = {4},
NUMBER = {2},
YEAR = {1980--1981},
PAGES = {497--529},
NOTE = {\textit{Proceedings of the third workshop
on {L}ie-admissible formulations} (4--9
August 1980, Boston). MR:613342. Zbl:0451.17002.},
ISSN = {0162-5519},
}
G. M. Benkart, D. J. Britten, and J. M. Osborn :
“Real flexible division algebras Canad. J. Math.
34 : 3
(1982 ),
pp. 550–588 .
MR
663304 Zbl
0469.17001 article
Abstract
People
BibTeX
In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the \( 3{\times} 3 \) complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [Benkart and Osborn 1981a, 1981b].
@article {key663304m,
AUTHOR = {Benkart, Georgia M. and Britten, Daniel
J. and Osborn, J. Marshall},
TITLE = {Real flexible division algebras},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal
Canadien de Math\'ematiques},
VOLUME = {34},
NUMBER = {3},
YEAR = {1982},
PAGES = {550--588},
DOI = {10.4153/CJM-1982-039-x},
NOTE = {MR:663304. Zbl:0469.17001.},
ISSN = {0008-414X},
}
G. Benkart :
“A Kac–Moody bibliography and some related references ,”
pp. 111–135
in
Lie algebras and related topics
(26 June–6 July 1984, Windsor, ON ).
Edited by D. J. Britten, F. W. Lemire, and R. V. Moody .
CMS Conference Proceedings 5 .
American Mathematical Society (Providence, RI ),
1986 .
MR
832196 Zbl
0578.17013 incollection
People
BibTeX
@incollection {key832196m,
AUTHOR = {Benkart, Georgia},
TITLE = {A {K}ac--{M}oody bibliography and some
related references},
BOOKTITLE = {Lie algebras and related topics},
EDITOR = {Britten, D. J. and Lemire, F. W. and
Moody, R. V.},
SERIES = {CMS Conference Proceedings},
NUMBER = {5},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1986},
PAGES = {111--135},
NOTE = {(26 June--6 July 1984, Windsor, ON).
MR:832196. Zbl:0578.17013.},
ISSN = {0731-1036},
ISBN = {9780821860090},
}
G. Benkart :
“Cartan subalgebras in Lie algebras of Cartan type ,”
pp. 157–187
in
Lie algebras and related topics
(26 June–6 July 1984, Windsor, ON ).
Edited by D. J. Britten, F. W. Lemire, and R. V. Moody .
CMS Conference Proceedings 5 .
American Mathematical Society (Providence, RI ),
1986 .
MR
832198 Zbl
0581.17006 incollection
Abstract
People
BibTeX
The known finite dimensional simple Lie algebras over an algebraically closed field of characteristic \( p\geq 7 \) are either classical (analogues of complex Lie algebras) or of Cartan type. Each Lie algebra \( L \) of Cartan type possesses a certain maximal subalgebra \( L_0 \) which is invariant under the automorphisms of \( L \) , and every Cartan subalgebra \( L_0 \) is a Cartan subalgebra of \( L \) . The main result of this work is to show that the Cartan subalgebras of \( L_0 \) are conjugate under the group generated by the generalized exponential mappings defined by
\[ E^z = 1 + \sum_{j=1}^{p-1} \left(\prod_{\nu = 1}^j (\nu \vert -\psi)^{-1}\right)(adz)^j \]
where \( z\in L_0 \) has \( (adz)^{p^k} = 0 \) , and where
\[ \psi = (adz)^p + \cdots + (adz)^{p^{k-1}} .\]
As a consequence, all the Cartan subalgebras of \( L \) in \( L_0 \) have the same dimension, and this dimension is computed for each type of algebra.
@incollection {key832198m,
AUTHOR = {Benkart, Georgia},
TITLE = {Cartan subalgebras in {L}ie algebras
of {C}artan type},
BOOKTITLE = {Lie algebras and related topics},
EDITOR = {Britten, D. J. and Lemire, F. W. and
Moody, R. V.},
SERIES = {CMS Conference Proceedings},
NUMBER = {5},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1986},
PAGES = {157--187},
NOTE = {(26 June--6 July 1984, Windsor, ON).
MR:832198. Zbl:0581.17006.},
ISSN = {0731-1036},
ISBN = {9780821860090},
}
G. M. Benkart, D. J. Britten, and F. W. Lemire :
Stability in modules for classical Lie algebras — a constructive approach Memoirs of the American Mathematical Society 430 .
American Mathematical Society (Providence, RI ),
1990 .
MR
1010997 Zbl
0706.17003 book
People
BibTeX
@book {key1010997m,
AUTHOR = {Benkart, G. M. and Britten, D. J. and
Lemire, F. W.},
TITLE = {Stability in modules for classical {L}ie
algebras---a constructive approach},
SERIES = {Memoirs of the American Mathematical
Society},
NUMBER = {430},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1990},
PAGES = {vi+165},
DOI = {10.1090/memo/0430},
NOTE = {MR:1010997. Zbl:0706.17003.},
ISSN = {0065-9266},
ISBN = {9780821824924},
}
G. Benkart, D. Britten, and F. Lemire :
“Projection maps for tensor products of \( \mathfrak{gl}(r,\mathbb{C}) \) -representations Publ. Res. Inst. Math. Sci.
28 : 6
(1992 ),
pp. 983–1010 .
MR
1203757 Zbl
0830.17004 article
Abstract
People
BibTeX
We investigate the tensor product
\[ \mathcal{T} = V(\lambda^1)\otimes \cdots \otimes V(\lambda^m) \]
of the finite dimensional irreducible \( \mathcal{G} = \mathfrak{gl}(r,\mathbb{C}) \) modules labelled by partitions \( \lambda^1,\dots \) , \( \lambda^m \) of \( m \) not necessarily distinct numbers \( n_1,\dots \) , \( n_m \) respectively. We determine the centralizer algebra \( \operatorname{End}_{\mathcal{G}}(\mathcal{T}) \) and the projection maps of \( \mathcal{T} \) onto its irreducible \( \mathcal{G} \) -summands and give an explicit construction of the corresponding maximal vectors. In the special case that \( n_i = 1 \) for \( i = 1,\dots \) , \( m \) , the results reduce to the well-known results of Schur and Weyl.
@article {key1203757m,
AUTHOR = {Benkart, Georgia and Britten, Daniel
and Lemire, Frank},
TITLE = {Projection maps for tensor products
of \$\mathfrak{gl}(r,\mathbb{C})\$-representations},
JOURNAL = {Publ. Res. Inst. Math. Sci.},
FJOURNAL = {Publications of the Research Institute
for Mathematical Sciences},
VOLUME = {28},
NUMBER = {6},
YEAR = {1992},
PAGES = {983--1010},
DOI = {10.2977/prims/1195167734},
NOTE = {MR:1203757. Zbl:0830.17004.},
ISSN = {0034-5318},
}
G. Benkart, D. Britten, and F. Lemire :
“Modules with bounded weight multiplicities for simple Lie algebras Math. Z.
225 : 2
(June 1997 ),
pp. 333–353 .
MR
1464935 Zbl
0884.17004 article
People
BibTeX
@article {key1464935m,
AUTHOR = {Benkart, Georgia and Britten, Daniel
and Lemire, Frank},
TITLE = {Modules with bounded weight multiplicities
for simple {L}ie algebras},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {225},
NUMBER = {2},
MONTH = {June},
YEAR = {1997},
PAGES = {333--353},
DOI = {10.1007/PL00004314},
NOTE = {MR:1464935. Zbl:0884.17004.},
ISSN = {0025-5874},
}
