I. M. Gel’fand, B. L. Feĭgin, and D. B. Fuks :
“Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of foliations Funkcional. Anal. i Priložen.
8 : 2
(1974 ),
pp. 13–29 .
An English translation was published in Funct. Anal. Appl. 8 :2 (1974)
MR
356082 article
People
BibTeX
@article {key356082m,
AUTHOR = {Gel\cprime fand, I. M. and Fe\u{\i}gin,
B. L. and Fuks, D. B.},
TITLE = {Cohomology of the {L}ie algebra of formal
vector fields with coefficients in its
dual space and variations of characteristic
classes of foliations},
JOURNAL = {Funkcional. Anal. i Prilo\v{z}en.},
FJOURNAL = {Funkcional\cprime nyi Analiz i ego Prilo\v{z}enija.
Akademija Nauk SSSR},
VOLUME = {8},
NUMBER = {2},
YEAR = {1974},
PAGES = {13--29},
URL = {http://mi.mathnet.ru/eng/faa2326},
NOTE = {An English translation was published
in \textit{Funct. Anal. Appl.} \textbf{8}:2
(1974). MR:356082.},
ISSN = {0374-1990},
}
I. M. Gel’fand, B. L. Feigin, and D. B. Fuks :
“Cohomologies of the Lie algebra of formal vector fields with coefficients in its adjoint space and variations of characteristic classes of foliations Funct. Anal. Appl.
8 : 2
(April 1974 ),
pp. 99–112 .
English translation of Russian original published in Funkcional. Anal. i Priložen. 8 :2 (1974)
Zbl
0298.57011 article
People
BibTeX
@article {key0298.57011z,
AUTHOR = {Gel\cprime fand, I. M. and Feigin, B.
L. and Fuks, D. B.},
TITLE = {Cohomologies of the {L}ie algebra of
formal vector fields with coefficients
in its adjoint space and variations
of characteristic classes of foliations},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {8},
NUMBER = {2},
MONTH = {April},
YEAR = {1974},
PAGES = {99--112},
DOI = {10.1007/BF01078594},
NOTE = {English translation of Russian original
published in \textit{Funkcional. Anal.
i Prilo\v{z}en.} \textbf{8}:2 (1974).
Zbl:0298.57011.},
ISSN = {0016-2663},
}
I. M. Gel’fand, B. L. Feigin, and D. B. Fuks :
“Cohomologies of infinite dimensional Lie algebras and Laplace operators Funkts. Anal. Prilozh.
12 : 4
(1978 ),
pp. 1–5 .
An English translation was published in Funct. Anal. Appl. 12 :4 (1979)
MR
515625 Zbl
0396.17008 article
People
BibTeX
@article {key515625m,
AUTHOR = {Gel\cprime fand, I. M. and Feigin, B.
L. and Fuks, D. B.},
TITLE = {Cohomologies of infinite dimensional
{L}ie algebras and {L}aplace operators},
JOURNAL = {Funkts. Anal. Prilozh.},
FJOURNAL = {Funktsional'ny\u{\i} Analiz i ego Prilozheniya},
VOLUME = {12},
NUMBER = {4},
YEAR = {1978},
PAGES = {1--5},
URL = {http://mi.mathnet.ru/eng/faa2023},
NOTE = {An English translation was published
in \textit{Funct. Anal. Appl.} \textbf{12}:4
(1979). MR:515625. Zbl:0396.17008.},
ISSN = {0374-1990},
}
B. L. Feĭgin and D. B. Fuks :
“Invariant differential operators on the line Funkts. Anal. Prilozh.
13 : 4
(1979 ),
pp. 91–92 .
An English translation was published in Funct. Anal. Appl. 13 :4 (1980)
MR
554429 Zbl
0425.58024 article
People
BibTeX
@article {key554429m,
AUTHOR = {Fe\u{\i}gin, B. L. and Fuks, D. B.},
TITLE = {Invariant differential operators on
the line},
JOURNAL = {Funkts. Anal. Prilozh.},
FJOURNAL = {Funktsional'ny\u{\i} Analiz i ego Prilozheniya},
VOLUME = {13},
NUMBER = {4},
YEAR = {1979},
PAGES = {91--92},
URL = {http://mi.mathnet.ru/eng/faa1955},
NOTE = {An English translation was published
in \textit{Funct. Anal. Appl.} \textbf{13}:4
(1980). MR:554429. Zbl:0425.58024.},
ISSN = {0374-1990},
}
I. M. Gel’fand, B. L. Feigin, and D. B. Fuks :
“Cohomology of infinite-dimensional Lie algebras and Laplace operators Funct. Anal. Appl.
12 : 4
(1979 ),
pp. 243–247 .
English translation of Russian original published in Funkts. Anal. Prilozh. 12 :4 (1978)
Zbl
0404.17008 article
Abstract
People
BibTeX
Although appreciable progress has been made in the last 10 years in calculating the cohomology of infinite-dimensional Lie algebras, some of the problems of this circle appear to be unapproachable to this day; this relates, in the first place, to the algebras of Hamiltonian and divergence-free vector fields and to the algebras \( L_1(n) \subset W_n \) . In the solutions of these problems there has been hardly any successful progress without the introduction of new methods.
Such methods could consist of the investigation of the Laplace operators induced by any metrics in the cochain complex. The impression is added that if the metric is introduced in a reasonable way, then the eigenvalues and eigenvectors of the Laplace operator, and hence also the cohomology, will turn out to be calculable. In the following papers we propose to investigate this possibility systematically. In this paper we will analyze one example in which the program indicated can be realized successfully all the way through. It is true that the question concerns a Lie algebra whose cohomology is known: the algebra \( L_1(1) \) of formal vector fields on the line having trivial 1-jet. In what follows we will denote this algebra by \( L_1 \) and whenever the question is of cohomology, we mean continuous cohomology with trivial coefficients.
The cohomology of the algebra \( L_1 \) , as well as of the algebras \( L_k(1) \) with \( k > 1 \) , was found in [Goncharova 1973]. Goncharova’s calculation is awkward and does not allow one to find the cohomology of the algebra \( L_1 \) without finding the cohomology of the other algebras \( L_k \) ).
Our paper significantly, as it seems to us, clarifies Goncharova’s theorem and contains some new results. In particular, we give an explicit description of cocycles representing the cohomology classes of the algebra \( L_1 \) (we also apply this method of describing cocycles to the algebras \( L_k \) ).
We recall that in the cohomology theory of infinite-dimensional Lie algebras, the cohomology of the algebra \( L_1 \) has special significance; the reason is explained in Goncharova’s paper. One can add that recently Bukhshtaber and Shokurov discovered a connection between these cohomologies and the Adams–Novikov spectral sequence in complex cobordism theory [Bukhshtaber and Shokurov 1978].
Our paper owes much to Goncharova’s paper: a whole series of our arguments is implicitly contained in it. It remains to indicate also the connection or in any case the analogy between what is presented below and the theory of Kats–Muda [1974].
@article {key0404.17008z,
AUTHOR = {Gel\cprime fand, I. M. and Feigin, B.
L. and Fuks, D. B.},
TITLE = {Cohomology of infinite-dimensional {L}ie
algebras and {L}aplace operators},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {12},
NUMBER = {4},
YEAR = {1979},
PAGES = {243--247},
DOI = {10.1007/BF01076378},
NOTE = {English translation of Russian original
published in \textit{Funkts. Anal. Prilozh.}
\textbf{12}:4 (1978). Zbl:0404.17008.},
ISSN = {0016-2663},
}
B. L. Feigin and D. B. Fuks :
“Invariant differential operators on the line Funct. Anal. Appl.
13 : 4
(1979 ),
pp. 314–315 .
Translation of Russian original published in Funkts. Anal. Prilozh. 13 :4 (1979)
Zbl
0434.58021 article
Abstract
People
BibTeX
The purpose of this note is to construct a new series of invariant differential operators. (Concerning the general theory of invariant differential operators, cf. the preprint [Krillov 1979] and the literature cited there.) Our operators act in tensor fields of various type on the line and are skew-symmetric in their arguments. The basic field is always \( C \) .
@article {key0434.58021z,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Invariant differential operators on
the line},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {13},
NUMBER = {4},
YEAR = {1979},
PAGES = {314--315},
DOI = {10.1007/BF01078385},
NOTE = {Translation of Russian original published
in \textit{Funkts. Anal. Prilozh.} \textbf{13}:4
(1979). Zbl:0434.58021.},
ISSN = {0016-2663},
}
B. L. Feĭgin and D. B. Fuks :
“Homology of the Lie algebra of vector fields on the line Funkts. Anal. Prilozh.
14 : 3
(1980 ),
pp. 45–60 .
An English translation was published in Funct. Anal. Appl. 14 :3 (1980)
MR
583800 Zbl
0482.57010 article
People
BibTeX
@article {key583800m,
AUTHOR = {Fe\u{\i}gin, B. L. and Fuks, D. B.},
TITLE = {Homology of the {L}ie algebra of vector
fields on the line},
JOURNAL = {Funkts. Anal. Prilozh.},
FJOURNAL = {Funktsional'ny\u{\i} Analiz i ego Prilozheniya},
VOLUME = {14},
NUMBER = {3},
YEAR = {1980},
PAGES = {45--60},
URL = {http://mi.mathnet.ru/eng/faa1830},
NOTE = {An English translation was published
in \textit{Funct. Anal. Appl.} \textbf{14}:3
(1980). MR:583800. Zbl:0482.57010.},
ISSN = {0374-1990},
}
B. L. Feĭgin and D. B. Fuks :
“Homology of the Lie algebra of vector fields on the line Funct. Anal. Appl.
14 : 3
(July 1980 ),
pp. 201–212 .
English translation of Russian original published in Funkts. Anal. Prilozh. 14 :3 (1980)
Zbl
0487.57011 article
People
BibTeX
@article {key0487.57011z,
AUTHOR = {Fe\u{\i}gin, B. L. and Fuks, D. B.},
TITLE = {Homology of the {L}ie algebra of vector
fields on the line},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {14},
NUMBER = {3},
MONTH = {July},
YEAR = {1980},
PAGES = {201--212},
DOI = {10.1007/BF01086182},
NOTE = {English translation of Russian original
published in \textit{Funkts. Anal. Prilozh.}
\textbf{14}:3 (1980). Zbl:0487.57011.},
ISSN = {0016-2663},
}
B. L. Feĭgin and D. B. Fuks :
“Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra Funct. Anal. Appl.
16 : 2
(1982 ),
pp. 114–126 .
English translation of Russian original published in Funkts. Anal. Prilozh. 16 :2 (1982)
MR
659165 Zbl
0505.58031 article
Abstract
People
BibTeX
The main result of this article is Theorem 1.1, which gives a complete classification of skew-symmetric differential operators, acting in tensor fields on the line and invariant with respect to diffeomorphisms of the line. The statement of this theorem was stated as a hypothesis in our note [1979].
@article {key0505.58031z,
AUTHOR = {Fe\u{\i}gin, B. L. and Fuks, D. B.},
TITLE = {Invariant skew-symmetric differential
operators on the line and {V}erma modules
over the {V}irasoro algebra},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {16},
NUMBER = {2},
YEAR = {1982},
PAGES = {114--126},
DOI = {10.1007/BF01081626},
NOTE = {English translation of Russian original
published in \textit{Funkts. Anal. Prilozh.}
\textbf{16}:2 (1982). Zbl:0505.58031.},
ISSN = {0016-2663},
}
B. L. Feĭgin and D. B. Fuks :
“Casimir operators in modules over the Virasoro algebra Dokl. Akad. Nauk SSSR
269 : 5
(1983 ),
pp. 1057–1060 .
An English translation was published in Sov. Math., Dokl. 27 (1983)
MR
701159 article
People
BibTeX
@article {key701159m,
AUTHOR = {Fe\u{\i}gin, B. L. and Fuks, D. B.},
TITLE = {Casimir operators in modules over the
{V}irasoro algebra},
JOURNAL = {Dokl. Akad. Nauk SSSR},
FJOURNAL = {Doklady Akademii Nauk SSSR},
VOLUME = {269},
NUMBER = {5},
YEAR = {1983},
PAGES = {1057--1060},
URL = {http://mi.mathnet.ru/eng/faa1577},
NOTE = {An English translation was published
in \textit{Sov. Math., Dokl.} \textbf{27}
(1983). MR:701159.},
ISSN = {0002-3264},
}
B. L. Feigin and D. B. Fuks :
“Verma modules over the Virasoro algebra Funkts. Anal. Prilozh.
17 : 3
(July 1983 ),
pp. 91–92 .
An English translation was published in Funct. Anal. Appl. 17 :3 (1983)
MR
714236 Zbl
0526.17010 article
People
BibTeX
@article {key714236m,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Verma modules over the {V}irasoro algebra},
JOURNAL = {Funkts. Anal. Prilozh.},
FJOURNAL = {Funktsional'ny\u{\i} Analiz i ego Prilozheniya},
VOLUME = {17},
NUMBER = {3},
MONTH = {July},
YEAR = {1983},
PAGES = {91--92},
URL = {http://mi.mathnet.ru/eng/faa1570},
NOTE = {An English translation was published
in \textit{Funct. Anal. Appl.} \textbf{17}:3
(1983). MR:714236. Zbl:0526.17010.},
ISSN = {0374-1990},
}
B. L. Feigin and D. B. Fuks :
“Verma modules over the Virasoro algebra Funct. Anal. Appl.
17 : 3
(1983 ),
pp. 241–242 .
English translation of Russian original published in Funkts. Anal. Prilozh. 17 :3 (1983)
Zbl
0529.17010 article
People
BibTeX
@article {key0529.17010z,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Verma modules over the {V}irasoro algebra},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {17},
NUMBER = {3},
YEAR = {1983},
PAGES = {241--242},
DOI = {10.1007/BF01078118},
NOTE = {English translation of Russian original
published in \textit{Funkts. Anal. Prilozh.}
\textbf{17}:3 (1983). Zbl:0529.17010.},
ISSN = {0016-2663},
}
B. L. Feigin and D. B. Fuks :
“Casimir operators in modules over the Virasoro algebra ,”
Sov. Math., Dokl.
27
(1983 ),
pp. 465–469 .
English translation of Russian original published in Dokl. Akad. Nauk SSSR 269 :5 (1983)
Zbl
0538.17007 article
People
BibTeX
@article {key0538.17007z,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Casimir operators in modules over the
{V}irasoro algebra},
JOURNAL = {Sov. Math., Dokl.},
FJOURNAL = {Soviet Mathematics. Doklady},
VOLUME = {27},
YEAR = {1983},
PAGES = {465--469},
NOTE = {English translation of Russian original
published in \textit{Dokl. Akad. Nauk
SSSR} \textbf{269}:5 (1983). Zbl:0538.17007.},
ISSN = {0197-6788},
}
B. L. Feigin and D. B. Fuks :
“Stable cohomology of the algebra \( W_n \) and relations of the algebra \( L_1 \) Funktsional. Anal. i Prilozhen.
18 : 3
(1984 ),
pp. 94–95 .
An English translation was published in Funct. Anal. Appl. 18 :3 (1984)
MR
757265 article
People
BibTeX
@article {key757265m,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Stable cohomology of the algebra \$W_n\$
and relations of the algebra \$L_1\$},
JOURNAL = {Funktsional. Anal. i Prilozhen.},
FJOURNAL = {Funktsional\cprime ny\u{\i} Analiz i
ego Prilozheniya},
VOLUME = {18},
NUMBER = {3},
YEAR = {1984},
PAGES = {94--95},
URL = {http://mi.mathnet.ru/eng/faa1490},
NOTE = {An English translation was published
in \textit{Funct. Anal. Appl.} \textbf{18}:3
(1984). MR:757265.},
ISSN = {0016-2663},
}
B. L. Feĭgin and D. B. Fuchs :
“Verma modules over the Virasoro algebra 230–245
in
Topology: General and algebraic topology, and applications
(Leningrad, 23–27 August 1982 ).
Edited by L. D. Faddeev and A. A. Mal’tsev .
Lecture Notes in Mathematics 1060 .
Springer (Berlin ),
1984 .
MR
770243 Zbl
0549.17010 incollection
People
BibTeX
@incollection {key770243m,
AUTHOR = {Fe\u{\i}gin, B. L. and Fuchs, D. B.},
TITLE = {Verma modules over the {V}irasoro algebra},
BOOKTITLE = {Topology: {G}eneral and algebraic topology,
and applications},
EDITOR = {Faddeev, L. D. and Mal\cprime tsev,
A. A.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1060},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1984},
PAGES = {230--245},
DOI = {10.1007/BFb0099939},
NOTE = {(Leningrad, 23--27 August 1982). MR:770243.
Zbl:0549.17010.},
ISSN = {0075-8434},
ISBN = {9780387133379},
}
B. L. Feigin and D. B. Fuks :
“Stable cohomology of the algebra \( W_n \) and relations of the algebra \( L_1 \) Funct. Anal. Appl.
18 : 3
(July 1984 ),
pp. 264–266 .
English translation of Russian original published in Funktsional. Anal. i Prilozhen. 18 :3 (1984)
Zbl
0559.17008 article
Abstract
People
BibTeX
Let \( W_n \) be the Lie algebra of formal vector fields in \( C^n \) and \( L_1 = L_1(n) \) be the subalgebra of this algebra composed of vector fields with trivial 1-jet. These algebras are graded, due to which their cohomology with coefficients in graded modules is also graded. We denote these gradings in cohomology by subscripts in parentheses. When we speak of cohomology we always mean continuous cohomology.
In [Fuks 1983] the stable cohomology of the Lie algebra \( L_1(n) \) with trivial coefficients is calculated, i.e.,
\[ H^r_{(m)}(L_1(n);C) ,\]
where \( n \) is sufficiently large compared with \( r \) and \( m \) ; some consequences of this calculation are given there. The origin of the restrictions on \( n \) in [1983] is the fact that the basis tensor invariants in \( C^n \) are linearly independent only if \( n \) is sufficiently large compared with the rank of the tensors considered; precisely what \( n \) should be is indicated by the “second fundamental theorem of invariant theory” (cf. [Weyl 1939]). The main content of this note is the remark that an entirely simple addition to the theorem indicated from the theory of invariants (possibly not new, although we have not come across it in the literature) allows us to extend essentially the range of those \( n \) for which the results of [1983] are valid. This improvement of the results of [1983] leads to a rather unexpected corollary on a defining system of generators and relations for the algebra \( L_1(n) \) .
@article {key0559.17008z,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Stable cohomology of the algebra \$W_n\$
and relations of the algebra \$L_1\$},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {18},
NUMBER = {3},
MONTH = {July},
YEAR = {1984},
PAGES = {264--266},
DOI = {10.1007/BF01086175},
NOTE = {English translation of Russian original
published in \textit{Funktsional. Anal.
i Prilozhen.} \textbf{18}:3 (1984).
Zbl:0559.17008.},
ISSN = {0016-2663},
}
B. L. Feigin and D. B. Fuks :
“Representations of the Virasoro algebra ,”
pp. 78–94
in
Metody topologii i rimanovoj geometrii v matematicheskoj fizike
[Methods of topology and Riemannian geometry in mathematical physics ]
(Druskininkai, Lithuania, 23–27 May 1983 ).
Edited by A. Matuzyavichyus .
Ministerstvo Vysshego i Srednego Spetsial’nogo Obrazovaniya Litovskoj SSR (Vilnius ),
1984 .
This was later developed in a 1986 Stockholm University research report and then further in Representation of Lie groups and related topics (1990)
Zbl
0642.17007 incollection
People
BibTeX
@incollection {key0642.17007z,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Representations of the {V}irasoro algebra},
BOOKTITLE = {Metody topologii i rimanovoj geometrii
v matematicheskoj fizike [Methods of
topology and {R}iemannian geometry in
mathematical physics]},
EDITOR = {Matuzyavichyus, A.},
PUBLISHER = {Ministerstvo Vysshego i Srednego Spetsial\cprime
nogo Obrazovaniya Litovskoj SSR},
ADDRESS = {Vilnius},
YEAR = {1984},
PAGES = {78--94},
NOTE = {(Druskininkai, Lithuania, 23--27 May
1983). This was later developed in a
1986 Stockholm University research report
and then further in \textit{Representation
of Lie groups and related topics} (1990).
Zbl:0642.17007.},
}
B. L. Feigin and D. B. Fuchs :
Representations of the Virasoro algebra .
Research report 25 ,
Department of Mathematics, Stockholm University ,
1986 .
A version of this was published in Representation of Lie groups and related topics (1990)
techreport
People
BibTeX
@techreport {key20810539,
AUTHOR = {Feigin, B. L. and Fuchs, D. B.},
TITLE = {Representations of the {V}irasoro algebra},
TYPE = {Research report},
NUMBER = {25},
INSTITUTION = {Department of Mathematics, Stockholm
University},
YEAR = {1986},
PAGES = {70},
NOTE = {A version of this was published in \textit{Representation
of Lie groups and related topics} (1990).},
}
F. G. Malikov, B. L. Feigin, and D. B. Fuks :
“Singular vectors in Verma modules over Kac–Moody algebras Funktsional. Anal. i Prilozhen.
20 : 2
(1986 ),
pp. 25–37 .
English translation of Russian original published in Funct. Anal. Appl. 20 :2 (1986)
MR
847136 article
People
BibTeX
@article {key847136m,
AUTHOR = {Malikov, F. G. and Feigin, B. L. and
Fuks, D. B.},
TITLE = {Singular vectors in {V}erma modules
over {K}ac--{M}oody algebras},
JOURNAL = {Funktsional. Anal. i Prilozhen.},
FJOURNAL = {Funktsional\cprime ny\u{\i} Analiz i
ego Prilozheniya},
VOLUME = {20},
NUMBER = {2},
YEAR = {1986},
PAGES = {25--37},
URL = {http://mi.mathnet.ru/eng/faa1269},
NOTE = {English translation of Russian original
published in \textit{Funct. Anal. Appl.}
\textbf{20}:2 (1986). MR:847136.},
ISSN = {0016-2663},
}
F. G. Malikov, B. L. Feigin, and D. B. Fuks :
“Singular vectors in Verma modules over Kac–Moody algebras Funct. Anal. Appl.
20 : 2
(April 1986 ),
pp. 103–113 .
English translation of Russian original published in Funktsional. Anal. i Prilozhen. 20 :2 (1986)
Zbl
0616.17010 article
Abstract
People
BibTeX
The basic results of this paper are explicit formulas for singular vectors in reducible Verma modules over Kac–Moody algebras. As is known, Verma modules \( M(\lambda) \) over a Kac–Moody algebra
\[ \mathfrak{g} = N_{-} \oplus H \oplus N_{+} \]
are parametrized by linear functionals \( \lambda \) on its Cartan subalgebra \( H \) . Reducible modules \( M(\lambda) \) correspond to functionals \( \lambda \) lying in the union of a countable number of hyperplanes in \( H^* \) ; these hyperplanes are enumerated by pairs \( (\alpha,n) \) , where \( \alpha \) is a positive root of the algebra \( \mathfrak{g} \) and \( n \) is a positive integer (Kac–Kazhdan theorem [1979]). If \( \lambda \) lies in exactly one such hyperplane and the root to which, this hyperplane corresponds is real, then the module \( M(\lambda) \) contains a unique (up to proportionality) singular vector (in different terminology a zero-vector or vector of highest weight), not proportional to a vacuum vector. We give an explicit formula for this vector which however has an unusual form: to the vacuum vector oneapplies monomials in the elements of the algebra \( N_{-} \) containing these elements to nonintegral and even nonreal powers. We start by explaining in Sec. 2 how to reduce such monomials to traditional form. Then we begin to derive the formulas for singular vectors. In Sec. 3 the case of the algebra \( \mathfrak{sl}(2)\hat{} \) , the simplest of the infinite-dimensional Kac–Moody algebras, is analyzed in detail; this section also contains important information about singular vectors in reducible Verma modules over \( \mathfrak{sl}(2)\hat{} \) corresponding to imaginary roots. In Sec. 4 the results of Sec. 3 are extended to the case of an arbitrary Kac–Moody algebra, where in the case of real roots one gets exhaustive info mation. In the concluding Sec. 5 we turn to the consideration of a special case, this time the case of the finite-dimensional Lie algebra \( \mathfrak{sl}(n) \) ; in this case we are able to reduce the answer to especially convenient form. We note that Verma modules over finite-dimensional simple Lie algebras are studied among other things in the recent paper of Zhelobenko [1986]; this paper also contains some expressions for singular vectors in Verma modules, which differ in form from those we find. The connection between our results and the results of [1986] is still unclear.
@article {key0616.17010z,
AUTHOR = {Malikov, F. G. and Feigin, B. L. and
Fuks, D. B.},
TITLE = {Singular vectors in {V}erma modules
over {K}ac--{M}oody algebras},
JOURNAL = {Funct. Anal. Appl.},
FJOURNAL = {Functional Analysis and its Applications},
VOLUME = {20},
NUMBER = {2},
MONTH = {April},
YEAR = {1986},
PAGES = {103--113},
DOI = {10.1007/BF01077264},
NOTE = {English translation of Russian original
published in \textit{Funktsional. Anal.
i Prilozhen.} \textbf{20}:2 (1986).
Zbl:0616.17010.},
ISSN = {0016-2663},
}
B. L. Feĭgin and D. B. Fuks :
“Cohomology of Lie groups and Lie algebras 121–209
in
Lie groups and Lie algebras–2 .
Seriya sovremennye problemy matematiki: Fundamental’nye napravleniya 21 .
VINITI (Moscow ),
1988 .
An English translation appeared in Lie groups and {L ie algebras, {II}} (2000)
MR
968446 Zbl
0653.17008 incollection
People
BibTeX
@incollection {key968446m,
AUTHOR = {Fe\u{\i}gin, B. L. and Fuks, D. B.},
TITLE = {Cohomology of {L}ie groups and {L}ie
algebras},
BOOKTITLE = {Lie groups and {L}ie algebras -- 2},
SERIES = {Seriya sovremennye problemy matematiki:
{F}undamental\cprime nye napravleniya},
NUMBER = {21},
PUBLISHER = {VINITI},
ADDRESS = {Moscow},
YEAR = {1988},
PAGES = {121--209},
URL = {http://mi.mathnet.ru/eng/intf93},
NOTE = {An English translation appeared in \textit{Lie
groups and {L}ie algebras, {II}} (2000).
MR:968446. Zbl:0653.17008.},
ISSN = {0233-6723},
}
B. L. Feigin, D. B. Fuks, and V. S. Retakh :
“Massey operations in the cohomology of the infinite dimensional Lie algebra \( L_1 \) 13–31
in
Topology and geometry: Rohlin semininar .
Edited by O. Ya. Viro .
Lecture Notes in Mathematics 1346 .
Springer (Berlin ),
1988 .
MR
970070 Zbl
0653.17010 incollection
People
BibTeX
@incollection {key970070m,
AUTHOR = {Feigin, B. L. and Fuks, D. B. and Retakh,
V. S.},
TITLE = {Massey operations in the cohomology
of the infinite dimensional {L}ie algebra
\$L_1\$},
BOOKTITLE = {Topology and geometry: {R}ohlin semininar},
EDITOR = {Viro, O. Ya.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1346},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1988},
PAGES = {13--31},
DOI = {10.1007/BFb0082769},
NOTE = {MR:970070. Zbl:0653.17010.},
ISSN = {0075-8434},
ISBN = {9783540459583},
}
B. L. Feigin and D. B. Fuks :
“Cohomology of some nilpotent subalgebras of the Virasoro and Kac–Moody Lie algebras J. Geom. Phys.
5 : 2
(1988 ),
pp. 209–235 .
Dedicated to I. M. Gelfand on his 75th birthday.
MR
1029428 Zbl
0692.17008 article
Abstract
People
BibTeX
@article {key1029428m,
AUTHOR = {Feigin, B. L. and Fuks, D. B.},
TITLE = {Cohomology of some nilpotent subalgebras
of the {V}irasoro and {K}ac--{M}oody
{L}ie algebras},
JOURNAL = {J. Geom. Phys.},
FJOURNAL = {Journal of Geometry and Physics},
VOLUME = {5},
NUMBER = {2},
YEAR = {1988},
PAGES = {209--235},
DOI = {10.1016/0393-0440(88)90005-8},
NOTE = {Dedicated to I.~M. Gelfand on his 75th
birthday. MR:1029428. Zbl:0692.17008.},
ISSN = {0393-0440},
}
B. L. Feigin and D. B. Fuchs :
“Representations of the Virasoro algebra ,”
pp. 465–554
in
Representation of Lie groups and related topics .
Edited by A. M. Vershik and D. P. Zhelobenko .
Advanced Studies in Contemporary Mathematics 7 .
Gordon and Breach (New York ),
1990 .
An earlier version of this appeared as a Stockholm University research report (1986) .
MR
1104280 Zbl
0722.17020 incollection
Abstract
People
BibTeX
@incollection {key1104280m,
AUTHOR = {Feigin, B. L. and Fuchs, D. B.},
TITLE = {Representations of the {V}irasoro algebra},
BOOKTITLE = {Representation of {L}ie groups and related
topics},
EDITOR = {Vershik, A. M. and Zhelobenko, D. P.},
SERIES = {Advanced Studies in Contemporary Mathematics},
NUMBER = {7},
PUBLISHER = {Gordon and Breach},
ADDRESS = {New York},
YEAR = {1990},
PAGES = {465--554},
NOTE = {An earlier version of this appeared
as a Stockholm University research report
(1986). MR:1104280. Zbl:0722.17020.},
ISSN = {0884-0016},
ISBN = {9782881246784},
}
B. L. Feigin and D. B. Fuchs :
“Cohomologies of Lie groups and Lie algebras ,”
pp. 125–223
in
Lie groups and Lie algebras, II .
Edited by E. B. Vinberg .
Encyclopaedia of Mathematical Sciences 21 .
Springer (Berlin ),
2000 .
Translation of Russian original published in Seriya sovremennye problemy matematiki 21 (1988)
MR
1756408 Zbl
0931.17014 incollection
People
BibTeX
@incollection {key1756408m,
AUTHOR = {Feigin, B. L. and Fuchs, D. B.},
TITLE = {Cohomologies of {L}ie groups and {L}ie
algebras},
BOOKTITLE = {Lie groups and {L}ie algebras, {II}},
EDITOR = {Vinberg, E. B.},
SERIES = {Encyclopaedia of Mathematical Sciences},
NUMBER = {21},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2000},
PAGES = {125--223},
NOTE = {Translation of Russian original published
in \textit{Seriya sovremennye problemy
matematiki} \textbf{21} (1988). MR:1756408.
Zbl:0931.17014.},
ISSN = {0938-0396},
ISBN = {9783540505853},
}
D. Fuchs and T. Ishkhanov :
“Invariants of Legendrian knots and decompositions of front diagrams Mosc. Math. J.
4 : 3
(July–September 2004 ),
pp. 707–717 .
To Borya Feigin, with love.
MR
2119145 Zbl
1073.53106 article
Abstract
People
BibTeX
The authors prove that the sufficient condition for the existence of an augmentation of the Chekanov–Eliashberg differential algebra of a Legendrian knot, which is contained in a recent work of the first author, is also necessary. As a by-product, the authors describe an algorithm for calculating Chekanov–Eliash invariants in terms of the front diagram of a Legendrian knot.
@article {key2119145m,
AUTHOR = {Fuchs, Dmitry and Ishkhanov, Tigran},
TITLE = {Invariants of {L}egendrian knots and
decompositions of front diagrams},
JOURNAL = {Mosc. Math. J.},
FJOURNAL = {Moscow Mathematical Journal},
VOLUME = {4},
NUMBER = {3},
MONTH = {July--September},
YEAR = {2004},
PAGES = {707--717},
DOI = {10.17323/1609-4514-2004-4-3-707-717},
NOTE = {To Borya Feigin, with love. MR:2119145.
Zbl:1073.53106.},
ISSN = {1609-3321},
}
