zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators. (English) Zbl 1156.17003
The authors study the universal enveloping algebra $U(\mathfrak{g})$ of the semidirect product $\mathfrak{g}$ of a semisimple Lie algebra $\mathfrak{s}$ with a solvable Lie algebra $\mathfrak{r}$. The authors determine certain elements in $U(\mathfrak{g})$ which commute with $\mathfrak{r}$. It turns out that, up to a functorial factor given by a Casimir operator of $\mathfrak{g}$, the commutators or these elements are similar to the commutators of generators of the algebra $\mathfrak{s}$. These elements generate what is called a virtual copy of $\mathfrak{s}$ in $U(\mathfrak{g})$ (as the normalizer version of these elements formally belongs only to the (skew)-field of fractions of $U(\mathfrak{g})$ and not to $U(\mathfrak{g})$ itself). Using this virtual copy the authors compute Casimir elements in $U(\mathfrak{g})$. As an application the authors obtain a formula for Casimir operators for the inhomogeneous Hamilton algebras $\mathrm{IHa}(N)$ and all its central extensions in arbitrary dimensions. The authors also study the behavior of virtual copies with respect to contractions.

MSC:
17B35Universal enveloping Lie (super)algebras
17B05Structure theory of Lie algebras
WorldCat.org
Full Text: DOI arXiv