98,425 research outputs found
How to obtain division algebras from a generalized Cayley-Dickson doubling process
New families of eight-dimensional real division algebras with large
derivation algebra are presented: We generalize the classical Cayley-Dickson
doubling process starting with a unital algebra with involution over a field F
by allowing the scalar in the doubling to be an invertible element in the
algebra. The resulting unital algebras are neither power-associative nor
quadratic. Starting with a quaternion division algebra D, we obtain division
algebras A for all invertible scalars chosen in D outside of F. This is
independent on where the scalar is placed inside the product and three pairwise
non-isomorphic families of eight-dimensional division algebras are obtained.
Over the reals, the derivation algebra of each such algebra A is isomorphic to
and the decomposition of A into irreducible su(2)-modules has
the form 1+1+3+3 (denoting an irreducible su(2)-module by its dimension). Their
opposite algebras yield more classes of pairwise non-isomorphic families of
division algebras of the same type. We thus give an affirmative answer to a
question posed by Benkart and Osborn in 1981.Comment: 23 pages; extended versio
Models of q-algebra representations: Matrix elements of the q-oscillator algebra
This article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is considered. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ``group operators'' on these representation spaces are computed. This ``local'' approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions 1F1 and the product of a 1F1 and a Bessel function. They are interpreted here as expansions of the matrix elements of a ``group operator'' (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillator algebra representations) in terms of the matrix elements in a reduced basis. As a by-product of this analysis an interesting new orthonormal basis was found for a q analog of the Bargmann–Segal Hilbert space of entire functions
Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry
We investigate whether there are unitary families of W-algebras with spin one
fields in the natural example of the Feigin-Semikhatov W^(2)_n-algebra. This
algebra is conjecturally a quantum Hamiltonian reduction corresponding to a
non-principal nilpotent element. We conjecture that this algebra admits a
unitary real form for even n. Our main result is that this conjecture is
consistent with the known part of the operator product algebra, and especially
it is true for n=2 and n=4. Moreover, we find certain ranges of allowed levels
where a positive definite inner product is possible. We also find a unitary
conformal field theory for every even n at the special level k+n=(n+1)/(n-1).
At these points, the W^(2)_n-algebra is nothing but a compactified free boson.
This family of W-algebras admits an 't Hooft limit that is similar to the
original minimal model 't Hooft limit. Further, in the case of n=4, we
reproduce the algebra from the higher spin gravity point of view. In general,
gravity computations allow us to reproduce some leading coefficients of the
operator product.Comment: 23 page
Weak Mirror Symmetry of Complex Symplectic Algebras
A complex symplectic structure on a Lie algebra \lie h is an integrable
complex structure with a closed non-degenerate -form. It is
determined by and the real part of the -form. Suppose that
\lie h is a semi-direct product \lie g\ltimes V, and both \lie g and
are Lagrangian with respect to and totally real with respect to .
This note shows that \lie g\ltimes V is its own weak mirror image in the
sense that the associated differential Gerstenhaber algebras controlling the
extended deformations of and are isomorphic. The geometry of
on the semi-direct product \lie g\ltimes V is also shown to be
equivalent to that of a torsion-free flat symplectic connection on the Lie
algebra \lie g. By further exploring a relation between with
hypersymplectic algebras, we find an inductive process to build families of
complex symplectic algebras of dimension from the data of the
-dimensional ones.Comment: 22 page
Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras
We classify all 4-dimensional first order bicovariant calculi on the
Jordanian quantum group GL_{h,g}(2) and all 3-dimensional first order
bicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases we
assume that the bicovariant bimodules are generated as left modules by the
differentials of the quantum group generators. It is found that there are 3
1-parameter families of 4-dimensional bicovariant first order calculi on
GL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariant
calculus on SL_{h}(2). This 3-dimensional calculus may be obtained through a
classical-like reduction from any one of the three families of 4-dimensional
calculi on GL_{h,g}(2). Details of the higher order calculi and also the
quantum Lie algebras are presented for all calculi. The quantum Lie algebra
obtained from the bicovariant calculus on SL_{h}(2) is shown to be isomorphic
to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian
universal enveloping algebra U_{h}(sl(2)) and also through a consideration of
the decomposition of the tensor product of two copies of the deformed adjoint
module. We also obtain the quantum Killing form for this quantum Lie algebra.Comment: 33 pages, AMSLaTeX, misleading remark remove
An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
A well-known ansatz (`trace method') for soliton solutions turns the
equations of the (noncommutative) KP hierarchy, and those of certain
extensions, into families of algebraic sum identities. We develop an algebraic
formalism, in particular involving a (mixable) shuffle product, to explore
their structure. More precisely, we show that the equations of the
noncommutative KP hierarchy and its extension (xncKP) in the case of a
Moyal-deformed product, as derived in previous work, correspond to identities
in this algebra. Furthermore, the Moyal product is replaced by a more general
associative product. This leads to a new even more general extension of the
noncommutative KP hierarchy. Relations with Rota-Baxter algebras are
established.Comment: 59 pages, relative to the second version a few minor corrections, but
quite a lot of amendments, to appear in J. Phys.
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