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 $su(2)\oplus F$ 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 $J$ with a closed non-degenerate $(2,0)$-form. It is determined by $J$ and the real part $\Omega$ of the $(2,0)$-form. Suppose that \lie h is a semi-direct product \lie g\ltimes V, and both \lie g and $V$ are Lagrangian with respect to $\Omega$ and totally real with respect to $J$. 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 $\Omega$ and $J$ are isomorphic. The geometry of $(\Omega, J)$ 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 $(J, \Omega)$ with hypersymplectic algebras, we find an inductive process to build families of complex symplectic algebras of dimension $8n$ from the data of the $4n$-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.