97 research outputs found
Notes on two-parameter quantum groups, (I)
A simpler definition for a class of two-parameter quantum groups associated
to semisimple Lie algebras is given in terms of Euler form. Their positive
parts turn out to be 2-cocycle deformations of each other under some
conditions. An operator realization of the positive part is given.Comment: 11 page
Principal Vertex Operator Representations for Toroidal Lie Algebras
In this paper we present the principal construction of the vertex operator
representation for toroidal Lie algebras.Comment: 29 pages, plain tex, no figure
Cross products, invariants, and centralizers
An algebra V with a cross product x has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from V-circle times n to V-circle times m that are invariant under the action of the automorphism group Aut(V, x) of V, which is a special orthogonal group when dim V = 3, and a simple algebraic group of type G(2) when dim V = 7. When m = n, this gives a graphical description of the centralizer algebra End(Aut(v, x))(V-circle times n), and therefore, also a graphical realization of the Aut(V, x)-invariants in V-circle times 2n equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group
McKay matrices for finite-dimensional Hopf algebras
For a finite-dimensional Hopf algebra , the McKay matrix of an -module encodes the relations for tensoring the simple -modules with . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of by relating them to characters. We show how the projective McKay matrix obtained by tensoring the projective indecomposable modules of with is related to the McKay matrix of the dual module of . We illustrate these results for the Drinfeld double of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of and in terms of several kinds of Chebyshev polynomials. For the matrix that encodes the fusion rules for tensoring with a basis of projective indecomposable -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions
Two-parameter quantum general linear supergroups
The universal R-matrix of two-parameter quantum general linear supergroups is
computed explicitly based on the RTT realization of
Faddeev--Reshetikhin--Takhtajan.Comment: v1: 14 pages. v2: published version, 9 pages, title changed and the
section on central extension remove
Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid
The parastatistics algebra is a superalgebra with (even) parafermi and (odd)
parabose creation and annihilation operators. The states in the parastatistics
Fock-like space are shown to be in one-to-one correspondence with the Super
Semistandard Young Tableaux (SSYT) subject to further constraints. The
deformation of the parastatistics algebra gives rise to a monoidal structure on
the SSYT which is a super-counterpart of the plactic monoid.Comment: Presented at the International Workshop "Differential Geometry,
Noncommutative Geometry, Homology and Fundamental Interactions" in honour of
Michel Dubois-Violette, Orsay, April 8-10, 200
The Equitable Basis for sl_2
This article contains an investigation of the equitable basis for the Lie
algebra sl_2. Denoting this basis by {x,y,z}, we have
[x,y] = 2x + 2y, [y,z] = 2y + 2z, [z, x] = 2z + 2x.
One focus of our study is the group of automorphisms G generated by exp(ad
x*), exp(ad y*), exp(ad z*), where {x*,y*,z*} is the basis for sl_2 dual to
{x,y,z} with respect to the trace form (u,v) = tr(uv). We show that G is
isomorphic to the modular group PSL_2(Z). Another focus of our investigation is
the lattice L=Zx+Zy+Zz. We prove that the orbit G(x) equals {u in L |(u,u)=2}.
We determine the precise relationship between (i) the group G, (ii) the group
of automorphisms for sl_2 that preserve L, (iii) the group of automorphisms and
antiautomorphisms for sl_2 that preserve L, and (iv) the group of isometries
for (,) that preserve L. We obtain analogous results for the lattice L*
=Zx*+Zy*+Zz*. Relative to the equitable basis, the matrix of the trace form is
a Cartan matrix of hyperbolic type; consequently,we identify the equitable
basis with the set of simple roots of the corresponding Kac-Moody Lie algebra
g. Then L is the root lattice for g and 1/2L* is the weight lattice, and G(x)
coincides with the set of real roots for g. Using L, L*, and G, we give several
descriptions of the isotropic roots for g and show that each isotropic root has
multiplicity 1. We describe the finite-dimensional sl_2-modules from the point
of view of the equitable basis. In the final section, we establish a connection
between the Weyl group orbit of the fundamental weights of g and Pythagorean
triples.Comment: Minor changes made to the introductory material, and a few typos
corrected. The final publication is available at http://www.springerlink.co
Commutator Leavitt path algebras
For any field K and directed graph E, we completely describe the elements of
the Leavitt path algebra L_K(E) which lie in the commutator subspace
[L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras
L_K(E) that satisfy L_K(E)=[L_K(E),L_K(E)]. We also show that these Leavitt
path algebras have the additional (unusual) property that all their Lie ideals
are (ring-theoretic) ideals, and construct examples of such rings with various
ideal structures.Comment: 24 page
Highest weight modules over quantum queer Lie superalgebra U_q(q(n))
In this paper, we investigate the structure of highest weight modules over
the quantum queer superalgebra . The key ingredients are the
triangular decomposition of and the classification of finite
dimensional irreducible modules over quantum Clifford superalgebras. The main
results we prove are the classical limit theorem and the complete reducibility
theorem for -modules in the category .Comment: Definition 1.5 and Definition 6.1 are changed, and a remark is added
in the new versio
Notes on two-parameter quantum groups, (II)
This paper is the sequel to [HP1] to study the deformed structures and
representations of two-parameter quantum groups
associated to the finite dimensional simple Lie algebras \mg. An equivalence
of the braided tensor categories \O^{r,s} and \O^{q} is explicitly
established.Comment: 21 page
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