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Nathan Geer

Publications and source records attributed to Nathan Geer.

At least 55 records · Page 3Linked to original sources
arXiv2013

Modified quantum dimensions and re-normalized link invariants

In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of link invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots.

math.QA↗
arXiv2013

G-links invariants, Markov traces and the semi-cyclic Uqsl2-modules

Kashaev and Reshetikhin proposed a generalization of the Reshetikhin-Turaev link invariant construction to tangles with a flat connection in a principal G-bundle over the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to sl2, defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi, and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.

math.QA↗
arXiv2013

Polynomial 6j-Symbols and States Sums

For q a root of unity of order 2r, we give explicit formulas of a family of 3-variable Laurent polynomials J_{i,j,k} with coefficients in Z[q] that encode the 6j-symbols associated with nilpotent representations of U_qsl_2. For a given abelian group G, we use them to produce a state sum invariant tau^r(M,L,h_1,h_2) of a quadruplet (compact 3-manifold M, link L inside M, homology class h_1\in H_1(M,Z), homology class h_2\in H_2(M,G)) with values in a ring R related to G. The formulas are established by a "skein" calculus as an application of the theory of modified dimensions introduced in [arXiv:0711.4229]. For an oriented 3-manifold M, the invariants are related to TV(M,L,f\in H^1(M,C^*)) defined in [arXiv:0910.1624] from the category of nilpotent representations of U_qsl_2. They refine them as TV(M,L,f)= Sum_h tau^r(M,L,h,f') where f' correspond to f with the isomorphism H_2(M,C^*) ~ H^1(M,C^*).

math.GT↗
arXiv2013

Topological invariants from non-restricted quantum groups

We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, respectively. In this case we show that these invariants are equal and extend to what we call a relative Homotopy Quantum Field Theory which is a branch of the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our main examples of relative spherical categories are the categories of finite dimensional weight modules over non-restricted quantum groups considered by C. De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories are not semi-simple and have an infinite number of non-isomorphic irreducible modules all having vanishing quantum dimensions. We also show that these categories have associated ribbon categories which gives rise to re-normalized link invariants. In the case of sl(2) these link invariants are the Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.

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arXiv2012

Traces on ideals in pivotal categories

We extend the notion of an ambidextrous trace on an ideal (developed by the first two authors) to the setting of a pivotal category. We show that under some conditions, these traces lead to invariants of colored spherical graphs (and so to modified 6j-symbols).

math.QA↗
arXiv2011

Ambidextrous objects and trace functions for nonsemisimple categories

We provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role in low-dimensional topology as well as in studying the category itself. In particular, we prove they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the $(1,p)$ minimal model in conformal field theory, and quantum groups at a root of unity.

math.RT↗
arXiv2011

Tetrahedral forms in monoidal categories and 3-manifold invariants

We introduce systems of objects and operators in linear monoidal categories called $\hat Ψ$-systems. A $\hat Ψ$-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture. We conjecture that all quantum groups at odd roots of unity give rise to $\hat Ψ$-systems and we verify this conjecture in the case of the Borel subalgebra of quantum $sl_2$.

math.GT↗
arXiv2010

Generalized trace and modified dimension functions on ribbon categories

In this paper we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are non-zero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.

math.RT↗
arXiv2009

Modified 6j-Symbols and 3-Manifold Invariants

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects.

math.GT↗
arXiv2007

On the Colored HOMFLY-PT, Multivariable and Kashaev Link Invariants

We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.

math.GT↗
arXiv2007

An invariant supertrace for the category of representations of Lie superalgebras

In this paper we give a re-normalization of the supertrace on the category of representations of Lie superalgebras of type I, by a kind of modified superdimension. The genuine superdimensions and supertraces are generically zero. However, these modified superdimensions are non-zero and lead to a kind of supertrace which is non-trivial and invariant. As an application we show that this new supertrace gives rise to a non-zero bilinear form on a space of invariant tensors of a Lie superalgebra of type I. The results of this paper are completely classical results in the theory of Lie superalgebras but surprisingly we can not prove them without using quantum algebra and low-dimensional topology.

math.RT↗
arXiv2007

Multivariable link invariants arising from Lie superalgebras of type I

This paper generalize [7](math.GT/0601291): We construct new links invariants from g, a type I basic classical Lie superalgebra. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module. Using this, we get a multivariable link invariant associated to any one parameter family of irreducible g-modules.

math.GT↗
arXiv2007

Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof-Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are "compatible with duality", i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof-Kazhdan quantization of some affine Lie superalgebras coincide with their Drinfeld-Jimbo-type quantizations.

math.QA↗
arXiv2007

Some remarks on quantized Lie superalgebras of classical type

In this paper we use the Etingof-Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld-Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: 1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and 2) a super version of the Drinfeld-Kohno Theorem.

math.QA↗
arXiv2006

Monodromy of trigonometric KZ equations

The famous Drinfeld-Kohno theorem for simple Lie algebras states that the monodromy representation of the Knizhnik-Zamolodchikov equations for these Lie algebras expresses explicitly via R-matrices of the corresponding Drinfeld-Jimbo quantum groups. This result was generalized by the second author to simple Lie superalgebras of type A-G. In this paper, we generalize the Drinfeld-Kohno theorem to the case of the trigonometric Knizhnik-Zamolodchikov equations for simple Lie superalgebras of type A-G. The equations contain a classical r-matrix on the Lie superalgebra, and the answer expresses through the quantum R-matrix of the corresponding quantum group. The proof is based on the quantization theory for Lie bialgebras developed by the first author and D. Kazhdan.

math.QA↗
arXiv2006

Multivariable link invariants arising from sl(2|1) and the Alexander polynomial

In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link invariant. This new invariant can be thought of as a multivariable version of the Links-Gould invariant. We also show that after a variable reduction our invariant specializes to the Conway potential function, which is a version of the multivariable Alexander polynomial.

math.GT↗