[Paper Review] Degenerate quantum general linear groups

Jin Cheng, Yan Wang+1 more|arXiv (Cornell University)|May 18, 2018

Algebraic structures and combinatorial models23 references3 citations

TL;DR

This paper introduces a new class of Hopf algebras, degenerate quantum general linear groups Uq(glm,n), by deforming the quantum group Uq(glm+n) through a degeneration of one quantum sl2 subalgebra. It develops a highest weight representation theory, classifies finite-dimensional simple modules via m+n−2 nonnegative integers and two nonzero scalars, constructs an R-matrix satisfying the Yang-Baxter equation, and shows this R-matrix yields the HOMFLY polynomial as a topological invariant of knots.

ABSTRACT

Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The finite dimensional simple modules are classified in terms of highest weights, which are essentially characterised by m+n-2 nonnegative integers and two arbitrary nonzero scalars. In the special case with m=2 and n=1, an explicit basis is constructed for each finite dimensional simple module. For all m and n, the degenerate quantum group has a natural irreducible representation acting on C(q)^(m+n). It admits an R-matrix that satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite. This in particular gives rise to isomorphisms between the two module structures of any tensor power of C(q)^(m+n) defined relative to the co-multiplication and its opposite respectively. A topological invariant of knots is constructed from this R-matrix, which reproduces the celebrated HOMFLY polynomial. Degenerate quantum groups of other classical types are briefly discussed.

Motivation & Objective

• To construct a new class of Hopf algebras as degenerate versions of Drinfeld-Jimbo quantum groups of type A.
• To develop a highest weight representation theory for these degenerate quantum groups.
• To classify finite-dimensional simple modules in terms of highest weights and determine their structure.
• To construct an R-matrix satisfying the Yang-Baxter equation and use it to define isomorphisms between tensor power module structures.
• To apply the R-matrix to construct a topological invariant of knots, showing it reproduces the HOMFLY polynomial.

Proposed method

• Define the degenerate quantum general linear group Uq(glm,n) as a Hopf algebra by modifying Serre relations involving one quantum sl2 subalgebra.
• Construct a natural module structure on C(q)^{m+n} and extend it to tensor powers.
• Introduce an R-matrix via the co-multiplication and its opposite, proving it satisfies the Yang-Baxter equation.
• Use the R-matrix to establish isomorphisms between Uq(glm,n)-module structures on V^{⊗r} defined by ͆ and ͆^{op}.
• Apply invariant theory to the R-matrix to construct a knot invariant via the trace of monodromy matrices.
• Generalize the construction to classical types B, C, and D using generalized Dynkin diagrams.

Experimental results

Research questions

• RQ1 Can a meaningful Hopf algebra be constructed by degenerating one quantum sl2 subalgebra of Uq(glm+n) while preserving structural and representation-theoretic properties?
• RQ2 How can the finite-dimensional simple modules of the degenerate quantum group Uq(glm,n) be classified in terms of highest weights?
• RQ3 Does the existence of an R-matrix satisfying the Yang-Baxter equation lead to isomorphisms between module structures defined by co-multiplication and its opposite?
• RQ4 Can the R-matrix be used to construct a topological invariant of knots, and does it reproduce known invariants like the HOMFLY polynomial?
• RQ5 What is the relationship between the degenerate quantum group Uq(glm,n) and the quantum supergroup Uq(glm|n)?

Key findings

• The finite-dimensional simple modules of Uq(glm,n) are classified by m+n−2 nonnegative integers and two arbitrary nonzero scalars in C(q)*.
• For m=2 and n=1, an explicit basis is constructed for each finite-dimensional simple module, providing a concrete realization.
• The R-matrix constructed satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite, ensuring isomorphism between the two module structures on any tensor power of C(q)^{m+n}.
• The R-matrix leads to a topological invariant of knots that coincides exactly with the HOMFLY polynomial, confirming its significance in low-dimensional topology.
• The degenerate quantum group Uq(glm,n) is not a deformation quantization of a universal enveloping algebra of a Lie algebra or superalgebra, indicating a distinct algebraic structure.
• The construction suggests a potential deep connection with quantum supergroups Uq(glm|n), particularly through quantum correspondences, though this remains to be fully established.