A rational approach to affine quantum algebras

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A basic mathematical operation we encounter early on is that of multiplication; we can multiply real numbers, rational numbers, integers, and so forth. This operation has an endless list of beautiful properties. For instance, every positive integer can be written uniquely as a product of prime numbers, a fact which has applications that transcend mathematics, such as in cryptography. Multiplication, however, also arises naturally in much more abstract settings. In linear algebra, one learns that we can multiply matrices - arrays of numbers which encode systems of equations. Abstracting this further, we can even multiply together vector spaces, examples of which include the real number line, the plane, three-dimensional space and higher dimension analogues. Why would we carry out an operation like this? It turns out that this type of multiplication of spaces, called a tensor product, arises in many remarkable, concrete settings; for instance, in analyzing certain lattice models of theoretical physics, and in the study of quantum analogues of important systems of differential equations. In these settings, the underlying multiplication is highly non-trivial, encoding a rich list of symmetries, and the entire setup may be viewed as arising from an abstract algebraic structure, often viewed as a type of symmetry algebra. It is in this context that the structures at the heart of my research, called quantum groups, arise. More precisely, I study the representation theory of those quantum algebras which are said to be of affine or double affine type. These were formally introduced in the 1980's, under the influence of the quantum inverse scattering method of theoretical physics, and have since become prolific mathematical objects whose theory frequently intertwines algebra, geometry and mathematical physics. A fascinating feature of these structures is that many of their key properties can be elegantly described in the language of rational functions. This is especially true for their tensor structure, and this leads to a wealth of interesting applications unique to affine quantum algebras. The goal of my research is to develop this language in several novel directions using algebraic tools, and then to apply it in order to address open problems and extend the existing theory in meaningful ways. Examples of specific problems that will be addressed include studying a variant of prime factorization of integers for tensor products in terms of the singularities of certain rational operators, constructing universal R and K-matrices (objects which arise in quantum integrability) for affine and twisted Yangians, and developing the emerging theory of affine quantum symmetric pairs. The results will be of interest to algebraists studying various topics in representation theory, geometers with interests in Lie theory and quiver varieties, and mathematical physicists studying a wide range of topics, including gauge theory and quantum integrable systems.