A Path Model for Quantum Skew-Symmetric Matrices by Aesha Siddiqui ’19, at the Mathematics Colloquium, today, 1 – 1:45 pm, Stetson Court Classroom 101

A Path Model for Quantum Skew-Symmetric Matrices by Aesha Siddiqui ’19, at the Mathematics Colloquium, today, 1 – 1:45 pm, Stetson Court Classroom 101

Abstract:

In 2014, Casteels developed a combinatorial model for the algebra of quantum matrices, and successfully used the model to develop new results in the theory of torus-invariant prime ideals. This combinatorial model allowed Casteels to embed the algebra of quantum matrices into its corresponding torus, a simpler structure more amenable to study. The model associates each generator of the algebra to a sum of path weights in a directed grid graph, where weights are elements of the algebra’s torus. We extend Casteels’ method to find an analogous model for the quantized coordinate ring of n x n skew-symmetric matrices over an infinite field. We want to show that this new, adapted model embeds the algebra of quantum skew-symmetric matrices into its corresponding torus, parallel to Casteels’ work. In doing so, we want to show that this combinatorial mapping is both injective and preserves the commutation relations of the initial algebra. These methods may lead to new insights on applying these path models to other quantum algebras, which have further applications, including quantum group theory, braided tensor categories, and knot theory. 

In 2014, Casteels developed a combinatorial model for the algebra of quantum matrices, and successfully used the model to develop new results in the theory of torus-invariant prime ideals. This combinatorial model allowed Casteels to embed the algebra of quantum matrices into its corresponding torus, a simpler structure more amenable to study. The model associates each generator of the algebra to a sum of path weights in a directed grid graph, where weights are elements of the algebra’s torus. We extend Casteels’ method to find an analogous model for the quantized coordinate ring of n x n skew-symmetric matrices over an infinite field. We want to show that this new, adapted model embeds the algebra of quantum skew-symmetric matrices into its corresponding torus, parallel to Casteels’ work. In doing so, we want to show that this combinatorial mapping is both injective and preserves the commutation relations of the initial algebra. These methods may lead to new insights on applying these path models to other quantum algebras, which have further applications, including quantum group theory, braided tensor categories, and knot theory.