This thesis deals with algebraic structures and their use in solving probabilistic models. Multiple deformations of known results are studied, generalizing the algebraic structures and algorithms used to solve them.

The first part studies how the Hopf square map, first studied in this way by Diaconis, Pang, and Ram, can be applied to a Hopf algebra that is non-commutative and non-cocommutative. A Markov chain is produced from the quantum deformation of the universal envelope of a Lie algebra. The chain's characteristics and phase transition are examined.

In the second part, the log-Gamma polymer of Seppäläinen is studied in the presence of an external field. In the point to point case, solved originally by Corwin, O'Connell, Seppäläinen, and Zygouras using the geometric RSK algorithm and Whittaker functions, the external field reduces to a scalar multiple. When the external field is applied to the point to line case studied by Bisi and Zygouras, deformations of these algebraic objects, refered to as twisting, are used. These deformed structures and used to produce integral formulas for the partition function.