Vicky Klima - Research Interests

Lie Algebra Representation Theory:

Many problems in physics and mathematics may be reduced to finding all the ways that a group of symmetries can be realized as a collection of matrices. These realizations are known as representations of the group, and searching for such realizations is an area of great interest in the study of Lie groups. The structure of a Lie group is reflected in the structure of its Lie algebra (tangent space to the Lie group at the identity). Questions regarding the representations of a Lie group may be restated as questions regarding the representations of its associated Lie algebra. One of the most basic questions one can ask concerning the structure of a particular Lie algebra is the following: What are the dimensions of the generalized eigenspaces (the so- called root spaces) of the algebra? My research addresses this question. Root multiplicities for finite type Kac-Moody algebras are all known, and root multiplicities of indefinite type Kac-Moody algebras are well understood. However, there are many interesting, open problems regarding the multiplicities of roots of indefinite type Kac-Moody algebras. In my research, I view certain indefinite type Kac-Moody algebras as modules over (i.e. representations of ) appropriate Kac-Moody algebras of affine or finite type.Kang’s multiplicity formula reduces my problem to one of calculating weight multiplicities in certain highest weight modules over particular affine or finite Kac-Moody algebras. If the calculations are over finite algebras, I am able to apply well known formulas. If the calculations are over indefinite type Kac-Moody algebras, I calculate the weight multiplicities using crystal basis theory for affine algebras.

Algebraic Music Theory:

Many musical properties of the traditional 12-fold pitch system of octave division are related to symmetry and have been studied algebraically. For example, the circle of fifths can be viewed as the integers modulo twelve whenthe group is generated by seven. I have been working with students to study how some of the well-known algebraic properties of traditional 12-tone music translate to other microtonal systems.