Algebra Commons^™

Hom Quandles, Alissa S. Crans, Sam Nelson

Alissa Crans

If A is an abelian quandle and Q is a quandle, the hom set Hom(Q,A) of quandle homomorphisms from Q to A has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom quandle structure. We generalize the result to the case of biquandles, collect observations and results about abelian quandles and the hom quandle, and show that the category of abelian quandles is symmetric monoidal closed.

Higher Dimensional Algebra Vi: Lie 2-Algebra, John C. Baez, Alissa S. Crans

Alissa Crans

The theory of Lie algebras can be categorified starting from a new notion of `2-vector space', which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, `linear functors' as morphisms and `linear natural transformations' as 2-morphisms. We define a `semistrict Lie 2-algebra' to be a 2-vector space L equipped with a skew-symmetric bilinear functor [ . , . ] : L x L -> L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the `Jacobiator', which in turn must satisfy a certain law of its …

Musical Actions Of Dihedral Groups, Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra

Alissa Crans

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

From Loop Groups To 2-Groups, John C. Baez, Danny Stevenson, Alissa S. Crans, Urs Schreiber

Alissa Crans

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having Lie(G) as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group …

Polynomial Knot And Link Invariants From The Virtual Biquandle, Alissa S. Crans, Allison Henrich, Sam Nelson

Alissa Crans

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gr\"obner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to …

Cohomology Of The Adjoint Of Hopf Algebras, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Masahico Saito

Alissa Crans

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.

Crossed Modules Of Racks, Alissa S. Crans, Friedrich Wagemann

Alissa Crans

We generalize the notion of a crossed module of groups to that of a crossed module of racks. We investigate the relation to categorified racks, namely strict 2-racks, and trunk-like objects in the category of racks, generalizing the relation between crossed modules of groups and strict 2-groups. Then we explore topological applications. We show that by applying the rack-space functor, a crossed module of racks gives rise to a covering. Our main result shows how the fundamental racks associated to links upstairs and downstairs in a covering fit together to form a crossed module of racks.

Cohomology Of Categorical Self-Distributivity, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Masahico Saito

Alissa Crans

We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provide solutions of the Yang–Baxter equation, and, conversely, solutions of the Yang–Baxter equation can be used to construct self-distributive operations in certain categories. Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, is analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All …

Cohomology Of Frobenius Algebras And The Yang-Baxter Equation, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Enver Karadayi, Masahico Saito

Alissa Crans

A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.

Enhancements Of Rack Counting Invariants Via Dynamical Cocycles, Alissa S. Crans, Sam Nelson, Aparna Sarkar

Alissa Crans

We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Alexander polynomial.

Never Underestimate A Theorem That Counts Something!, Tyler J. Evans

Tyler Evans