Introduction E9F5FC

Questions FFFFC0

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• What is a coalgebra?
• What is the antipode?
• What does comultiplication look like for elements of the forgotten basis?
• Consider the symmetric functions of the eigenvalues of a matrix. Interpret it as a Hopf algebra.
  • What does comultiplication look like for elements of the various bases? (see page 46 in Grinberg and Reiner)

Thoughts

• Hopf algebra is important for duality because it relates generation of internal structure (comultiplication) with generation of external structure (multiplication)
• From W: Coalgebra
  • In general, algebra and coalgebra are dual notions (meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are dual objects (meaning that a coalgebra is the dual object of an algebra and conversely).
  • In finite dimensions, {$(A ⊗ A)^∗$} and {$A^∗ ⊗ A^∗$} are isomorphic.
  • A primitive element is an element x that satisfies Δ(x) = x ⊗ 1 + 1 ⊗ x. The primitive elements of a Hopf algebra form a Lie algebra.
• From Hopf algebras in combinatorics
  • Hopf’s motivation: the (co-)homology of a compact Lie group carries bialgebra structure that explains why it takes a certain form.
  • Note how the cocommutative Hopf algebrasT(V), Sym(V) have much of their structure controlled by their k-submodules V, which consist of primitive elements only (although, in general, not of all their primitive elements). This is not far from the truth in general, and closely related to Lie algebras.
  • Symmetric algebras Sym(V) are always cocommutative and commutative. Homology and cohomology of H-spaces are always cocommutative and commutative in the topologist’s sense where one reinterprets that twist map A⊗AT→A⊗A to have the extra sign

Examples of coalgebras

Symmetric functions

• Product is multiplication of functions f(x)g(x)
• Coproduct is separation of inputs into two sets f(x1...xj yj+1...yn)
• {$\Delta(m_\lambda)=\sum_{\lambda=\mu \bigcup \nu} m_\mu \otimes m_\nu$} (breaking up the rows)

The divided power coalgebra

• Polynomial ring {$K[X]$}
• {$\Delta(X^n) = \sum_{k=0}^n \dbinom{n}{k} X^k\otimes X^{n-k}$}
  • like mapping {$X \rightarrow X_{\text{left}}+X_{\text{right}}$} where {$+$} means "or" and taking tensor products to the {$n$}th power
• {$\epsilon(X^n)=\begin{cases}1& \mbox{if } n=0\\0& \mbox{if } n>0 \end{cases}$}
  • like setting {$X \rightarrow 0$}

The trigonometric coalgebra

• {$C$} is a two-dimensional {$K$}-vector space with basis {$\{s,c\}$} ("sine", "cosine")
• {$Δ(s) = s⊗c + c⊗s$}
• {$Δ(c) = c⊗c − s⊗s$}
• thus {$Δ$} maps in a way that splits the angle {$\theta=\alpha + \beta$} so that {$Δ(s(\theta)) = s(\alpha)⊗c(\beta) + c(\alpha)⊗s(\beta)$}
• {$ε:C → K$} is given by
• {$ε(s) = 0$}
• {$ε(c) = 1$}

The incidence coalgebra

• Given a locally finite poset {$P$} with set of intervals {$J$}, define {$J$} to be the basis
• {$\Delta[x,z] = \sum_{x < y < z} [x,y] \otimes [y,z] $}
• Here {$Δ$} maps so as to split the interval in all possible ways. In a locally finite poset, there are only finitely many ways.
• Whereas the incidence algebra consists of functions from intervals to real numbers with product given by
• {$(f * g)(x,y):=\sum_{x \leq z \leq y} f(x,z) g(z,y)$}

The tensor algebra - the algebra on words

The exterior algebra

The symmetric algebra - the algebra on multisets (or words of commutative letters for which {$xy=yx$})

The singular homology of a topological space forms a graded coalgebra whenever the Künneth isomorphism holds, e.g. if the coefficients are taken to be a field.

Lie bialgebra

• Vidas Regelskis at Vilnius University is expert at Hopf algebras.

Readings

• W: Hopf algebras

Videos

https://en.wikipedia.org/wiki/Combinatorics_and_physics

• Combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer,[4] showing that the renormalization of Feynman diagrams can be described by a Hopf algebra.