[Paper Review] On blocks of defect two and one simple module, and Lie algebra structure of $HH^1$

David J. Benson, Radha Kessar+1 more|City Research Online (City University London)|Apr 15, 2016

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper computes the Lie algebra structure of the first Hochschild cohomology $HH^1$ for quantum complete intersections over a field of odd characteristic $p$, proving that such algebras are solvable with derived subalgebra nilpotent and socle of dimension $2e$. As a consequence, it establishes an upper bound of $2\sqrt{I}$ on the dimension of $J(B)/J(B)^2$ for blocks of finite group algebras with defect group of order $p^2$ and a unique simple module, under splitting field assumptions.

ABSTRACT

Let $k$ be a field of odd prime characteristic $p$. We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over $k$. As a consequence, we prove that if $B$ is a defect $2$-block of a finite group algebra $kG$ whose Brauer correspondent $C$ has a unique isomorphism class of simple modules, then a basic algebra of $B$ is a local algebra which can be generated by at most $2\sqrt I$ elements, where $I$ is the inertial index of $B$, and where we assume that $k$ is a splitting field for $B$ and $C$.

Motivation & Objective

• To determine the Lie algebra structure of $HH^1(A)$ for a class of quantum complete intersection algebras $A$ over a field of odd characteristic $p$.
• To establish bounds on the number of loops in the quiver of symmetric split algebras using the socle of $HH^1(A)$ as a $Z(A)$-module.
• To apply these results to modular representation theory, particularly to blocks of finite group algebras with defect group of order $p^2$ and a unique simple module.
• To investigate whether the largest ${\mathcal{O}}$-free commutative quotient of a basic algebra is symmetric, providing evidence against a characterization of nilpotent blocks.

Proposed method

• Compute $HH^1(A)$ as a Lie algebra for $A = k\langle x,y \mid x^p = y^p = 0, yx = qxy\rangle$, where $q$ is a root of unity of order $e \geq 2$ dividing $p-1$.
• Analyze the derived Lie subalgebra $\mathcal{L}' = [\mathcal{L}, \mathcal{L}]$ and show it is nilpotent and $p$-toral with $({\mathcal{L}}')^{[p]} = \{0\}$.
• Use the socle of $\mathcal{L}$ as a $Z(A)$-module to bound the dimension of $J(B)/J(B)^2$ for related symmetric algebras $B$.
• Apply stable equivalences of Morita type to transfer bounds from $A$ to blocks $B$ of finite group algebras.
• Construct the largest ${\mathcal{O}}$-free commutative quotient of the $p$-adic lift $\hat{A}$ of $A$, showing it is not symmetric.
• Use Chebyshev polynomials $f_n(u)$ to describe the $\mathcal{O}$-structure of $\hat{A}$ and its commutator ideal.

Experimental results

Research questions

• RQ1 What is the Lie algebra structure of $HH^1(A)$ for quantum complete intersections $A$ with $p$-power relations and $q$-commutation?
• RQ2 How does the socle of $HH^1(A)$ as a $Z(A)$-module constrain the quiver structure of symmetric split algebras?
• RQ3 Can the number of loops in the quiver of a symmetric split algebra be bounded using $HH^1$ and its socle?
• RQ4 What is the dimension of $J(B)/J(B)^2$ for a block $B$ of a finite group algebra with defect group of order $p^2$ and a unique simple module?
• RQ5 Is the largest ${\mathcal{O}}$-free commutative quotient of a non-nilpotent block’s basic algebra symmetric?

Key findings

• The dimension of $HH^1(A)$ is $2(p + ((p-1)/e)^2)$, where $e$ is the order of $q$ in $k^\times$.
• The center of $HH^1(A)$ is trivial: $Z(\mathcal{L}) = \{0\}$.
• There exists a 2-dimensional maximal toral subalgebra $\mathcal{H}$ such that $\mathcal{L} = \mathcal{H} \oplus \mathcal{L}'$, with $\mathcal{L}'$ nilpotent.
• The derived subalgebra $\mathcal{L}'$ is abelian if and only if $e = p-1$, and $\dim_k(Z(\mathcal{L}')) = 2e + 2$.
• The socle $\mathrm{soc}_{Z(A)}(\mathcal{L})$ has dimension $2e$ and is contained in $Z(\mathcal{L}')$.
• The quotient $\mathcal{L}/\mathcal{L}'$ has dimension 2, and $J(Z(A))\mathcal{L} = \mathcal{L}'$.