Skew Group Algebras: Structure and Applications

• Skew group algebras are noncommutative structures formed by twisting the tensor product of an algebra with a group algebra through a finite group action, serving as a fundamental tool in representation theory and noncommutative geometry.
• They preserve key homological invariants, including global and finitistic dimensions as well as Koszul properties, via spectral sequences and Morita equivalence under appropriate conditions.
• They underpin rich deformation theories with PBW deformations and enable explicit computations in Hochschild cohomology, facilitating advanced studies in modular and nonmodular settings.

A skew group algebra is a noncommutative ring constructed from an associative algebra $A$ (typically over a field $k$) equipped with an action by a finite group $G$ via algebra automorphisms. The group action is incorporated into the structure of $A$ by converting $A$ into a new algebra—denoted $A G$ or $A \rtimes G$—whose multiplication is twisted by the group action. Skew group algebras occupy a central role across representation theory, noncommutative geometry, homological algebra, deformation theory, and the paper of group actions on both commutative and noncommutative algebras.

1. Algebraic Definition and Fundamental Structure

Given a finite group $G$ acting by automorphisms on an associative $k$-algebra $A$, the skew group algebra $A G$ is defined as the tensor product $A \otimes_k kG$ as a $k$-vector space, endowed with multiplication

$(a \otimes g)(b \otimes h) = a\,g(b) \otimes gh$

for $a, b \in A$ and $g, h \in G$ (Li, 2012). Usually, elements are written as $a g := a \otimes g$; the product rule reads $a g \cdot b h = a \cdot g(b) \cdot (gh)$. The algebra $A$ embeds as $A \otimes 1$, while $kG$ embeds as $1 \otimes kG$, with $kG$ acting on $A$ by $g a = g(a)$. This construction generalizes both group rings and tensor products of $A$ and $kG$ (the case where $G$ acts trivially).

2. Representation Theory and Homological Invariants

The module theory of $A G$ is intimately linked to both the $A$-module structure and the group action. If $H \leq G$ is any subgroup, the induction–restriction formalism plays a central role. The key facts—specialized to modular settings—include:

• Existence of complete sets of primitive orthogonal idempotents $E$ in $A$ closed under subgroup action, necessary for comparison of module categories and homological invariants (Li, 2013, Li, 2012).
• Under a free action of a Sylow $p$-subgroup $S$ on $E$ (if $\operatorname{char}k=p>0$), global and finitistic dimensions, as well as strong global dimension and representation type, are preserved: e.g., $$\operatorname{gldim}(A G) = \operatorname{gldim}(A) = \operatorname{gldim}(A^S)$$ (Li, 2013).
• The generalized Koszul property is preserved under forming $A G$ provided the group action respects grading: $A$ is (generalized) Koszul if and only if $A G$ is, and $$\operatorname{Ext}^*_{A G}((A G)_0,(A G)_0) \simeq \operatorname{Ext}^*_A(A_0,A_0)\,G$$ (Li, 2012).

If $|G|$ is invertible in $k$, these results simplify substantially: the skew group algebra construction commutes with many classical homological invariants and often provides Morita equivalences.

3. Cohomology, Support Varieties, and (Fg) Transfer

For an augmented $k$-algebra $A$ with finite group $G$ acting by automorphisms, the cohomology of the skew group algebra $A G$ is connected to that of $A$ and $G$ via a Lyndon–Hochschild–Serre spectral sequence: $$E_2^{p, q} = H^p(G, H^q(A, k)) \implies H^{p+q}(A G, k)$$ (Nguyen et al., 2013). Under additional hypotheses (existence of a polynomial subring in the image of restriction, $H^*(A, k)$ being free and finitely generated over it with a stable basis), $H^*(A G, k)$ is Noetherian, yielding good support variety theory (Sandøy, 25 Nov 2024).

When $|G|$ is invertible in $k$, Linckelmann's separable equivalence machinery ensures that $A$ and $A G$ are "separably equivalent" and share finite generation of Hochschild cohomology (the $(Fg)$ property): $\Lambda$ is $(Fg)$ if and only if $\Lambda G$ is $(Fg)$ (Sandøy, 25 Nov 2024).

4. Deformation Theory, PBW Deformations, and Hochschild Cohomology

Skew group algebras support rich deformation theories, especially for group actions on polynomial or symmetric algebras. In positive characteristic (the modular case), new classes of PBW (Poincaré–Birkhoff–Witt) deformations arise—distinct from those possible in characteristic $0$ (Shepler et al., 2013, Grimley et al., 7 Nov 2024):

• One considers deformations $H_{a, \kappa}$ where relations such as $g v - ^g v = X(g, v)$ and $v w - w v = \kappa(v, w)$ hold, with parameter functions $X$ and $\kappa$ subject to intricate non-linear constraints ensuring that the associated graded algebra is $S(V) \rtimes G$ (Grimley et al., 7 Nov 2024).
• The full classification of deformations for cyclic transvection groups in characteristic $p$ connects PBW deformation conditions with explicit combinatorial systems and solutions in the group algebra (Grimley et al., 7 Nov 2024); in characteristic zero, Lusztig– and Drinfeld–type deformations are isomorphic, but in the modular case new phenomena arise (Shepler et al., 2013).
• Hochschild cohomology provides the natural home for first-order deformation parameters; explicit double complexes (such as the tensor product of the Koszul and bar resolutions) and chain maps yield practical methods for understanding liftings and obstructions (Shepler et al., 2013, Shepler et al., 2019).

The Gerstenhaber bracket, a Lie bracket on Hochschild cohomology, controls obstructions to deforming the algebra. Twisted product resolutions offer computation-friendly models for explicit computation of Gerstenhaber brackets in modular settings (Shepler et al., 2019).

5. Morita Reductions, Quivers, and Galois Coverings

The structure of skew group algebras is further elucidated by their Morita reduction to basic algebras, often described as path algebras of specific quivers with relations:

• For a skew group algebra $T_S(M)*G$ formed from a quiver $Q$ and a group action on its vertices/arrows, Morita equivalence to $k Q_G$ holds, where $Q_G$ is the "Demonet quiver" whose vertices are pairs $(i, U)$ with $i$ a $G$-orbit representative and $U$ an irreducible representation of the stabilizer $G_i$ (Meur, 2018). Explicit formulas using intertwiners and monoidal categories allow for computational decompositions of elements as linear combinations of paths (Meur, 2018).
• Galois semi–covering functors, pushing down modules from $A$ to $A G$, preserve indecomposability and irreducible morphisms in favorable cases; almost split sequences and radical filtrations are transferred under these functors, preserving stable ranks and enabling detailed paper of the Auslander–Reiten structure in $A G$ (Sardar et al., 27 Jul 2025).

6. Quasi-Hereditary, Stratification, and Borel Subalgebras

Skew group algebras preserve quasi-hereditary and highest weight structures under compatible group actions. If a $G$-equivariant partial order on the simple modules exists, then $A$ is quasi-hereditary if and only if $A G$ is (Rasmussen, 2023). Exact Borel subalgebra structures, directedness, strong quasi-heredity, and related filtration invariants also lift to the skew group algebra. The induction functors interact compatibly with standard and pseudo-standard modules. Such structural stability is fundamental in contexts where stratifications and highest-weight paradigms govern homological and categorical behavior, including in categorification and representation theory of wreath products.

7. Extensions, Generalizations, and Related Constructions

Several generalizations position skew group algebras within broader categorical and algebraic frameworks:

• Azumaya and maximal order conditions: For a skew group ring (possibly crossed product), $A*G$ is Azumaya if and only if $A$ is Azumaya and $G$ acts freely on $Z(A)$ (Crawford, 2017). Quantum Kleinian singularities and their skew group algebras have Azumaya and maximal order properties after suitable localizations. Classical theorems (e.g., Auslander's Theorem) generalize to these contexts (Crawford, 2017, Gaddis et al., 2017).
• $\infty$-Categorical perspectives: The skew group dg-algebra $A G$ models the homotopy colimit (group quotient) of the group action in the Morita model structure of dg-categories, producing equivalences in the derived $\infty$-categorical setting (Christ, 23 Jan 2025). Orbit dg-categories and extensions to ring spectra connect skew group constructions with colimits in stable homotopy theory.
• Connections to Hecke and quantum algebras: Skew Hecke algebras $\mathcal{H}_R(G, H, A, \alpha)$ generalize both skew group and classical Hecke algebras (Waldron et al., 2023). When $H = 1$, one recovers $A \rtimes G$; structural decompositions and isomorphisms to corners and invariant algebras clarify the interplay with standard constructions in representation theory (Waldron et al., 2023).
• Skew-gentle and orbifold algebras: Skew-gentle algebras, described as skew-group algebras of gentle algebras by a $\mathbb{Z}_2$ action, have rich connections to surfaces, orbifolds, and their derived categories, with geometric classification via winding numbers of line fields on orbifolds (Amiot et al., 2019).

Summary Table: Core Structural Features

Structure	Skew Group Algebra ($A G$) Expression	Key Homological/Representation Features
Underlying vector space	$A \otimes_k kG$, $a g \cdot b h = a g(b) (gh)$	Incorporates group action directly into algebra
Global/finitistic dim.	$$\operatorname{gldim}(A G) = \operatorname{gldim}(A)$$ (under freeness)	Invariance under suitable group action
Koszul property	Holds iff $A$ is (generalized) Koszul under graded $G$-action	Double Ext algebra: $$\operatorname{Ext}^*(A G) \cong \operatorname{Ext}^*(A) \, G$$
Hochschild cohomology	Spectral sequence: $$E_2^{p,q} = H^p(G, H^q(A, k)) \implies H^{p+q}(A G, k)$$	Finite generation (Fg) and support varieties transfer
Morita reduction	$e A G e \cong k Q_G$, $e$ chosen from group–quiver data	Path algebra quiver for reduced algebra
Deformations	PBW deformations via $X$, $\kappa$; modular case admits new solutions	Classified by PBW conditions and Hochschild classes

The skew group algebra framework unifies group actions on algebras with module theory, homological invariants, noncommutative geometry, deformation theory, and categorical quotients, while supporting explicit computations and structural classifications in modular and nonmodular settings alike.