Maximal Subalgebras of Finite-Dimensional Algebras Alex Sistko Joint work with Miodrag Iovanov Department of Mathematics University of Iowa November 20, 2017 Conference on Geometric Methods in Representation Theory University of Iowa Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Logistics and Set-Up Conventions: k is a field. 1 Unless otherwise noted, all algebras are associative, 2 unital, and finite-dimensional over k . We say two subalgebras A , A ′ of B are conjugate if there 3 is a k -algebra automorphism α of B such that α ( A ) = A ′ . A subalgebra A ⊂ B is maximal if A � = B and for any 4 subalgebra A ′ , A ⊂ A ′ ⊂ B implies A = A ′ or A ′ = B . For a fixed algebra B , we wish to answer: Can we classify the maximal subalgebras of B ? 1 Does B share any interesting representation-theoretic 2 information with its maximal subalgebras? Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Previous Work Related and Similar Results Upper bounds for maximal commutative subalgebras of M n ( C ) : Schur [24]; Gerstenhaber [7]. Upper bound for maximal subalgebras of M n ( C ) : Agore [1]. Classification for maximal subalgebras of central separable algebras: Racine [20], [21]. Classical work in Lie Theory: Dynkin [3]; Malcev [14]. Maximal subalgebras of other non-associative algebras: Elduque [4], [5]; Martinez and Zelmanov [15]; Racine [22]. Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
To the best of our knowledge... No classification of maximal subalgebras in the general semisimple case. No attempt to classify maximal subalgebras of arbitrary finite-dimensional associative algebras. No attempt to see if “maximality” has representation-theoretic consequences for finite-dimensional associative algebras. Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Classification: Split vs. Semisimple Fix an algebra B . Let A be a maximal subalgebra of B . Split Type A is said to be of split type if there exists an ideal of I of B contained in A ∩ J ( B ) , such that J ( B ) / I is a simple A -bimodule and B / I is isomorphic to the trivial extension of A / I by J ( B ) / I . Semisimple/Separable Type A is said to be of semisimple (or separable) type if J ( B ) ⊂ A . Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Classification: Split vs. Semisimple (Cont.) Theorem (I-S 2017) Let B be a finite-dimensional k-algebra, A ⊂ B a maximal subalgebra. Then A is either of split type or separable type. Note If B / J ( B ) is separable over k , and B 0 ∼ = B / J ( B ) is a subalgebra of B such that B = B 0 ⊕ J ( B ) , then one of the following holds: If A is of split type, then A is conjugate under an inner 1 automorphism to an algebra of the form B 0 ⊕ I , where I ⊂ J ( B ) is a maximal B -subbimodule (and in particular, J ( B ) 2 ⊂ I ). If A is of separable type, then A is conjugate under an inner 2 automorphism to an algebra of the form A ′ ⊕ J ( B ) , where A ′ is a maximal subalgebra of B 0 . Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Classification: Split vs. Semisimple (Cont.) Maximal Sub-bimodules of J(B) Difficult to classify (non-unital) subalgebras of (non-unital) 1 nilpotent algebras. Results in small dimension [8]. 2 Nice interpretation when k = k and B is basic. 3 Maximal Subalgebras of Semisimple Algebras We classify maximal subalgebras of semisimple algebras 1 in the general case. Extends previous work of Racine [21] (central separable 2 case). Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Classification: Simple Case Let B = M n ( D ) , D a division k -algebra with dim k D < ∞ . 1 There are three types of maximal subalgebras of B . 2 The first two were noticed by Racine (the third type does 3 not occur in the central separable case). Subalgebra Types S1 : Conjugate to block-upper-triangular matrices with two 1 blocks, contain Z ( D ) . S2: Simple, contain Z ( D ) . 2 S3: Simple, do not contain Z ( D ) . 3 Theorem (I-S 2017) Let B = M n ( D ) as before, and A ⊂ B a maximal subalgebra. Then A is conjugate under an inner automorphism to a maximal subalgebra of type S1, S2, or S3. Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Classification: Semisimple Case Quick Note The image of the diagonal map M n ( D ) → M n ( D ) × M n ( D ) is a maximal subalgebra of M n ( D ) × M n ( D ) , which we call ∆ 2 ( n , D ) . Theorem (I-S 2017) Let B = � t i = 1 M n i ( D i ) , where each D i is a division k-algebra, and n 1 ≤ n 2 ≤ . . . ≤ n t . Then any maximal subalgebra of B is conjugate to an algebra of one of the following two types: There is an i < t such that n i = n i + 1 and D i ∼ = D i + 1 , and 1 �� � �� � × ∆ 2 ( n i , D i ) × A = j < i M n j ( D j ) j > i + 1 M j ( D j ) . �� � �� � For some i ≤ t, A = j < i M n j ( D j ) × A i × j > i M n j ( D j ) , 2 where A i ⊂ M n i ( D i ) is a subalgebra of type S1, S2, or S3. If each block is central simple over k, then this automorphism can be chosen to be an inner automorphism. Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Some Consequences Assume for this slide that k = k (of course some of these statements hold in far greater generality!). Observations from Classification If B ∼ = kQ / I is basic, the classification works very well. 1 Split Type: Codim-1 subspaces of parallel arrows. 1 Separable Type: Identifying vertices of Q . 2 Maximal subalgebras of semisimple algebras are 2 representation-finite and have gl . dim ≤ 1. If B / J ( B ) = � t i = 1 M n i ( k ) and n 1 ≤ . . . ≤ n t , then the 3 maximal dimension of a subalgebra is dim k B − 1 − max { n 1 − 2 , 0 } (extends [1]). Can “find all maximal subalgebras” using inner 4 automorphisms. Can construct separable and split (in fact trivial) 5 extensions. Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Separable and Split Extensions Go back to k arbitrary. Separable Functors Definition: A functor F : C → D is separable if 1 � � � � F ( α ) α η := X − → Y �→ F ( X ) − − − → F ( Y ) has a section, i.e. a natural transformation σ with σ ◦ η = 1 C ( − , − ) . A ⊂ B is a separable extension ⇔ Res B A is separable. 2 A ⊂ B is a split extension ⇔ − ⊗ A B is separable. 3 General Principle Separable extensions allow you to transfer 1 representation-theoretic properties from A to B . Split extensions allow you to transfer 2 representation-theoretic properties from B to A . Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Maximal Subalgebras and Separable Functors Theorem (I-S 2017) Let B be a finite-dimensional k-algebra, and A ⊂ B a maximal subalgebra of separable type. If B / J ( B ) is a separable k-algebra, then A ⊂ B is separable. Theorem (I-S 2017) Let B be a finite-dimensional k-algebra, and A ⊂ B a maximal subalgebra of split type. Suppose that I 1 � I 2 is a minimal extension of ideals with I 1 ⊂ A and I 2 �⊂ A. Then the following hold: If ( I 2 / I 1 ) 2 = 0 , then A / I 1 ⊂ B / I 1 is trivial. 1 If the simple B-modules are 1 -dimensional, A / I 1 ⊂ B / I 1 is 2 split. Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Current/Future Work Question Does the poset of subalgebras of B say anything about its representation theory? In particular, does “maximality” mean anything? Some Notation P k ( B ) = poset of k -subalgebras of B . 1 Iso k ( A , B ) = { A ′ ∈ P k ( B ) | A ′ ∼ = A } . 2 AbCat = small abelian categories with additive functors. 3 Restriction and Induction Restriction and induction yield functors 1 Res : P k ( B ) op → AbCat, Ind : P k ( B ) → AbCat. Fact: There may be A , A ′ ∈ P k ( B ) with A ∼ = A ′ , which yield 2 non-isomorphic functors Mod- A = Mod- A ′ ↔ Mod- B . Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Current/Future Work (Cont.) Question If we fix A ∈ P k ( B ) , can we classify the different functors between Mod- A and Mod- B which arise from Res/Ind in this fashion? Automorphisms and Embeddings Aut k ( B ) acts on P k ( B ) in a natural way, and Iso k ( A , B ) is 1 an Aut k ( B ) -invariant subset. If A ∈ P k ( B ) has the property that Iso k ( A , B ) = Aut k ( B ) · A , 2 then there is only one Res/Ind functor (up to isomorphism). More generally, if Iso k ( A , B ) is a finite union of 3 Aut k ( B ) -orbits, then only finitely-many functors arise. Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Current/Future Work (Cont.) Geometric Interpretation For m ≤ n := dim k B , the GL ( B ) -action on Gr m ( B ) restricts 1 to the closed subgroup Aut k ( B ) . AlgGr m ( B ) = { m -dimensional subalgebras of B } form a 2 closed, Aut k ( B ) -invariant subset of Gr m ( B ) . Goal: For any closed subgroup G ≤ Aut k ( B ) , classify the 3 G -orbits of AlgGr m ( B ) and relate to isoclasses. Early Results ( k = k , B = kQ / I ) Finitely-many Aut k ( kQ ) -orbits on AlgGr n − 1 ( kQ ) . 1 Finitely-many Inn k ( B ) -orbits on AlgGr n − 1 ( B ) ⇔ Q has no 2 parallel arrows. In type A , Iso k ( A , kQ ) = Aut k ( kQ ) · A for all 3 A ∈ AlgGr n − 1 ( kQ ) . Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
Thanks Thanks for listening! Alex Sistko Maximal Subalgebras of Finite-Dimensional Algebras
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