Deformations of Azumaya Algebras
نویسنده
چکیده
In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold (C or complex analytic). Deformation theory of associative algebras was initiated by Gerstenhaber in [G]. A deformation of an associative algebra A over an Artinian ring a is an a-linear associative algebra structure on A⊗ a such that, for the maximal ideal m of a, A ⊗ m is an ideal, and the quotient algebra on A is the original one. Gerstenhaber showed that the Hochschild cochain complex of an associative algebra A has a structure of a differential graded Lie algebra (DGLA), and that deformations of A over an Artinian ring a are classified by Maurer-Cartan elements of the DGLA C(A,A)[1] ⊗ m. A Maurer-Catan element of a DGLA L with the differential δ is by definition an element λ of L satisfying
منابع مشابه
Azumaya Monads and Comonads
The definition of Azumaya algebras over commutative rings R requires the tensor product of modules over R and the twist map for the tensor product of any two R-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category A by considering a monad (F,m, e) on A endowed with a dis...
متن کاملQuasi-elementary H-Azumaya Algebras Arising from Generalized (Anti) Yetter-Drinfeld Modules
Let H be a Hopf algebra with bijective antipode, α, β ∈ AutHopf (H) and M a finite dimensional (α, β)-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k,H), the Brauer group of H .
متن کاملThe Differential Azumaya Algebras and Non-commutative Picard–Vessiot Cocycles
A differential Azumaya algebra, and in particular a differential matrix algebra, over a differential field K with constants C is trivialized by a Picard–Vessiot (differential Galois) extension E. This yields a bijection between isomorphism classes of differential algebras and Picard–Vessiot cocycles Z(G(E/K), PGLn(C)) which cobound in Z (G(E/K), PGLn(E)).
متن کاملAzumaya Structure on D-branes and Deformations and Resolutions of a Conifold Revisited: Klebanov-strassler-witten vs. Polchinski-grothendieck
In this sequel to [L-Y1], [L-L-S-Y], and [L-Y2] (respectively arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], and arXiv:0901.0342 [math.AG]), we study a D-brane probe on a conifold from the viewpoint of the Azumaya structure on D-branes and toric geometry. The details of how deformations and resolutions of the standard toric conifold Y can be obtained via morphisms from Azumaya points are...
متن کامل