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\section*{deformation theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{formal_geometry}{}\paragraph*{{Formal geometry}}\label{formal_geometry}

[[!include formal geometry -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{Modules}{Modules, derivations and K\"a{}hler differentials}\dotfill \pageref*{Modules} \linebreak
\noindent\hyperlink{cotangent_complex}{Cotangent complex}\dotfill \pageref*{cotangent_complex} \linebreak
\noindent\hyperlink{further_categorification}{Further categorification}\dotfill \pageref*{further_categorification} \linebreak
\noindent\hyperlink{deformation_theory_via_differential_graded_lie_algebras}{Deformation theory via differential graded Lie algebras}\dotfill \pageref*{deformation_theory_via_differential_graded_lie_algebras} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

Deformation theory studies problems of extending structures to extensions of their domains. \textbf{Formal deformation theory}, is the part of the deformation theory where the extensions are \emph{[[infinitesimal object|infinitesimal]]}.

A typical problem in formal deformation theory has the structure that

\begin{itemize}%
\item a [[morphism]] $f : X \to Y$ of certain [[space]]s is given,


\item and [[infinitesimal object|infinitesimal]] thickenings $\tilde X$ and $\tilde Y$ of $X$ and $Y$ are prescribed, with injection morphisms $X \to \tilde X$ and $Y \to \tilde Y$



\end{itemize}
and asks whether a bottom horizontal morphism $\tilde f$ in the diagram

\begin{displaymath}
\itexarray{
    X &\stackrel{f}{\to}& Y
    \\
    \downarrow && \downarrow
    \\
    \tilde X &\stackrel{\tilde f}{\to}& \tilde Y
  }
\end{displaymath}
may be found. This morphism $\tilde f$ would be called an \emph{infinitesimal deformation} of $f$.

In other words:

Formal deformation theory studies the \textbf{[[obstruction|obstruction theory]]} of extensions to \textbf{infinitesimal} thickenings.

A typical example of an \emph{infinitesimal thickening} is a \emph{square-0-extension} of a [[ring]]:

let $R$ be a [[ring]], to be thought of as the ring of functions on the [[space]] $X$ in the above diagram. Let furthermore $N$ be an $R$-module, to be thought of as the $R$-module of [[section]]s of a [[vector bundle]] over $X$.

Then consider the new ring, whose underlying group is the [[direct sum]] $R \oplus N$, equipped with the product structure

\begin{displaymath}
(r_1, n_1) \cdot (r_2, n_2) = (r_1 r_2, r_1 n_2 + n_1 r_2)
  \,.
\end{displaymath}
This is the \textbf{square 0-extension} of $R$ by $N$. It should be thought of as the algebra of functions that consists of elements of $R$ and $N$, where the elements in $N$ are thought of as functions with values in [[infinitesimal object|infinitesimal quantities]], so that their would-be product ``$n_1 \cdot n_2$'' vanishes.

So the ring $R \oplus N$ may be thought of as the ring of functions on the infinitesimal extension $\tilde X$ of $X$, which is the space obtained by adding to $X$ all the \emph{vectors of infinitesimal length} in the vector bundle over $X$.

There is a canonical ring homomorphism $R\oplus N \to R$ that is the identity on $R$ and $0$ on $N$. This is to be thought of as the pullback of functions on spaces along the inclusion of spaces $X \to \tilde X$ (which in turn may be thought of as the 0-[[section]] of the vector bundle on $X$).

Similarly, let $R_2$ be another ring with module $N_2$ and square-0 extension $R_2 \oplus N_2$, thought of, respectively, as the ring of functions on a space $Y$, the module of sections of a vector bundle on $Y$ and the ring of functions on the space of infinitesimal vectors of this vector bundle.

In terms of these function rings, a morphism $f : X \to Y$ of spaces corresponds to a ring homomorphism $R_1 \leftarrow R_2 : f^*$. Hence we have a situation

\begin{displaymath}
\itexarray{
    R_1 &\stackrel{f^*}{\leftarrow}& R_2
    \\
    \uparrow && \uparrow
    \\
    R_1 \oplus N_1 && R_2 \oplus N_2
  }
  \,.
\end{displaymath}
The obvious obstruction problem now is whether we can \textbf{deform} $f^*$ to a morphism $R_1 \oplus N_1 \leftarrow R_2 \oplus N_2 : \tilde f^*$ of rings, such that we get a commuting diagram

\begin{displaymath}
\itexarray{
    R_1 &\stackrel{f^*}{\leftarrow}& R_2
    \\
    \uparrow && \uparrow
    \\
    R_1 \oplus N_1 &\stackrel{\tilde f^*}{\leftarrow}& R_2 \oplus N_2
  }
  \,.
\end{displaymath}
The obstruction to the existence of such lifts is measured by [[cohomology]] with coefficients in the [[cotangent complex]] of $R_1$.

This is the archetypical problem that deformation theory deals with. As always, after studying this a bit it turns out that in order to obtain a good theory, one needs to adopt the [[nPOV]]. Problems as above may be stated in the [[category]] [[Ring]] of rings, but they may have good answers only in [[vertical categorification|categorifications]] of this for instance to the [[(∞,1)-category]] of [[E-∞-ring]]s.

\hypertarget{Modules}{}\subsubsection*{{Modules, derivations and K\"a{}hler differentials}}\label{Modules}

In order to better see the structure of the above archetypical problem of deformation theory, we describe some aspects of the canonical [[bifibration]] of ring modules in a way that nicely organizes all the concepts [[module]], [[derivation]], [[Kähler differential]] in a single picture that lends itself to [[vertical categorification]]. (Following [[Deformation Theory|DefTheory]].)

With [[Ring]] denoting the [[category]] of (commutative, unital) [[rings]], write

\begin{displaymath}
p : Mod \to Ring
\end{displaymath}
for the [[bifibration]] of [[modules]] over [[rings]]: objects of $Mod$ are pairs consisting of a ring $R$ an an $R$-[[module]] $N$, and morphism $(R_1,N_1) \to (R_2, N_2)$ are pairs consisting of a ring homomorphism $f : R_1 \to R_2$ and a morphism $F : N_1 \to N_2 \otimes_f R_2$ of $R_2$-modules.

(Recall for instance from the discussion at [[Sweedler coring]]) that this bifibration is a way to think of the [[stack]] of algebraic [[vector bundles]].)

But there is also another functor $G : Mod \to Ring$ of interest: for $N$ any $R$-module, we may form the ring $G(N) := R \oplus N$ called the \textbf{square 0-extension} of $R$, in which multiplication is given by

\begin{displaymath}
(r_1,n_1) \cdot (r_2, n_2) := (r_1 r_2, n_1 r_2 + n_2 r_1)
  \,.
\end{displaymath}
Moreover, there is a [[natural transformation|natural]] morphism of rings $G(N) \to R$ given by sending $(r,n) \mapsto r$. A [[section]] $v : R \to G(n)$ of this morphism is precisely a [[derivation]] of $R$ with values in the module $N$.

This may be organized into a single functor

\begin{displaymath}
Mod \to [I,Ring]
\end{displaymath}
into the [[arrow category]] of [[Ring]], that sends to the $R$-module $N$ to the morphism $G(N) \to R$. The original bifibration factors through this morphism by the right endpoint evaluation

\begin{displaymath}
\itexarray{
    Mod &&\stackrel{p}{\to}&& Ring
    \\
    & \searrow && \nearrow_{\mathrlap{d_1}}
    \\
    && [I,Ring]
  }
  \,.
\end{displaymath}
Finally notice that the [[functor]] $G$ has a [[left adjoint|left]] [[adjoint functor]]

\begin{displaymath}
\Omega : Ring \to Mod
\end{displaymath}
that sends a ring $R$ to the $R$-module $\Omega_K(R)$ of [[Kähler differentials]], i.e. to the module that encodes the [[cotangent bundle]].

\hypertarget{cotangent_complex}{}\subsubsection*{{Cotangent complex}}\label{cotangent_complex}

Using the module of K\"a{}hler differentials is not appropriate in general; instead we need to take its derived version. To talk about the nonabelian derived functors, Quillen introduced a model category structure on the category of simplicial commutative rings. Given a morphism $f: A\to B$ of rings, which makes $B$ an $A$-algebra, the category $AbGr(A-Alg/B)$ of abelian group objects in the slice category $A$-$Alg/B$ of $A$-algebras over $B$ is equivalent both to the category of $B$-modules and the trivial (= square zero) extensions of $A$ by $B$-modules. In particular we can consider the forgetful functor $AbGr(A-Alg/B)\to A-Alg/B$ which has a left adjoint $Ab_{B/A} : A-Alg/B\to AbGr(A-Alg/B)\cong {}_B Mod$. All said is true for simplicial commutative rings as well. Now the \textbf{relative cotangent complex} $L_{B/A}$ is the value on $B$ of the left [[derived functor]] $\mathbb{L} Ab_{B/A}(B)$. Regarding that the left adjoint at the nonderived level (and for usual rings) can be expressed via K\"a{}hler differentials, this explains the phrase ``derived version of the module K\"a{}hler differentials''.

The above situation generalizes from the category [[Ring]] to an arbitrary presentable [[(∞,1)-category]] $C$ by replacing the [[bifibration]] $Mod \to Ring$ by the [[stabilization]] $T_C \to C$ of the [[codomain fibration]] of $C$: the [[tangent (∞,1)-category]] of $C$.

The projection $p : T_C \to C$ still has a [[left adjoint]]

\begin{displaymath}
\Omega : C \to T_C
\end{displaymath}
for which a representative which is also a section (in a strict sense) of $p$ may be taken; any such representative is called the [[cotangent complex]] functor for $C$. The special property section property, like in the motivating example above, says that the composition

\begin{displaymath}
C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C
\end{displaymath}
is the identity [[(∞,1)-functor]].

\hypertarget{further_categorification}{}\subsubsection*{{Further categorification}}\label{further_categorification}

\ldots{}

\hypertarget{deformation_theory_via_differential_graded_lie_algebras}{}\subsection*{{Deformation theory via differential graded Lie algebras}}\label{deformation_theory_via_differential_graded_lie_algebras}

Over a field of characteristic zero, there is an approach to deformation theory via [[differential graded Lie algebras]] (or more generally [[L-infinity algebras]]). One can find some exposition about this approach in the Kontsevich and Lurie references below. See also discussion at MathOverflow: \href{http://mathoverflow.net/questions/385/deformation-theory-and-differential-graded-lie-algebras}{def theory and dgla-s}.

In this approach, one begins with an object $X$ (for example a scheme, or a complex manifold, or a vector bundle, or an associative algebra, or a dg category, or \ldots{}) that one would like to deform. Then the general principle is that there exists a dgLa $L_X$ with the property that the functor $Def_{L_X} : Art \to Set$, which sends a local Artin algebra $(A,m)$ to the set of [[Maurer-Cartan equation|Maurer-Cartan solutions]] in $(L_X \otimes m)^1$ modulo the gauge action of $(L_X \otimes m)^0$, is isomorphic to the functor which sends a local Artin algebra $(A,m)$ to the set of isomorphism classes of deformations of $X$ over $\operatorname{Spec} A$. Note the similarity with Schlessinger's theory: both here and in Schlessinger's work, we deal with functors from Artin algebras to sets.

In the case of a compact complex manifold, the dgLa in question is given by the so-called [[Kodaira-Spence theory|Kodaira-Spencer]] dgLa: holomorphic vector fields tensor $(0,q)$-forms (this is just the [[Dolbeault cohomology|Dolbeault resolution]] of the sheaf of holomorphic vector fields). In the case of an associative algebra (or a dg algebra, or an A-infinity algebra, or a dg category, or an A-infinity category), the appropriate dgLa is the Hochschild complex with the Hochschild differential and the Gerstenhaber bracket.

In this language, the [[Tian-Todorov theorem]] on the unobstructedness of deformations of [[Calabi-Yau manifold]]s translates to the statement that the [[Kodaira-Spencer theory|Kodaira-Spencer]] dgLa of a Calabi-Yau manifold is homotopy abelian --- that is, it is quasi-isomorphic to an abelian dg Lie algebra. Barannikov-Kontsevich proved more generally that the dgLa given by holomorphic polyvector fields tensor $(0,q)$-forms on a Calabi-Yau manifold is homotopy abelian. The deformation-theoretic consequence is that the ``extended deformations'' of Calabi-Yau manifolds are unobstructed. These ``extended deformations'' should be realized by certain $A_\infty$ deformations of (a dg enhancement of) the derived category of coherent sheaves on the Calabi-Yau.

The following paper is a good introduction to these ideas:

\begin{itemize}%
\item Marco Manetti, \emph{Deformation theory via differential graded Lie algebras} , \textdegree{}(\href{http://arxiv.org/abs/math/0507284}{arXiv:0507284})

\end{itemize}
The Kontsevich and Soibelman references below are also good.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Kuranishi deformation theory]]


\item [[Kodaira-Spencer theory]]


\item [[Artin-Mazur formal group]]


\item [[deformation context]], [[formal moduli problem]], [[Lie differentiation]]


\item [[tangent complex]], [[André-Quillen cohomology]], [[Hochschild cohomology]]


\item [[cotangent complex]], [[André-Quillen homology]], [[Hochschild homology]]


\item [[derived deformation theory]], [[derived algebraic geometry]]


\item [[formal scheme]], [[formal smoothness]]


\item [[deformation quantization]]


\item [[Lubin-Tate theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Michael Schlessinger]], [[Jim Stasheff]], \emph{The Lie algebra structure of tangent cohomology and deformation theory}, Journal of pure and applied algebra 38 (1985) 313-322


\item [[Michael Schlessinger]], [[Jim Stasheff]], \emph{Deformation theory and rational homotopy type} (\href{https://arxiv.org/abs/1211.1647}{arXiv:1211.1647})


\item C. Doran, S. Wong, “Deformation Theory: An Historical Annotated Bibliography,” Chapter 2 in Deformation of Galois Representations, to appear in the AMS-IP Studies in Advanced Mathematics Series.


\item M. Doubek, M. Markl, P. Zima, \emph{Deformation theory (lecture notes)}, Archivum mathematicum \textbf{43} (5), 2007, 333--371, \href{http://arxiv.org/abs/0705.3719}{arXiv:0705.3719}


\item [[Martin Markl]], \emph{Deformation theory of algebras and their diagrams}, 129 pp, CBMS \textbf{116}, AMS 2012, \href{http://www.ams.org/bookstore-getitem/item=CBMS-116}{book page}


\item Wikipedia: \href{http://en.wikipedia.org/wiki/Deformation_theory}{deformation theory}, \href{http://en.wikipedia.org/wiki/Cotangent_complex}{cotangent complex}


\item E. Sernesi, \emph{An overview of classical deformation theory}, \href{http://www.mat.uniroma3.it/users/sernesi/sernesioverviewdefth.pdf}{pdf}


\item [[Alexander Grothendieck]], \emph{Cat\'e{}gories cofibr\'e{}es additives et complexe cotangent relatif}, Lecture Notes in Mathematics 79


\item [[Luc Illusie]], \emph{Complexe cotangent et d\'e{}formations I}, Lec. Notes Math. \textbf{239}, Springer 1971, xv+355 pp.; \emph{II}, LNM \textbf{283}, Springer 1972. vii+304 xv+355 pp.


\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Deformation theory I} (\href{http://www.math.ksu.edu/~soibel/Book-vol1.ps}{ps}); \emph{Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I}, \href{http://arxiv.org/abs/math/0606241}{math.AG/0606241} -- two parts of large unfinished books on the subject; [[Yan Soibelman]], \emph{Lectures on deformation theory and mirror symmetry} (\href{http://www.math.ksu.edu/~soibel/ipam-final.ps}{ps})


\item [[Maxim Kontsevich]], \emph{Topics in deformation theory} (A rough write up of a Berkeley course, early 90-s), \href{http://www.math.uchicago.edu/~mitya/langlands/kontsdef.ps}{ps}


\item [[Jacob Lurie]], [[Deformation Theory]] (\href{http://arxiv.org/abs/0709.3091}{arXiv:0709.3091}) -- describes a very setup for deformation theory over any [[(∞,1)-category]] is described. Then as an application the deformation theory of [[E-∞-ring]]s is developed. An application: J. Lurie, \emph{Moduli problems for ring spectra}, \href{http://www.math.harvard.edu/~lurie/papers/moduli.pdf}{moduli.pdf}.


\item E. Sernesi, \emph{Deformations of algebraic schemes} (monograph) Grundlehren der Math. Wiss. \textbf{334}, Springer 2006. xii+339 pp. \href{http://www.ams.org/mathscinet-getitem?mr=2008e:14011}{MR2008e:14011}


\item Alexander I. Efimov, [[Valery Lunts|Valery A. Lunts]], [[Dmitri Orlov|Dmitri O. Orlov]], \emph{Deformation theory of objects in homotopy and derived categories}

\begin{itemize}%
\item \emph{I: general theory} \href{http://arxiv.org/abs/math/0702838}{arXiv:math/0702838};
\item \emph{II: pro-representability of the deformation functor} \href{http://arxiv.org/abs/math/0702839}{arXiv:math/0702839};
\item \emph{III: abelian categories} \href{http://arxiv.org/abs/math/0702840}{arXiv:math/0702840}

\end{itemize}

\item [[Martin C. Olsson]], \emph{Deformation theory of representable morphisms of algebraic stacks}, Mathematische Zeitschrift\_\_253\_\_, n. 1, 25--62 (2006) \href{http://dx.doi.org/10.1007/s00209-005-0875-9}{doi}; \emph{Tangent spaces and obstructon theories}, lectures, \href{http://math.berkeley.edu/~molsson/MSRISummer07.pdf}{MSRISummer07.pdf}


\item B. Fantechi, M. Manetti, \emph{Obstruction calculus for functors of Artin rings I}, J. Algebra \textbf{202} (1998), no. 2, 541--576, \href{http://www.ams.org/mathscinet-getitem?mr=99f:14004}{MR99f:14004}.


\item Domenico Fiorenza, Marco Manetti, Elena Martinengo, \emph{Semicosimplicial DGLAs in deformation theory}, \href{http://arxiv.org/abs/0803.0399}{arxiv/0803.0399}


\item S. Merkulov, B. Vallette, \emph{Deformation theory of properads}, \href{http://arxiv.org/abs/0707.0889}{arXiv:0707.0889}


\item [[Vladimir Hinich]], \emph{Deformations of homotopy algebras}, Communication in Algebra, 32 (2004), 473-494, \href{http://www.ams.org/mathscinet-getitem?mr=2101417}{MR2005m:18009}, \href{http://dx.doi.org/10.1081/AGB-120027907}{doi}, \href{http://math.haifa.ac.il/hinich/WEB/mypapers/dha.pdf}{pdf}; \emph{Deformations of sheaves of algebras}, Adv. Math. \textbf{195} (2005), no. 1, 102--164, \href{http://www.ams.org/mathscinet-getitem?mr=2145794}{MR2007d:13021}, \href{http://dx.doi.org/10.1016/j.aim.2004.07.007}{doi}, \href{http://math.haifa.ac.il/hinich/WEB/mypapers/dsa.pdf}{pdf}; \emph{DG coalgebras as formal stacks}, J. Pure Appl. Algebra \textbf{162} (2001), no. 2-3, 209--250 (), \href{http://arxiv.org/abs/math.AG/9812034}{math.AG/9812034}; \emph{Formal deformations of sheaves of algebras}, video of a talk at MSRI 2002, \href{http://www.msri.org/publications/ln/msri/2002/hodgetheory/hinich/1/index.html}{link}


\item V. Hinich, V. Schechtman, \emph{Deformation theory and Lie algebra homology} I. Algebra Colloq. 4 (1997), no. 2, 213--240, \href{http://www.ams.org/mathscinet-getitem?mr=1682729}{MR2000c:14006a}; part II., Algebra Colloq. 4 (1997), no. 3, 291--316, \href{http://www.ams.org/mathscinet-getitem?mr=1681547}{MR2000c:14006b}; preprint version \href{http://arxiv.org/abs/alg-geom/9405013}{alg-geom/9405013};\_Homotopy Lie algebras\_, I.M. Gelfand Seminar, Adv. in Sov. Math., 16 (1993), Part 2, 1-28.


\item W. Lowen, M. Van den Bergh, \emph{Deformation theory of abelian categories}, Trans. Amer. Math. Soc. \textbf{358} (2006), no. 12, 5441--5483; \href{http://arxiv.org/abs/math/0405226}{arXiv:math.CT/0405226}.


\item M. Van den Bergh, \emph{Notes on formal deformations of abelian categories}, \href{http://arxiv.org/abs/1002.0259}{arXiv:1002.0259}


\item M. Talpo, [[Angelo Vistoli|A. Vistoli]], \emph{Deformation theory from the point of view of fibered categories}, \href{http://arxiv.org/abs/1006.0497}{arxiv/1006.0497}


\item B. Mazur, \emph{Perturbations, deformations, and variations (and ``Near-misses'') in Geometry, Physics, and Number Theory}, \href{http://www.ams.org/bull/2004-41-03/S0273-0979-04-01024-9/S0273-0979-04-01024-9.pdf}{BAMS 41(3), 307-336}


\item M. Artin, \emph{Deformations of singularities}, TATA Lecture Notes vol. 54.


\item M. Artin, \emph{Versal deformations and algebraic stacks}, Invent. Math. 1974


\item K. Kodaira, L. Nirenberg, D. C. Spencer, \emph{On the existence of deformation of complex analytic structures}, Ann. Math. \textbf{68}, 450-459 (1958).


\item K. Kodaira, D. C. Spencer, \emph{On deformation of complex analytic structures}, I II, Ann. Math. \textbf{67}, 328-466 (1958).


\item M. Schlessinger, \emph{Functors of Artin rings}, Trans. AMS 130, 208-222 (1968) -- this was a groundbreaking article at the time, still much cited.


\item B. Osserman, \emph{Deformation theory and moduli in algebraic geometry}, \href{http://www.msri.org/people/members/defthy07/lectures/brian.pdf}{pdf}


\item [[Robin Hartshorne]], \emph{Deformation theory}, Grad. texts in math. Springer 2010, viii+234 pp. (draft of the chap. I-IV: \href{http://math.berkeley.edu/~robin/math274root.pdf}{pdf}), \href{http://books.google.hr/books?id=bwhEX01JlXkC&lpg=PP1&ots=2zkSvT3Bvz&dq=Robin%20Hartshorne%20deformation%20theory&pg=PP1#v=onepage&q&f=false}{gBooks}, \href{http://dx.doi.org/10.1007/978-1-4419-1596-2}{doi}


\item \href{http://math.stanford.edu/~vakil/727}{homepage} of Ravi Vakil's graduate Stanford class on deformation theory and moduli spaces


\item [[Kai Behrend]], [[Barbara Fantechi|B. Fantechi]], \emph{The intrinsic normal cone}, Invent. Math. \textbf{128} (1997), no. 1, 45--88, \href{http://www.ams.org/mathscinet-getitem?mr=98e:14022}{MR98e:14022} \href{http://arxiv.org/abs/alg-geom/9601010}{arXiv:alg-geom/9601010}


\item [[Dennis Gaitsgory]], \emph{Lie theory from the point of view of derived algebraic geometry}, Lecture notes from a mini-course, Nantes, 2014, \href{http://www.math.harvard.edu/~gaitsgde/Nantes14/}{web}.


\item Vladimir Dotsenko, Sergey Shadrin, [[Bruno Vallette]], \emph{Pre-Lie deformation theory}, \href{http://arxiv.org/abs/1502.03280}{arxiv/1502.03280}


\item M. Gerstenhaber, S. D. Schack, \emph{Algebras, bialgebras, quantum groups, and algebraic deformations}, in: Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), 51-92, Contemp. Math. \textbf{134}, Amer. Math. Soc. 1992.


\item Gregory Ginot, Sinan Yalin, \emph{Deformation theory of bialgebras, higher Hochschild cohomology and formality}, \href{http://arxiv.org/abs/1606.01504}{arxiv/1606.01504} [[!redirects deformation]] [[!redirects deformations]]



\end{itemize}


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