Algebra over a field

An algebra over a field $k$ is a vector space $A$ over $k$ equipped with a bilinear multiplication $A\times A\to A$.

A Poisson algebra is a commutative associative algebra with a Lie bracket that is a derivation in each argument. Smooth functions on a symplectic manifold are its standard example. The linear span of moment-map components may form a finite-dimensional Lie subalgebra, while closing those components under multiplication normally produces a much larger Poisson algebra.

Let $h$ be central and a non-zero-divisor in an associative algebra, with commutative quotient modulo $h$. Dividing commutators by $h$ and reducing modulo $h$ is independent of the lifts, since changing a lift by $hc$ changes the quotient by a commutator, zero modulo $h$. Antisymmetry, the Leibniz rule and the Jacobi identity descend from the corresponding commutator identities. For the order filtration of a Weyl algebra, the convention is $\{{\xi }_i,x_j\}={\delta }_{ij}$, with the opposite sign for $\{x_i,{\xi }_j\}$.

An algebra isomorphism is a bijective linear map preserving multiplication and identity between two algebras over the same field. In classification of a square-zero extension of an algebra, an equivalence must additionally induce the identity on the prescribed quotient and kernel.

A division algebra over a field $k$ is a division ring equipped with a unital algebra structure over $k$, with $k$ mapping into its center of an associative algebra. Every nonzero element is invertible; it need not be finite-dimensional over $k$.

A finite-dimensional associative division algebra over $\mathbb{R}$ is one of the three displayed algebras. A nonreal element has an irreducible quadratic minimal polynomial and yields $i^2=-1$. Its centralizer is a division algebra over central $\mathbb{C}$, hence is $\mathbb{C}$ by splitting minimal polynomials. Decompose the whole algebra into the plus and minus eigenspaces of conjugation by $i$. If the minus space is zero, the algebra is $\mathbb{C}$; otherwise it is $\mathbb{C}j$, since multiplying by $j^{-1}$ takes that space into the centralizer. One has $jz=\overline{z}j$ and $j^2\in \mathbb{R}$. Division forces $j^2<0$, which can be rescaled to $-1$, yielding $\mathbb{H}$. Of these, only $\mathbb{R},\mathbb{H}$ have center $\mathbb{R}$.

A central division algebra over $K$ is a finite-dimensional division algebra whose center is exactly $K$. It is a central simple algebra. Every Brauer group class has a unique central division algebra representative up to isomorphism.

Let $D$ be a finite-dimensional division algebra over an algebraically closed field $k$. Left multiplication by $a\in D$ has an eigenvalue $\lambda \in k$ and a nonzero eigenvector $v$. Thus $(a-\lambda )v=0$; multiplying by $v^{-1}$ gives $a=\lambda$. Every element is scalar, so $D=k$.

A quaternion is $a+bi+cj+dk$ with real coefficients, where $i^2=j^2=k^2=ijk=-1$. They form a noncommutative real division algebra. The norm squared is $a^2+b^2+c^2+d^2$, and every nonzero quaternion has inverse its conjugate divided by that squared norm. The unit quaternions form the Lie group $S^3$.

The displayed complex matrices, with real coefficients, form a four-dimensional real associative algebra. Their three imaginary basis matrices square to minus the identity and satisfy $JK=L$, $KL=J$, $LJ=K$, with the reversed products negative. For $\alpha =aI+bJ+cK+dL$, quaternionic conjugation gives $\alpha \overline{\alpha }=(a^2+b^2+c^2+d^2)I$. Every nonzero matrix therefore has inverse $\overline{\alpha }/(a^2+b^2+c^2+d^2)$ inside this algebra. This realizes the real division algebra of quaternions within $M_2(\mathbb{C})$; it is not a complex vector subspace.

A quaternion of norm $1$ is a unit quaternion. They form the topological group $\mathrm{Sp}(1)$ and identify with the sphere $S^3\subset {\mathbb{R}}^4$. Multiplication and inversion preserve the norm. The Hurewicz theorem identifies ${\pi }_3(S^3)$ with $H_3(S^3;\mathbb{Z})$; hence homotopy-group addition in a topological group shows that $q\mapsto q^m$ has mapping degree $m$, rather than $m^2$.

A linear subspace of an algebra over a field closed under multiplication is a subalgebra. If a unital subalgebra is intended, it must also contain the ambient identity. A closed subalgebra of a normed algebra is closed in the norm topology.

For a field $k$, a homomorphism of $k$-algebras is a unital ring homomorphism preserving scalar multiplication by every element of $k$. For unital algebras it is equivalently a ring homomorphism commuting with their structural maps from $k$. Pullback by a morphism of algebraic varieties gives such a homomorphism between their coordinate rings.

An $R$-point of a unital algebra over a field $A$ is a unital algebra homomorphism over a field $A\to R$. For the quantum plane over the complex numbers, scalar points lie on the two coordinate axes when $q\ne 1$, while matrix-valued points are pairs $X,Y$ with $YX=qXY$.

For finite matrices satisfying $YX=qXY$, with $q$ not a root of unity, $Y$ sends the generalized eigenspace of $X$ at $\lambda$ into the one at $q^{-1}\lambda$. Finiteness of the spectrum makes $Y$ nilpotent on the sum of the nonzero $X$ generalized eigenspaces. Exchanging $X,Y$ gives the corresponding statement for $X$. Splitting the zero generalized eigenspace of $X$ by that of $Y$ gives the displayed invariant direct sum; both operators are nilpotent on $U$.

For a basis $(e_i)$ of a finite-dimensional algebra, its structure constants are the scalars defined byThey express multiplication as polynomial coordinate functions.

A $k$-algebra is finitely generated when finitely many elements generate it under addition, multiplication, and scalar multiplication. Equivalently, it is a quotient of a polynomial ring $k[x_1,\dots ,x_n]$.

Let $A$ be a commutative algebra over a field $k$, with a homomorphism $A\to k$ restricting to the identity on $k$, and let $I$ be its kernel. If $A$ is generated by finitely many elements, subtracting their images in $k$ gives finitely many generators in $I$. Every polynomial in those generators, modulo $I^2$, is a linear combination of their classes. Thus $I/I^2$, the algebraic cotangent space at this point, is finite-dimensional. For $A=k+xk[x,y]$ and $I=xk[x,y]$, the classes $x{y}^j$, $j\ge 0$, form an infinite basis modulo $I^2=x^{2}k[x,y]$. This is a subalgebra of a polynomial ring which is not a finitely generated algebra.

A presentation of an algebra over a field with finitely many generators and relations. The Hilbert basis theorem ensures that every finitely generated algebra over a field admits such a presentation. Its module of Kähler differentials is presented by the Jacobian matrix of the relations.

Suppose a $k$-algebra $A$ has elements $f_i,g_i$ with ${\sum }_i{g}_i{f}_i=1$, and each $A_{f_i}$ is a finitely generated algebra. Let $B\subseteq A$ contain the $f_i,g_i$ and numerators of finite generating sets of all $A_{f_i}$. Then $B$ is finitely generated and $B_{f_i}=A_{f_i}$. For $a\in A$, clear denominators so that all $f_i^{N}a$ belong to $B$. A unit-ideal identity for powers gives $1={\sum }_i{c}_i{f}_i^N$ in $B$, hence $a={\sum }_i{c}_i{f}_i^{N}a\in B$. Thus $A=B$.

Let $A\subseteq B\subseteq C$ be rings. If $A$ is Noetherian, $C$ is a finitely generated $A$-algebra, and $C$ is a finite $B$-module, then $B$ is a finitely generated $A$-algebra.

Let $L/k$ be a field extension. If $A{\otimes }_{k}L$ is a finitely generated algebra over $L$, then $A$ is finitely generated over $k$. Choose finitely many generators of $A{\otimes }_{k}L$ and collect the finitely many coefficients from $A$ occurring in them. The $k$-subalgebra $A_0$ generated by those coefficients satisfies $(A/A_0){\otimes }_{k}L=0$, and faithful flatness of $L/k$ gives $A=A_0$.

For a vector space with quadratic form $Q$, its Clifford algebra is generated by vectors subject to $v^2=Q(v)1$. In a basis this is equivalently ${\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2{\eta }^{\mu \nu }1$.

After complexifying a nondegenerate metric in $D=2k$ dimensions, pair gamma matrices into $k$ fermionic creation and annihilation operators. Their binary occupation states span $2^k$ dimensions, and the gamma matrices generate all endomorphisms of this space. The Dirac module is therefore irreducible as a Clifford module. The even algebra preserves the chirality grading, splitting its Lorentz-spinor representation into two irreducible Weyl modules of dimension $2^{k-1}$. In $D=2k+1$, adjoining the old chirality matrix as the extra gamma gives an irreducible $2^k$-dimensional Spin group representation with no Weyl splitting. The two possible signs of that extra gamma distinguish Clifford modules but give equivalent Spin group modules.

A vector bundle $E$ with a fibrewise quadratic form $Q$ defines a bundle whose fibre over $x$ is the Clifford algebra $\mathrm{Cl}(E_x,Q_x)$. When the polar form is a nondegenerate bilinear form, local orthonormal frames identify its fibres with a fixed Clifford algebra, and orthogonal transition maps induce the algebra transition maps. A Clifford module bundle is a bundle of modules over these algebras, rather than the algebra bundle itself. For a Riemannian manifold, use its cotangent bundle and the metric quadratic form, with the sign convention specified.

Clifford multiplication is the action of a Clifford algebra on a module. The displayed relation uses the negative Riemannian metric convention; in a compatible Hermitian inner product, $c(v{)}^{\ast }=-c(v)$ for real $v$. Another convention writes $C(v)=ic(v)$, giving self-adjoint generators with positive anticommutator. Switching between these conventions switches a geometric Dirac operator between self-adjoint and skew-adjoint realizations.

For $2n$ hermitian Euclidean Clifford generators, the commuting hermitian involutions $J_j=i{\gamma }^{2j-1}{\gamma }^{2j}$ admit a common eigenvector. Each odd-indexed generator flips one joint eigenvalue. The $2^n$ resulting vectors are nonzero and mutually orthogonal. Thus any nonzero invariant complex representation has dimension at least $2^n$.

For hermitian generators satisfying $\{{\gamma }^I,{\gamma }^J{\}}_{+}=2{\delta }^{IJ}$ and any nonzero vector $v$, the vectors ${\gamma }^{I}v$ are linearly independent over the real numbers. Indeed, $A={\sum }_I{a}_I{\gamma }^I$ with real $a_I$ obeys $A^2=({\sum }_I{a}_I^2)1$, so $\Vert Av{\Vert }^2=({\sum }_I{a}_I^2)\Vert v{\Vert }^2$. Complex coefficients need a different argument.

Gamma matrices give a matrix representation of the Clifford algebra of spacetime.

For gamma matrices with $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}$ and ${\gamma }^{ab}=[{\gamma }^a,{\gamma }^b]/2$, contracted Clifford algebra gives ${\gamma }_a{\gamma }^m{\gamma }^a=(2-D){\gamma }^m$. Expanding the two antisymmetrized factors gives ${\gamma }_a{\gamma }_b{\gamma }^m{\gamma }^a{\gamma }^b-D{\gamma }^m$. The five-gamma product equals $[2D-(D-2{)}^2]{\gamma }^m$, by moving the middle contracted gamma through and using the three-gamma contraction twice. Subtraction yields the displayed identity. A convention without the factor one-half in each bivector multiplies the result by four.

Feynman slash notation contracts a four-vector with the gamma matrices. With mostly-minus signature, the Clifford algebra gives $/p^2=p^{2}I$, so $(/p-m{)}^{-1}=(/p+m)/(p^2-m^2)$ away from the mass shell.

For signature $+---$, use ${\gamma }^0=\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ and ${\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$, where ${\sigma }^i$ are Pauli matrices. Their Clifford algebra follows directly from the Pauli anticommutator. The chirality matrix is ${\gamma }^5=\mathrm{diag}(-I_2,I_2)$, separating the two chiral components of a Dirac spinor.

A Hermitizing matrix for a set of gamma matrices is an invertible Hermitian matrix $H$ satisfying ${\gamma }^{a†}=H{\gamma }^a{H}^{-1}$. Equivalently, each $H{\gamma }^a$ is Hermitian. It defines a covariant adjoint $\bar{\psi }={\psi }^{†}H$. If ${\gamma }^{\prime a}=M{\gamma }^a{M}^{-1}$, then $H^{\prime }=M^{-†}H{M}^{-1}$ has the same property. In a standard Lorentzian Dirac basis, $H={\gamma }^0$ is a conventional choice.

In the standard Dirac representation of the gamma matrices, $({\gamma }^0{)}^{†}={\gamma }^0$ and $({\gamma }^i{)}^{†}=-{\gamma }^i$, so ${\gamma }^{a†}={\gamma }^0{\gamma }^a{\gamma }^0$. The transpose signs in that representation are positive for $a=0,2$ and negative for $a=1,3$. The charge-conjugation matrix expresses the basis-independent intertwining relation $C^{-1}{\gamma }^{a}C=-({\gamma }^a{)}^T$. Under an arbitrary nonunitary similarity transformation, the Hermitizing matrix changes by inverse Hermitian congruence; it need not equal the transformed temporal gamma matrix.

With metric $(+---)$, ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and ${ϵ}^{0123}=+1$, odd ordinary gamma matrix traces vanish, $\mathrm{Tr}({\gamma }^{\mu }{\gamma }^{\nu })=4g^{\mu \nu }$, andThe first follows by repeated Clifford algebra anticommutation and cyclicity of the trace. The second is an antisymmetric tensor, whose normalization is fixed by the $0123$ component.

With metric $(+---)$, the chirality matrix has $({\gamma }^5{)}^2=1$ and anticommutes with each gamma matrix. Its eigenspaces define chirality; $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ are the corresponding projectors. These identities follow directly by moving a gamma matrix through the ordered product and using the Clifford algebra.

With signature $(+,-,-,-)$, one standard representation is ${\tilde{\gamma }}^0=\mathrm{diag}(I_2,-I_2)$ and ${\tilde{\gamma }}^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$. Block multiplication and the Pauli matrix multiplication law give time-time anticommutator $2I_4$, mixed anticommutators zero, and spatial anticommutators $-2{\delta }^{ij}{I}_4$. Thus these gamma matrices represent the spacetime Clifford algebra.

For the Dirac representation of the gamma matrices, the unitary matrixsends ${\tilde{\gamma }}^0$ to $\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ while commuting with every spatial gamma matrix. This is the standard Weyl representation of the gamma matrices with spatial blocks $(0,{\sigma }^i;-{\sigma }^i,0)$. The similarity transformation preserves the Clifford algebra relations. Changing block-order or chirality conventions can change the displayed transformation, so its direction must be specified.

A Hopf algebra is a compatible algebra and coalgebra equipped with a counit and antipode. Coordinate rings of affine algebraic groups are commutative Hopf algebras.

An element $x$ of positive degree is primitive when $\Delta x=x\otimes 1+1\otimes x$. Graded commutators of primitives are primitive, so they form a graded Lie algebra. In a universal enveloping algebra over $\mathbb{Q}$, the primitive elements of a Hopf algebra are exactly its given Lie algebra: the associated graded algebra of the word-length filtration is symmetric, where no nonzero homogeneous polynomial of length greater than one is additive under the diagonal.

An algebra over a field acted on by a bialgebra is a module algebra when multiplication and unit are module maps: the displayed product identity holds and $h1=\epsilon (h)1$. This expresses the product rule determined by comultiplication.

Over a field with $q\ne 0$ and $q^2\ne 1$, this Hopf algebra has generators $E,F,K,K^{-1}$ and relations $KE=q^{2}EK$, $KF=q^{-2}FK$ and $EF-FE=(K-K^{-1})/(q-q^{-1})$. One comultiplication convention is $\Delta E=E\otimes K+1\otimes E$, $\Delta F=F\otimes 1+K^{-1}\otimes F$ and $\Delta K=K\otimes K$. Its counit takes $E,F$ to zero and $K$ to one. The antipode takes $E$ to $-E{K}^{-1}$, $F$ to $-KF$ and $K$ to $K^{-1}$.

If $Ev=0$ and $Kv=\lambda v$, induction using the defining commutator gives the displayed identity. When $q$ is not a root of unity, a finite string of length $n+1$ in a simple module has $\lambda =ϵq^n$, $ϵ=\pm 1$. Its distinct $K$ eigenvalues make the string vectors independent, and the nonzero raising coefficients prove simplicity.

The symmetric quantum integer is the displayed Laurent polynomial, with $[0{]}_q=0$. It obeys $[n+1{]}_q=q^{-1}[n{]}_q+q^n$ and $[r{]}_q[s+1{]}_q-[s{]}_q[r+1{]}_q=[r-s{]}_q$. At a primitive odd $\ell$th root of unity, $[n{]}_q=0$ exactly when $\ell$ divides $n$.

Given a Yang-Baxter R-matrix, take generators $t_{ij}$ subject to the displayed matrix relation, with $T_1=T\otimes I$, $T_2=I\otimes T$. The coproduct $\Delta t_{ij}={\sum }_k{t}_{ik}\otimes t_{kj}$ and counit $\epsilon (t_{ij})={\delta }_{ij}$ give a bialgebra. An antipode requires additional inverses or an appropriate determinant quotient; an arbitrary RTT bialgebra is not automatically a Hopf algebra.

Given an invertible braiding $C$ on $V\otimes V$, the universal bialgebra coacting on $V$ for which $C$ is a comodule map is generated by the matrix coefficients $t_{ij}$. Its comultiplication is $\Delta t_{ij}={\sum }_k{t}_{ik}\otimes t_{kj}$. Colinearity gives quadratic relations. Writing $R=\tau C$ turns the relation into the familiar RTT construction with the appropriate ordered tensor factors. A Yang-Baxter R-matrix supplies a scalar coquasitriangular structure.

For the two-by-two RTT construction with the standard Yang-Baxter R-matrix, the displayed determinant commutes with the matrix generators and satisfies $\Delta D_q=D_q\otimes D_q$ and $\epsilon (D_q)=1$. Imposing $D_q=1$ gives the coordinate Hopf algebra of quantum SL2; adjoining its inverse instead gives the quantum general-linear coordinate algebra.

The generators $a,b,c,d$ obey $ab=qba$, $ac=qca$, $bd=qdb$, $cd=qdc$, $bc=cb$, $ad-da=(q-q^{-1})bc$ and $ad-qbc=1$. Matrix multiplication gives the coproduct, and the antipode matrix is $\left(\begin{array}{cc}d& -q^{-1}b\\ -qc& a\end{array}\right)$. For $q\ne 0,\pm 1$, its quotient with $c=0$, $a=u$, $d=u^{-1}$ and $b=v$ is the quantum Laurent-plane algebra $uv=qvu$, proving noncommutativity. The same quotient distinguishes $\Delta b$ from its flipped coproduct.

A Hopf algebra is cocommutative when flipping the two tensor factors leaves its coproduct unchanged. A group algebra has coproduct $g\mapsto g\otimes g$, so it is cocommutative even for a nonabelian group, whose group algebra is not commutative.

A Hopf algebra is commutative when its multiplication is commutative. This is independent of whether its coproduct is cocommutative; function algebras of finite nonabelian groups give commutative but noncocommutative examples.

Functions on a finite group form a Hopf algebra under pointwise multiplication, with $(\Delta f)(g,h)=f(gh)$, $\epsilon (f)=f(1)$ and $(Sf)(g)=f(g^{-1})$. The coproduct is cocommutative exactly when the group is abelian, since delta functions distinguish distinct products $gh,hg$.

For a left module over a Hopf algebra, a vector $v$ is invariant when $hv=\epsilon (h)v$. In the internal hom for modules over a Hopf algebra, invariant maps are exactly the R-module homomorphisms for the acting algebra.

The full ${\mathrm{Hom}}_k(M,N)$ carries the action $(hf)(u)=\sum h_{(1)}f(S(h_{(2)})u)$, making it the internal hom right adjoint to $-\otimes M$. No finite-dimensionality or inverse antipode is needed.

The convolution inverse $S$ of the identity on a bialgebra: $m(S\otimes 1)\Delta =j\epsilon =m(1\otimes S)\Delta$. Existence makes the bialgebra a Hopf algebra; $S$ reverses multiplication.

An associative algebra is an algebra over a field whose multiplication satisfies $(ab)c=a(bc)$. Unless stated otherwise, the algebras considered here have a multiplicative identity.

A primitive ideal of a unital associative algebra is the annihilator of a module for a nonzero simple module. Every maximal proper two-sided ideal is primitive: its quotient is a simple ring, and any simple module over that quotient has zero annihilator there. In a unital Banach algebra, a simple module can be realized as $A/L$ for a maximal left ideal; this ideal is closed, making the representation a bounded algebraically irreducible action on a Banach space.

An Ore extension uses a unital endomorphism $\sigma$ of $A$ and a $\sigma$-derivation $\delta$, satisfying $\delta (ab)=\sigma (a)\delta (b)+\delta (a)b$. Its elements have unique expressions ${\sum }_i{a}_i{t}^i$ and multiplication is determined by the displayed relation. Applying the relation to $t(ab)$ shows why the endomorphism and derivation conditions are exactly the compatibility conditions for associativity. A differential Ore extension has $\sigma =\mathrm{id}$. Adjoining ordinary partial derivatives successively to a polynomial ring gives the ordered-monomial construction of a Weyl algebra.

An algebra involution on a complex associative algebra is a conjugate-linear map satisfying $(xy{)}^{\ast }=y^{\ast }{x}^{\ast }$ and $(x^{\ast }{)}^{\ast }=x$. It fixes the identity when one exists. A star algebra is an algebra equipped with such a map. These algebraic identities do not themselves impose norm continuity or the C-star identity.

An associative algebra with a two-sided multiplicative identity. For a subalgebra of bounded operators, the term unital here means that it contains the identity of the represented Hilbert space.

A complex associative algebra with a conjugate-linear involution satisfying the displayed product-reversal identities. An operator algebra closed under the Hilbert-space adjoint is a star-algebra. A C-star algebra additionally has a complete norm with $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$.

A Hermitian algebra element of a star-algebra is an element fixed by its algebra involution. The identity and zero are Hermitian algebra elements. For every $x$, both $x{x}^{\ast }$ and $x^{\ast }x$ are Hermitian algebra elements. A real spectrum for all such elements is an extra condition in a general Banach algebra, rather than a consequence of the involution identities alone.

A linear subalgebra closed under the involution. In bounded operators, this means closure under the adjoint operator; neither norm closure nor operator-topology closure is part of the definition. These distinctions matter in the Von Neumann double commutant theorem and the Kaplansky density theorem.

The cocenter of a finite-dimensional algebra is the quotient by the vector subspace spanned by additive commutators $ab-ba$; this subspace need not be an ideal. For a group algebra, its basis is indexed by conjugacy classes: every commutator of basis elements is a difference of conjugates, and every difference of conjugates is a commutator. For $M_n(k)$, off-diagonal matrix units and $E_{ii}-E_{jj}$ show that the quotient is one-dimensional, identified by the trace.

In characteristic $p>0$, the displayed map on the cocenter of a finite-dimensional algebra is additive and Frobenius-semilinear. Expand a $p$th power into words: nonconstant cyclic-word orbits have length $p$ and cancel modulo commutators. Also $(ab{)}^p$ and $(ba{)}^p$ have the same cocenter class, so the map kills additive commutators and is well defined.

Over a perfect field with split semisimple quotient, the stable image of this map has dimension equal to the number of simple modules of the algebra. Indeed, the kernel of $C(A)\to C(A/J(A))$ has representatives in the nilpotent Jacobson radical, so an iterate kills it. On the semisimple algebra quotient, each matrix-block coordinate is a field trace, and $\mathrm{tr}(a^p)=\mathrm{tr}(a{)}^p$. For a split semisimple quotient, Frobenius is bijective on every coordinate. In a group algebra over an algebraically closed field, the stable image has the p-regular element conjugacy classes as its basis, proving the simple-module counting theorem.

For a finitely generated associative algebra, choose matrices for its generators and impose its polynomial relations. This defines an affine variety. With prescribed orthogonal idempotents $1={\sum }_i{e}_i$, require their images to be the standard projections on ${⨁}_i{k}^{n_i}$. The resulting variety is ${\mathrm{Rep}}_A(\mathbf{n})$, acted on by ${\prod }_i{\mathrm{GL}}_{n_i}(k)$. Without a chosen idempotent decomposition, the usual variety is ${\mathrm{Rep}}_A(r)={\mathrm{Hom}}_{k\text{-alg}}(A,M_r(k))$. Its orbits are module isomorphism classes, and their stabilizers are automorphism groups.

The unit group $E^{\times }$ is a nonempty open subset of the affine space $E$, detected by the determinant of left multiplication. It is therefore an irreducible connected linear algebraic group. Its map to $(E/J(E){)}^{\times }$ has kernel $1+J(E)$, a closed unipotent subgroup since the radical is nilpotent.

The free associative algebra has a basis of finite words in its generators, including the empty word as identity. Multiplication concatenates words without commutativity relations. It is the path algebra of the quiver with r loops. With one generator it is a polynomial algebra.

A finite-dimensional unital associative algebra over a field admitting a linear functional $\epsilon$ whose pairing $(a,b)\mapsto \epsilon (ab)$ is nondegenerate. The dual numbers with functional extracting the nilpotent coefficient are an example; semisimplicity is unnecessary.

The same underlying module as an associative algebra, with multiplication $a{\cdot }_{\mathrm{op}}b=ba$. Right multiplications identify $H^{\mathrm{op}}$ with ${\mathrm{End}}_H(H)$ for the left regular module.

A right ideal $I$ in a unital associative algebra $A$ is a vector subspace with $IA\subseteq I$, equivalently a submodule of the right regular module. It need not be a left ideal. Its finite generation refers to right multiplication by elements of $A$.

A proper right ideal $L$ is maximal if no proper right ideal strictly contains it. Equivalently, $A/L$ is a simple module. The Jacobson radical is the intersection of all maximal right ideals.

A formal associative deformation is a unital $k[[t]]$-bilinear product continuous for the adic topology on $A[[t]]$ of the form $a\ast b=ab+{\sum }_{r\ge 1}{t}^r{\mu }_r(a,b)$. Each coefficient ${\mu }_r$ is a $k$-bilinear map $A\otimes A\to A$, and the whole product satisfies associativity. For an infinite-dimensional algebra, $A[[t]]$ is the completed tensor product ${\underleftarrow{\lim }}_{n}A\otimes k[t]/(t^n)$ and generally differs from the ordinary $A\otimes k[[t]]$.

The algebra over $k[t]$ defined by $yx-xy=t$ has ordered basis $x^i{y}^j$, by its differential Ore-extension construction. Its $t$-adic completion is a formal associative formal associative deformation of $k[x,y]$. In characteristic zero its first-order multiplication term is $f(a,b)={\partial }_{y}a{\partial }_{x}b$. The cocycle is not a coboundary: a degree-two coboundary on a commutative algebra is symmetric, whereas $f(y,x)-f(x,y)=1$.

The order-$\epsilon$ part of associativity is $af(b,c)-f(ab,c)+f(a,bc)-f(a,b)c=0$, so $f$ is a Hochschild cocycle. A change of coordinates $T=1+\epsilon g$ changes $f$ to $f+\delta g$. Therefore equivalence classes of first-order associative deformations of a fixed underlying algebra are its second Hochschild cohomology. Unital deformations use normalized cochains.

Expanding associativity for $a\ast b=ab+tf(a,b)+t^2{\mu }_2(a,b)$ modulo $t^3$ gives the displayed equation. If $\delta f=0$, its right side is a degree-three Hochschild cocycle. Its class in $H{H}^3(R,R)$ must vanish to extend the first-order formal associative deformation to second order. A nonzero class in $H{H}^2$ therefore need not integrate to a full formal associative deformation.

A formal associative deformation is trivial if a $k[[t]]$-linear automorphism continuous for the adic topology $T=\mathrm{id}(\mathrm{mod}t)$ fixing the identity transports the deformed multiplication to the original one. This is equivalence by a formal change of coordinates, stronger than merely an abstract algebra isomorphism between unspecified middle terms.

If $H{H}^2(A,A)=0$, every formal associative deformation on the adic completion of a module $A[[t]]$ is trivial. After killing lower coefficients, associativity makes the order-$r$ coefficient a Hochschild cocycle ${\mu }_r=\delta g_r$. Transport by $\mathrm{id}+t^r{g}_r$ subtracts $\delta g_r$ and kills it. The successive transformations stabilize modulo each $t^n$ and converge to an invertible change of coordinates. Completeness, and equivalence congruent to the identity modulo $t$, are essential to this argument.