symbols.tex

% \newsymbol{label}{description}{symbol}

\newsymbol{lie bracket}
{Lie bracket}
{$[\ph, \ph]$}

\newsymbol{lie algebra}
{Lie algebras}
{$\glie, \hlie, \dotsc$}

\newsymbol{general linear lie algebra matrix}
{general linear Lie~algebra of~$n \times n$ matrices}
{$\gllie(n, \kf)$}

\newsymbol{general linear lie algebra endomorphism}
{general linear Lie~algebra of~$V$}
{$\gllie(V)$}

\newsymbol{heisenberg lie algebra}
{Heisenberg Lie~algebra}
{$\heisenberglie(n, \kf)$}

\newsymbol{heisenberg basis}
{basis of Heisenberg Lie~algebra}
{$p_1, \dotsc, p_n, q_1, \dotsc, q_n, c$}

\newsymbol{standard basis matrix}
{standard basis matrix of~$\Mat(n \times m, \kf)$}
{$E_{ij}$}

\newsymbol{standard basis vector}
{standard basis vector of~$\kf^n$}
{$e_i$}

\newsymbol{special linear lie algebra matrices}
{special linear Lie~algebra of~$n \times n$ matrices}
{$\sllie(n, \kf)$}

\newsymbol{special linear lie algebra endomorphism}
{special linear Lie~algebra of~$V$}
{$\sllie(V)$}

\newsymbol{upper triangular lie algebra}
{Lie~algebra of upper triangular matrices of size~$n$}
{$\trianglie(n, \kf)$}

\newsymbol{strictly upper triangular lie algebra}
{Lie~algebra of strictly upper triangular matrices of size~$n$}
{$\upperlie(n, \kf)$}

\newsymbol{diagonal lie algebra}
{Lie~algebra of diagonal matrices of size~$nn$}
{$\diaglie(n, \kf)$}

\newsymbol{affine transformation lie algebra}
{Lie~algebra of affine transformations of~$\kf^n$}
{$\afflie(n, \kf)$}

\newsymbol{commutator space}
{commutator space of two subsets~$X$,~$Y$}
{$[X, Y]$}

\newsymbol{centralizer of subset}
{centralizer of subset~$X$}
{$\centerlie_{\glie}(X)$}

\newsymbol{centralizer of element}
{centralizer of an element~$x$}
{$\centerlie_{\glie}(x)$}

\newsymbol{center}
{center of~$\glie$}
{$\centerlie(\glie)$}

\newsymbol{standard basis of sl2}
{standard basis of~$\sllie(2, \kf)$}
{$e$,~$h$,~$f$}

\newsymbol{normalizer}
{normalizer of a linear subspace~$U$}
{$\normallie_{\glie}(U)$}

\newsymbol{product of lie algebras}
{product of the Lie~algebras~$\glie_\lambda$}
{$\prod_{\lambda \in \Lambda} \glie$}

\newsymbol{external direct sum of lie algebras}
{(external) direct sum of the Lie~algebras~$\glie_\lambda$}
{$\bigoplus_{\lambda \in \Lambda} \glie_\lambda$}

\newsymbol{quotient of lie algebra}
{quotient of~$\glie$ by its ideal~$I$}
{$\glie/I$}

\newsymbol{abelianization of lie algebra}
{abelianization of~$\glie$}
{$\glie^{\ab}$}

\newsymbol{extension of scalars of lie algebra}
{extension of scalars of~$\glie$ from~$\kf$ to~$A$}
{$A \tensor_{\kf} \glie$,~$\glie \tensor_{\kf} A$}

\newsymbol{loop lie algebra}
{loop Lie~algebra of~$\glie$}
{$\looplie(\glie)$}

\newsymbol{opposite algebra}
{opposite algebra of~$A$}
{$A^{\op}$}

\newsymbol{opposite element}
{element of~$A^\op$ corresponding to an element~$a$ of~$A$}
{$a^{\op}$}

\newsymbol{opposite lie algebra}
{opposite Lie~algebra of~$\glie$}
{$\glie^{\op}$}

\newsymbol{adjoint map}
{homomorphism of the adjoint representation}
{$\ad$}

\newsymbol{category lie algebras}
{category of~\liealgebras{$\kf$}}
{$\cLie{\kf}$}

\newsymbol{category algebras}
{category of~\algebras{$\kf$}}
{$\cAlg{\kf}$}

\newsymbol{category abelian lie algebras}
{category of abelian~\liealgebras{$\kf$}}
{$\cLieab{\kf}$}

\newsymbol{derivations}
{Lie~algebra of derivations of the \enquote{algebra}~$A$}
{$\Der(A)$}

\newsymbol{inner derivations}
{space of inner derivations of~$\glie$}
{$\InnDer(\glie)$}

\newsymbol{outer derivations}
{space of outer derivations of~$\glie$}
{$\OutDer(\glie)$}

\newsymbol{exponential}
{exponential of~$f$}
{$\exp(f)$}

\newsymbol{automorphisms}
{group of Lie~algebra homomorphisms of~$\glie$}
{$\Aut(\glie)$}

\newsymbol{inner automorphisms}
{group of inner autorphisms of~$\glie$}
{$\InnAut(\glie)$}

\newsymbol{outer automorphisms}
{group of outer derivations of~$\glie$}
{$\OutAut(\glie)$}

\newsymbol{action on an element}
{action of~$x$ on~$m$}
{$x \act m$}

\newsymbol{product of representations}
{product of the representations~$M_\lambda$}
{$\prod_{\lambda \in \Lambda} M_\lambda$}

\newsymbol{direct sum of representations}
{direct sum of the representations~$M_\lambda$}
{$\bigoplus_{\lambda \in \Lambda} M_\lambda$}

\newsymbol{tensor product of representations}
{tensor product of representations~$M_1, \dotsc, M_r$}
{$M_1 \tensor \dotsb \tensor M_r$}

\newsymbol{internal hom representation}
{$\Hom$-representation of~$M$ and~$N$}
{$\Hom_{\kf}(M, N)$}

\newsymbol{dual space representation}
{dual representation of~$M$}
{$M^*$}

\newsymbol{certain subrepresentation}
{the subrepresentation of~$M$ spanned by all~$x \act m$}
{$\glie M$}

\newsymbol{invariants of representation}
{space of invariants of~$M$}
{$M^{\glie}$}

\newsymbol{quotient representation}
{quotient representation of~$M$ by~$N$}
{$M/N$}

\newsymbol{exterior power}
{\howmanyth{$d$} exterior power of~$V$}
{$\Exterior^d(V)$}

\newsymbol{symmetric power}
{\howmanyth{$d$} symmetric power of~$V$}
{$\Symm^d(V)$}

\newsymbol{hom of representations}
{space of homomorphisms of representations from~$M$ to~$N$}
{$\Hom_{\glie}(M,N)$}

\newsymbol{end of representations}
{algebra of endomorphisms of representations of~$M$}
{$\End_{\glie}(M)$}

\newsymbol{category lie algebra representation}
{category of representations of~$\glie$}
{$\cRep{\glie}$}

\newsymbol{category lie algebra trivial representations}
{category of trivial representations of~$\glie$}
{$\cReptriv{\glie}$}

\newsymbol{abelian extensions}
{class of abelian extensions of~$\glie$}
{$\AbEx(\glie)$}

\newsymbol{abelian extensions by h}
{class of abelian extensions of~$\glie$ by~$\hlie$}
{$\AbEx(\glie, \hlie)$}

\newsymbol{abelian extensions by h via theta}
{class of abelian extensions of~$\glie$ by~$\hlie$ via~$\theta$}
{$\AbEx(\glie, \hlie, \theta)$}

\newsymbol{abelian extensions associated to M}
{class of abelian extensions of~$\glie$ associated to the representation~$M$}
{$\AbEx(\glie, M)$}

\newsymbol{internal semidirect product}
{internal semidirect product of~$\glie$ by~$\hlie$}
{$\glie \ltimes \hlie$}

\newsymbol{external semidirect product}
{external semidirect product of~$\glie$ by~$\hlie$ along~$\theta$}
{$\glie \ltimes_\theta \hlie$}

\newsymbol{cyclic sum}
{abbrev. for the cyclic sum~$F(a,b,c) + F(b,c,a) + F(c,a,b)$}
{$\sum_{\cyc} F(a,b,c)$}

\newsymbol{chain complex}
{chain complex (of vector spaces)}
{
	\begin{tabular}[t]{@{}l@{}}
		$((X_n)_{n \in \Integer}, (d_n)_{n \in \Integer})$, \\
		$(X_\bullet, d_\bullet)$,\,~$X_\bullet$
	\end{tabular}
}
\newsymbol{cochain complex}
{cochain complex (of vector spaces)}
{
	\begin{tabular}[t]{@{}l@{}}
		$((X^n)_{n \in \Integer}, (d^n)_{n \in \Integer})$, \\
		$(X^\bullet, d^\bullet)$,\,~$X^\bullet$
	\end{tabular}
}

\newsymbol{differential}
{differential of a (co)chain complex}
{$d$}

\newsymbol{alternating maps}
{space of alternating maps from~$V \times \dotsb \times V$ to~$W$}
{$\Alt^n(V,W)$}

\newsymbol{lie algebra chain complex}
{Lie algebra chain complex of~$\glie$ with coefficients in~$M$}
{$\Chain_\bullet(\glie, M)$}

\newsymbol{lie algebra cochain complex}
{Lie algebra cochain complex of~$\glie$ with coefficients in~$M$}
{$\Chain^\bullet(\glie, M)$}

\newsymbol{cycles}
{space of~\cycles{$n$} of~$X_\bullet$}
{$\Cycle_n(X_\bullet)$}

\newsymbol{boundaries}
{space of~\boundaries{$n$} of~$X_\bullet$}
{$\Boundary_n(X_\bullet)$}

\newsymbol{homology}
{\howmanyth{$n$} homology of~$X_\bullet$}
{$\Homology_n(X_\bullet)$}

\newsymbol{cocycles}
{space of~\cocycles{$n$} of~$X^\bullet$}
{$\Cycle^n(X^\bullet)$}

\newsymbol{coboundaries}
{space of~\coboundaries{$n$} of~$X^\bullet$}
{$\Boundary^n(X^\bullet)$}

\newsymbol{cohomology}
{\howmanyth{$n$} cohomology of~$X^\bullet$}
{$\Homology^n(X^\bullet)$}

\newsymbol{lie algebra homology}
{\howmanyth{$n$} Lie~algebra homology of~$\glie$ with coefficients in~$M$}
{$\Homology_n(\glie, M)$}

\newsymbol{lie algebra cohomology}
{\howmanyth{$n$} Lie~algebra cohomology of~$\glie$ with coefficients in~$M$}
{$\Homology^n(\glie, M)$}

\newsymbol{homomorphism of chain complexes}
{homomorphism of chain complexes}
{$f_\bullet, g_\bullet, \dotsc$}

\newsymbol{induced map on homology}
{map induced on the~\howmanyth{$n$} homology by~$f_\bullet$}
{$\Homology_n(f_\bullet)$}

\newsymbol{homomorphism of cochain complexes}
{homomorphism of cochain complexes}
{$f^\bullet, g^\bullet, \dotsc$}

\newsymbol{induced map on cohomology}
{map induced on the~\howmanyth{$n$} cohomology by~$f^\bullet$}
{$\Homology^n(f^\bullet)$}

\newsymbol{homomorphism of lie algebra chain complex}
{homomorphism of Lie algebra chain complexes induced by~$f$}
{$\Chain_\bullet(f)$}

\newsymbol{homomorphism of lie algebra homology}
{map induced on the~\howmanyth{$n$} Lie~algebra homology by~$f$}
{$\Homology_n(f)$}

\newsymbol{homomorphism of lie algebra cochain complex}
{homomorphism of Lie algebra chain cocomplexes induced by~$f$}
{$\Chain^\bullet(f)$}

\newsymbol{homomorphism of lie algebra cohomology}
{map induced on the~\howmanyth{$n$} Lie~algebra cohomology by~$f$}
{$\Homology^n(f)$}

\newsymbol{chain complex image}
{image of the homomorphism of chain complexes~$f_\bullet$}
{$\im(f_\bullet)$}

\newsymbol{chain complex kernel}
{kernel of the homomorphism of chain complexes~$f_\bullet$}
{$\ker(f_\bullet)$}

\newsymbol{cochain complex image}
{image of the homomorphism of cochain complexes~$f^\bullet$}
{$\im(f^\bullet)$}

\newsymbol{cochain complex kernel}
{kernel of the homomorphism of cochain complexes~$f^\bullet$}
{$\ker(f^\bullet)$}

\newsymbol{coinvariants}
{space of coinvariants of a~\representation{$\glie$}~$M$}
{$M_{\glie}$}

\newsymbol{derivations of module}
{space of derivations of a~\representation{$\glie$}~$M$}
{$\Der(\glie, M)$}

\newsymbol{inner derivations of module}
{space of inner derivations of a~\representation{$\glie$}~$M$}
{$\InnDer(\glie, M)$}

\newsymbol{outer derivations of module}
{space of outer derivations of a~\representation{$\glie$}~$M$}
{$\OutDer(\glie, M)$}

\newsymbol{tensor product of chain complex and module}
{image of~$X_\bullet$ under~$(\ph) \tensor Y$}
{$X_\bullet \tensor Y$}

\newsymbol{category modules}
{category of left~\modules{$A$}}
{$\cMod{A}$}

\newsymbol{simple tensor}
{alternative notation for the simple tensor~$v_1 \tensor \dotsb \tensor v_n$}
{$(v_1, \dotsc, v_n)$}

\newsymbol{tensor power}
{\howmanyth{$d$} tensor power of~$V$}
{$V^{\tensor d}$}

\newsymbol{tensor algebra}
{tensor algebra of~$V$}
{$\Tensor(V)$}

\newsymbol{tensor algebra on morphisms}
{extension of~$f$ to a homomorphism between tensor algebras}
{$\Tensor(f)$}

\newsymbol{noncommutative polynomial algebra}
{noncommutative polynomial in the variables~$X_i$}
{$\kf\gen{ X_i \suchthat i \in I}$}

\newsymbol{simple symmetric tensor}
{symple symmetric tensor of~$v_1, \dotsc, v_d$ in~$\Symm^d(V)$}
{$v_1 \dotsm v_d$}

\newsymbol{symmetric algebra}
{symmetric algebra of~$V$}
{$\Symm(V)$}

\newsymbol{symmetric algebra on morphisms}
{extension of~$f$ to a homomorphism between symmetric algebras}
{$\Symm(f)$}

\newsymbol{category commutative algebras}
{category of commutative~\algebras{$\kf$}}
{$\cCAlg{\kf}$}

\newsymbol{exterior algebra}
{exterior algebra of~$V$}
{$\Exterior(V)$}

\newsymbol{universal enveloping algebra}
{universal enveloping algebra of~$\glie$}
{$\Univ(\glie)$}

\newsymbol{element in universal enveloping algebra}
{image of~$x$ in the universal enveloping algebra}
{$\class{x}$}

\newsymbol{universal enveloping algebra on morphisms}
{extension of~$\varphi$ to an (anti-)homomorphism between universal enveloping algebras}
{$\Univ(\varphi)$}

\newsymbol{antipode}
{antipode of~$\Univ(\glie)$}
{$S$}

\newsymbol{augumentation}
{augumentation of a~\algebra{$\kf$}}
{$\varepsilon$}

\newsymbol{augumented algebra}
{augumented~\algebra{$\kf$}}
{$(A, \varepsilon)$}

\newsymbol{counit}
{counit of~$\Univ(\glie)$}
{$\varepsilon$}

\newsymbol{comultiplication}
{comultiplication of~$\Univ(\glie)$}
{$\Delta$}

\newsymbol{free lie algebra set}
{free Lie~algebra on the set~$X$}
{$\freelieset(X)$}

\newsymbol{free lie algebra vector space}
{free Lie~algebra on the vector space~$V$}
{$\freelievect(V)$}

\newsymbol{free lie algebra set on morphisms}
{induced homomorphism between free Lie~algebras}
{$\freelieset(f)$}

\newsymbol{free lie algebra vector space on morphisms}
{induced homomorphism between free Lie~algebras}
{$\freelievect(f)$}

\newsymbol{homogeneous part}
{\howmanyth{$p$} homogeneous part of the graded algebra~$A$}
{$A_p$}

\newsymbol{homogeneous summand}
{\howmanyth{$p$} homogeneous summand of the element~$x$}
{$x_p$}

\newsymbol{restricted homomorphism of graded algebras}
{\howmanyth{$p$} component of a homomorphism of graded algebras~$\Phi$}
{$\Phi_p$}

\newsymbol{category graded algebras}
{category of graded~\algebras{$\kf$}}
{$\cgAlg{\kf}$}

\newsymbol{filtered part}
{\howmanyth{$p$} term of the filtration of~$A$}
{$A_{(p)}$}

\newsymbol{restricted homomorphism of filtered algebras}
{\howmanyth{$p$} component of a homomorphism of filtered algebras~$\Phi$}
{$\Phi_{(p)}$}

\newsymbol{category filtered algebras}
{category of filtered~\algebras{$\kf$}}
{$\cfAlg{\kf}$}

\newsymbol{length function}
{length function of~$G$ with respect to~$S$}
{$\ell_S$}

\newsymbol{degree in filtered algebra}
{degree of an element~$x$ of a filtered~\algebra{$\kf$}~$A$}
{$\deg(x)$}

\newsymbol{graded residue class}
{residue class of~$x$ in~$\gr[p](A)$}
{$\fclass{x}_p$}

\newsymbol{homogeneous part of associated graded algebra}
{\howmanyth{$p$} homogeneous part of the associated graded algebra of~$A$}
{$\gr[p]{A}$}

\newsymbol{associated graded algebra}
{associated graded algebra of a filtered algebra~$A$}
{$\gr(A)$}

\newsymbol{canonical map into associated graded algebra}
{canonical map from~$A$ to~$\gr(A)$ for a filtered algebra~$A$}
{$\gamma$}

\newsymbol{associated graded algebra on morphisms}
{induced homomorphism between associated graded algebras}
{$\gr(\Phi)$}

\newsymbol{induced homogeneous ideal}
{homogeneous ideal induced by~$I$}
{$\gr(I)$}

\newsymbol{skew polynomial algebra}
{skew polynomial algebra}
{$R[t; \delta]$}

\newsymbol{mod elements}
{$v_1$ and~$v_2$ are equal up to an element of~$W$}
{$v_1 \equiv v_2 \pmod{W}$}

\newsymbol{ordered n tuples}
{set of ordered~\tuples{$n$} with entries in~$I$}
{$I^n$}

\newsymbol{ordered leq n tuples}
{set of ordered tuples with entries in~$I$ of length~$\leq n$}
{$I^{(n)}$}

\newsymbol{ordered tuples}
{set of all ordered tuples with entries in~$I$}
{$I^*$}

\newsymbol{ordered monomial}
{ordered monomial associated to~$\alpha \in I^*$}
{$x_\alpha$}

\newsymbol{ordered concatination}
{ordered concatination of the tuples~$\alpha$ and~$\beta$}
{$\alpha \cdot \beta$}

\newsymbol{element leq tuple}
{$i$ is less or equal to the first entry of~$\alpha$}
{$i \leq \alpha$}

\newsymbol{quotient chain complex}
{quotient chain complex}
{$X_\bullet / Y_\bullet$}

\newsymbol{directed union of chain complexes}
{directed union of the chain complexes~$X_\bullet^{(i)}$}
{$\bigcup_{i \in I} X_\bullet^{(i)}$}

\newsymbol{direct sum of chain complexes}
{direct sum of chain complexes}
{$\bigoplus_{i \in I} X_\bullet^{(i)}$}

\newsymbol{right derived functor}
{right derived functor(s) of~$F$}
{$\Right^\bullet F$}