Quantized Enveloping Algebras and Quantum Serre Relations

Let $R$ be a commutative ring. A bialgebra over $R$ is a unital associative $R$-algebra $(A,m,\eta )$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms) together with $R$-algebra homomorphisms $\mathrm{\Delta }:A\to A{\otimes }_{R}A$ (the coproduct) and $\epsilon :A\to R$ (the counit) such that $\mathrm{\Delta }$ is coassociative and the counit axioms hold:

$(\mathrm{\Delta }\otimes {\mathrm{id}}_A)\mathrm{\Delta }=({\mathrm{id}}_A\otimes \mathrm{\Delta })\mathrm{\Delta },(\epsilon \otimes {\mathrm{id}}_A)\mathrm{\Delta }={\mathrm{id}}_A=({\mathrm{id}}_A\otimes \epsilon )\mathrm{\Delta }\mathrm{.}$

The counit equations use the canonical identifications $R{\otimes }_{R}A\cong A\cong A{\otimes }_{R}R$. The target $A{\otimes }_{R}A$ has the $R$-algebra structure of The tensor product of $R$-algebras has multiplication $(a\otimes b)(a^{\mathrm{\prime }}\otimes b^{\mathrm{\prime }})=a{a}^{\mathrm{\prime }}\otimes b{b}^{\mathrm{\prime }}$.

A Hopf algebra over $R$ is a bialgebra $(A,m,\eta ,\mathrm{\Delta },\epsilon )$ together with an $R$-linear antipode $S:A\to A$ satisfying both convolution-inverse equations

$m(S\otimes {\mathrm{id}}_A)\mathrm{\Delta }=\eta \circ \epsilon =m({\mathrm{id}}_A\otimes S)\mathrm{\Delta },$

where $(\eta \circ \epsilon )(a)=\epsilon (a)1_A$. We do not assume that $S$ is invertible, involutive or multiplicative. Throughout, $\mathrm{\Delta }$ and $\epsilon$ are algebra homomorphisms, and the displayed order of the antipode factors is our convention.

Let $k$ be a field of characteristic $0$. A Lie bialgebra is a Lie algebra $\mathfrak{a}$ over $k$ together with a linear map ${\delta }_{\mathfrak{a}}:\mathfrak{a}\to {\mathrm{\Lambda }}^2\mathfrak{a}$ such that

$\mathrm{Alt}({\delta }_{\mathfrak{a}}\otimes \mathrm{id}){\delta }_{\mathfrak{a}}=0$

and ${\delta }_{\mathfrak{a}}([x,y])=[x,{\delta }_{\mathfrak{a}}(y)]-[y,{\delta }_{\mathfrak{a}}(x)]$ for all $x,y\in \mathfrak{a}$. The bracket action on ${\mathrm{\Lambda }}^2\mathfrak{a}$ is $[x,u\wedge v]=[x,u]\wedge v+u\wedge [x,v]$.

Let $Q$ be an additive abelian group. Suppose $\mathfrak{a}={⨁}_{\alpha \in Q}{\mathfrak{a}}_{\alpha }$ and $\mathfrak{c}={⨁}_{\alpha \in Q}{\mathfrak{c}}_{\alpha }$ are $Q$-graded vector spaces with finite-dimensional graded pieces, and suppose that, within each space separately, only finitely many pairs of nonzero graded pieces have degrees summing to any fixed degree. A pairing between them is degreewise perfect if ${\mathfrak{a}}_{\alpha }$ pairs perfectly with ${\mathfrak{c}}_{-\alpha }$ and all other degree pairs are orthogonal. It then identifies $\mathfrak{c}$ with the restricted graded dual ${\mathfrak{a}}^{\mathrm{g}\mathrm{r}\mathrm{\prime }}:={⨁}_{\alpha \in Q}{\mathfrak{a}}_{\alpha }^{\ast }$. The induced pairing on exterior powers is the determinant pairing: $⟨x_1\wedge \cdots \wedge x_m,y_1\wedge \cdots \wedge y_m⟩=\det (⟨x_i,y_j⟩{)}_{i,j=1}^m$.

For the Lie bialgebra duality below, require both brackets to preserve degree: $[{\mathfrak{a}}_{\alpha },{\mathfrak{a}}_{\beta }]\subseteq {\mathfrak{a}}_{\alpha +\beta }$ and $[{\mathfrak{c}}_{\alpha },{\mathfrak{c}}_{\beta }]\subseteq {\mathfrak{c}}_{\alpha +\beta }$. Two such Lie bialgebras are dual when these pairings are degreewise perfect and their cobrackets are transposes of the opposite brackets: $⟨{\delta }_{\mathfrak{a}}(x),y\wedge z⟩=⟨x,[y,z]⟩$ and $⟨x\wedge x^{\mathrm{\prime }},{\delta }_{\mathfrak{c}}(y)⟩=⟨[x,x^{\mathrm{\prime }}],y⟩$. For $x\in {\mathfrak{a}}_{\gamma }$, degree preservation and orthogonality make $⟨x,[{\mathfrak{c}}_{\alpha },{\mathfrak{c}}_{\beta }]⟩$ vanish unless $\alpha +\beta =-\gamma$. There are only finitely many such nonzero pairs, and their finite-dimensional perfect pairings identify the transpose with a unique element of $({\mathrm{\Lambda }}^2\mathfrak{a}{)}_{\gamma }$ under the determinant pairing. The same argument applies with the two algebras exchanged, and linear extension handles arbitrary elements. Thus the transposes lie in the ordinary exterior squares rather than formal infinite sums.

A root-graded Manin triple is a $Q$-graded Lie algebra $\mathfrak{d}={⨁}_{\alpha \in Q}{\mathfrak{d}}_{\alpha }$ with finite-dimensional graded pieces, a symmetric invariant bilinear form $B$ pairing ${\mathfrak{d}}_{\alpha }$ perfectly with ${\mathfrak{d}}_{-\alpha }$ and satisfying $B({\mathfrak{d}}_{\alpha },{\mathfrak{d}}_{\beta })=0$ whenever $\alpha +\beta \mathrm{\ne }0$, and graded Lie subalgebras ${\mathfrak{d}}_{+}$ and ${\mathfrak{d}}_{-}$ such that $\mathfrak{d}={\mathfrak{d}}_{+}\oplus {\mathfrak{d}}_{-}$ as a vector space and $B({\mathfrak{d}}_{+},{\mathfrak{d}}_{+})=B({\mathfrak{d}}_{-},{\mathfrak{d}}_{-})=0$. The cross pairing is degreewise perfect: a vector in $({\mathfrak{d}}_{+}{)}_{\alpha }$ annihilating $({\mathfrak{d}}_{-}{)}_{-\alpha }$ also annihilates $({\mathfrak{d}}_{+}{)}_{-\alpha }$ by isotropy, so it is zero by perfectness on the double. The same argument applies with the two halves exchanged, and finite dimensionality gives perfectness of the cross pairing. All other degree pairs are orthogonal by the condition on $B$. Require finite degree decompositions within each of ${\mathfrak{d}}_{+}$ and ${\mathfrak{d}}_{-}$ separately, as above; a triple satisfying this requirement is called locally finite. No finite-decomposition condition is imposed on the whole double. This permits opposite Borels with infinitely many roots, since each Borel has support in one root cone. The finite-dimensional Manin-triple definition is the special case with finitely many nonzero graded pieces.

A symmetrizable Cartan datum for a quantum group consists of a finite nonempty index set $I=\{1,\dots ,n\}$, a symmetrizable generalized Cartan matrix $A=(a_{ij}{)}_{i,j\in I}$ (Generalized cartan matrix, Symmetrizable generalized cartan matrix) and a chosen diagonal symmetrizer $D=\mathrm{diag}(d_1,\dots ,d_n)$ with $d_i\in {\mathbb{Z}}_{>0}$ and $d_i{a}_{ij}=d_j{a}_{ji}$ for all $i,j$; free abelian groups $P^{\vee }$ and $P$ (Free abelian group on a set) with an integer-valued bilinear pairing $⟨\cdot ,\cdot ⟩:P\times P^{\vee }\to \mathbb{Z}$; simple coroots $h_i\in P^{\vee }$ that are linearly independent in $P^{\vee }{\otimes }_{\mathbb{Z}}\mathbb{Q}$; and simple roots ${\alpha }_i\in P$ that freely generate the root lattice $Q={⨁}_{i\in I}\mathbb{Z}{\alpha }_i\subseteq P$ (Kac Moody root lattice height and positive cone) and satisfy $⟨{\alpha }_j,h_i⟩=a_{ij}$ for all $i,j$ (Realization of a generalized cartan matrix, Minimal realizations exist and are unique up to isomorphism).

The index set is finite and $A$ need not be nonsingular. We do not require the simple coroots to span $P^{\vee }{\otimes }_{\mathbb{Z}}\mathbb{Q}$, and the pairing is not required to be perfect; choosing $P$ and $P^{\vee }$ is part of the datum, not something determined by the matrix alone.

We fix an indeterminate $q$ over $\mathbb{Q}$, work over $\mathbb{Q}(q)$, and set $q_i=q^{d_i}$. For $h\in P^{\vee }$, $⟨{\alpha }_i,h⟩\in \mathbb{Z}$ is the exponent in $q^{⟨{\alpha }_i,h⟩}$. We use the row convention $⟨{\alpha }_j,h_i⟩=a_{ij}$ throughout. No fundamental weights with prescribed values on all $h_i$ are part of this definition.

Let $n\ge 1$ and let $B=(b_{ij})$ be a real symmetric $n\times n$ matrix of rank $r$. For $J\subseteq \{1,\dots ,n\}$, write $B_J=(b_{ij}{)}_{i,j\in J}$ for its principal submatrix. Use the convention that the empty matrix has determinant $1$ and is invertible. There is a subset $J$ with $\mathrm{\mid }J\mathrm{\mid }=r$ for which $B_J$ is nonsingular.

If in addition $A$ is a real $n\times n$ matrix and $D=\mathrm{diag}(d_1,\dots ,d_n)$ has $d_i>0$ with $B=DA$, then for the same $J$,

$\det B_J=\left({\prod }_{i\in J}{d}_i\right)\det A_J\mathrm{.}$

Thus $A_J$ is also nonsingular and has size $r$, and $\mathrm{\mid }\{1,\dots ,n\}\setminus J\mathrm{\mid }=n-r$. In particular, if $A$ is symmetrizable and has corank one, then $B=DA$ has corank one and the conclusion gives a nonsingular $(n-1)\times (n-1)$ principal submatrix.

Symmetry is essential: $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ has rank $1$ but no nonsingular one-by-one principal submatrix. Berkeley's Gabber–Kac example in Ch. 10 §10.4.2.6 assumes the positive-semidefinite corank-one case; the principal-minor argument above needs only symmetry and therefore also applies to indefinite symmetrizable matrices.

Let $(I,A,D,P,P^{\vee },q)$ be a symmetrizable Cartan datum for a quantum group (Symmetrizable Cartan data for quantum groups) and put $q_i=q^{d_i}$. For $m\in \mathbb{N}$, define the $q_i$-integer

$[m{]}_i:=\frac{q_i^m-q_i^{-m}}{q_i-q_i^{-1}},[0{]}_i:=0,$

and the $q_i$-factorial

$[m{]}_i!:={\prod }_{k=1}^m[k{]}_i,[0{]}_i!:=1.$

For $0\le r\le m$, define the Gaussian binomial ${\left(\genfrac{}{}{0px}{}{m}{r}\right)}_i:=[m{]}_i!\mathrm{/}([r{]}_i![m-r{]}_i!)$, and set it to $0$ when $r<0$ or $r>m$. In the published one-parameter convention, write ${\left(\genfrac{}{}{0px}{}{m}{r}\right)}_t:={\left(\genfrac{}{}{0px}{}{m}{r,m-r}\right)}_t$ for the two-part $q$-multinomial coefficient of The $q$-integer, $q$-factorial and $q$-multinomial coefficients.

For an element $x$ of a unital $\mathbb{Q}(q)$-algebra and $m\ge 0$, define its divided power at $q_i$ by $x^{(m)}:=x^m\mathrm{/}[m{]}_i!$, so $x^{(0)}=1$ and $x^{(1)}=x$.

The symmetric convention and the published asymmetric convention are related, for $m\ge 0$ and $0\le r\le m$, by

$[m{]}_i=q_i^{-(m-1)}[m{]}_{q_i^2},{\left(\genfrac{}{}{0px}{}{m}{r}\right)}_i=q_i^{-r(m-r)}{\left(\genfrac{}{}{0px}{}{m}{r}\right)}_{q_i^2}\mathrm{.}$

In particular $[m{]}_i$ is invariant under $q_i\mapsto q_i^{-1}$. These quantities depend on the symmetrizer only through $q_i=q^{d_i}$.

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field of characteristic $0$ (Field, The characteristic of a ring: the least $n\ge 1$ with $n\cdot 1_R=0$ when one exists, and $0$ otherwise) and let $\mathfrak{l}$ be a Lie algebra over $k$ (Lie algebras over a field). Equip its universal enveloping algebra $U(\mathfrak{l})$ with the standard cocommutative Hopf structure (Hopf-algebra structure on U(g)), with coproduct $\mathrm{\Delta }(x)=x\otimes 1+1\otimes x$ for $x\in \mathfrak{l}$ and counit $\epsilon (x)=0$. If $J⊴U(\mathfrak{l})$ is a two-sided ideal such that

$\mathrm{\Delta }(J)\subseteq J\otimes U(\mathfrak{l})+U(\mathfrak{l})\otimes J,J\subseteq \ker \epsilon ,$

then $\mathfrak{j}:=J\cap \mathfrak{l}$ is a Lie ideal and $J=U(\mathfrak{l})\mathfrak{j}=\mathfrak{j}U(\mathfrak{l})$. Thus $J$ is uniquely determined by $\mathfrak{j}$ among two-sided ideals satisfying both displayed conditions.

• The zero Lie algebra is included: then $U(0)=k$, $\ker \epsilon =0$, and the only admissible ideal is $J=0$.
• The augmentation hypothesis is necessary. For nonzero $\mathfrak{l}$, the ideal $J=U(\mathfrak{l})$ satisfies the coproduct containment, but $J\cap \mathfrak{l}=\mathfrak{l}$ while $U(\mathfrak{l})(J\cap \mathfrak{l})=\ker \epsilon \mathrm{\ne }U(\mathfrak{l})$.
• The unrestricted “largest ideal with fixed primitive part” claim is false: for $\mathfrak{l}=kx$, $U(\mathfrak{l})=k[x]$, the ideals $0$ and $(x^2)$ have the same intersection $0$ with $\mathfrak{l}$, and $0⊊(x^2)$.
• AC is used to choose and well-order a basis of arbitrary $\mathfrak{l}$, as required by the supplied general PBW theorem, and to split the induced filtration of $J$ in step 1.3. These are the only nonconstructive choices in this proof; the characteristic-zero use is exactly the invertibility of $n$ in step 2.1. No assertion is made in positive characteristic.

Let $(A,m,\eta ,\mathrm{\Delta },\epsilon )$ be a bialgebra over a commutative ring $R$ (Bialgebras, counits and antipodes over a commutative ring) and let $S,S^{\mathrm{\prime }}:A\to A$ be antipodes. Then $S=S^{\mathrm{\prime }}$. Thus a bialgebra admits at most one Hopf algebra structure with its fixed multiplication, unit, coproduct and counit. Every antipode also satisfies $S(1_A)=1_A$.

Let $k$ be a field of characteristic $0$, and let $(\mathfrak{d},{\mathfrak{d}}_{+},{\mathfrak{d}}_{-};B)$ be a locally finite root-graded Manin triple in the sense of Lie bialgebras, degreewise duality, and root-graded Manin triples. Define ${\delta }_{+}:{\mathfrak{d}}_{+}\to {\mathrm{\Lambda }}^2{\mathfrak{d}}_{+}$ and ${\delta }_{-}:{\mathfrak{d}}_{-}\to {\mathrm{\Lambda }}^2{\mathfrak{d}}_{-}$ by transposing the opposite brackets under the degreewise perfect pairing induced by $B$:

$⟨{\delta }_{+}(x),y\wedge z⟩=B(x,[y,z]),⟨u\wedge v,{\delta }_{-}(y)⟩=B([u,v],y)\mathrm{.}$

Then these maps are well defined and make ${\mathfrak{d}}_{+}$ and ${\mathfrak{d}}_{-}$ dual Lie bialgebras.

Let $(I,A,D,P,P^{\vee },q)$ be a symmetrizable Cartan datum for a quantum group (Symmetrizable Cartan data for quantum groups) and put $q_i=q^{d_i}$. Write $K_i:=K_{d_i{h}_i}$ and $K_i^{-1}:=K_{-d_i{h}_i}$. Let $V$ be the $\mathbb{Q}(q)$-vector space with basis the symbols $\{E_i,F_i{\}}_{i\in I}\cup \{K_h{\}}_{h\in P^{\vee }}$, and let $T:=T(V)$ be its free unital associative $\mathbb{Q}(q)$-algebra (Tensor algebra of a vector space, Universal property of the tensor algebra). Let $I_{\mathrm{D}\mathrm{J}}$ be the two-sided ideal of $T$ generated by (The ideal generated by a subset and principal ideals) the following relations:

$K_0-1,K_h{K}_{h^{\mathrm{\prime }}}-K_{h+h^{\mathrm{\prime }}}(h,h^{\mathrm{\prime }}\in P^{\vee });$

$K_h{E}_i-q^{⟨{\alpha }_i,h⟩}{E}_i{K}_h,K_h{F}_i-q^{-⟨{\alpha }_i,h⟩}{F}_i{K}_h(h\in P^{\vee },i\in I);$

$E_i{F}_j-F_j{E}_i-{\delta }_{ij}\frac{K_i-K_i^{-1}}{q_i-q_i^{-1}}(i,j\in I);$

${\mathrm{S}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{e}}_{ij}^{+}:={\sum }_{r=0}^{1-a_{ij}}(-1{)}^r{\left(\genfrac{}{}{0px}{}{1-a_{ij}}{r}\right)}_i{E}_i^{1-a_{ij}-r}{E}_j{E}_i^r,{\mathrm{S}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{e}}_{ij}^{-}:={\sum }_{r=0}^{1-a_{ij}}(-1{)}^r{\left(\genfrac{}{}{0px}{}{1-a_{ij}}{r}\right)}_i{F}_i^{1-a_{ij}-r}{F}_j{F}_i^r(i\mathrm{\ne }j),$

where the Gaussian binomials are those of Quantum integers, factorials, Gaussian binomials and divided powers at $q_i$. The Drinfeld-Jimbo quantized enveloping algebra is

$U_q(\mathfrak{g}):=T\mathrm{/}{I}_{\mathrm{D}\mathrm{J}}$

(The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$), with $E_i,F_i,K_h$ denoting the images of the generators. In this quotient, $K_i^{-1}$ is the two-sided inverse of $K_i$.

It is graded by the root lattice $Q={⨁}_{i\in I}\mathbb{Z}{\alpha }_i$ (Kac Moody root lattice height and positive cone), with $\mathrm{deg}{E}_i={\alpha }_i$, $\mathrm{deg}{F}_i=-{\alpha }_i$ and $\mathrm{deg}{K}_h=0$; thus $U_q(\mathfrak{g})={⨁}_{\beta \in Q}{U}_q(\mathfrak{g})[\beta ]$.

For every unital associative $\mathbb{Q}(q)$-algebra $B$, any assignment of elements $E_i^{\mathrm{\prime }},F_i^{\mathrm{\prime }},K_h^{\mathrm{\prime }}\in B$ satisfying these relations extends uniquely to a $\mathbb{Q}(q)$-algebra homomorphism $U_q(\mathfrak{g})\to B$ (A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring).