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The Drinfeld-Jimbo quantized enveloping algebra by generators and relations

Definition

Let $(I, A, D, P, P^{\lor}, q)$ 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 $Q (q)$ -vector space with basis the symbols ${E_{i}, F_{i}}_{i \in I} \cup {K_{h}}_{h \in P^{\lor}}$ , and let $T : = T (V)$ be its free unital associative $Q (q)$ -algebra (Tensor algebra of a vector space, Universal property of the tensor algebra). Let $I_{D J}$ be the two-sided ideal of $T$ generated by (The ideal generated by a subset and principal ideals) the following relations:

$K_{0} - 1, K_{h} K_{h^{'}} - K_{h + h^{'}} (h, h^{'} \in P^{\lor});$

$K_{h} E_{i} - q^{⟨ α_{i}, h ⟩} E_{i} K_{h}, K_{h} F_{i} - q^{- ⟨ α_{i}, h ⟩} F_{i} K_{h} (h \in P^{\lor}, i \in I);$

$E_{i} F_{j} - F_{j} E_{i} - δ_{i j} \frac{K_{i} - K_{i}^{- 1}}{q_{i} - q_{i}^{- 1}} (i, j \in I);$

${S e r r e}_{i j}^{+} : = \sum_{r = 0}^{1 - a_{i j}} (- 1)^{r} {(\binom{1 - a_{i j}}{r})}_{i} E_{i}^{1 - a_{i j} - r} E_{j} E_{i}^{r}, {S e r r e}_{i j}^{-} : = \sum_{r = 0}^{1 - a_{i j}} (- 1)^{r} {(\binom{1 - a_{i j}}{r})}_{i} F_{i}^{1 - a_{i j} - r} F_{j} F_{i}^{r} (i \neq j),$

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

$U_{q} (g) : = T / I_{D J}$

(The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ), 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 $Q = ⨁_{i \in I} Z α_{i}$ (Kac Moody root lattice height and positive cone), with $\deg E_{i} = α_{i}$ , $\deg F_{i} = - α_{i}$ and $\deg K_{h} = 0$ ; thus $U_{q} (g) = ⨁_{β \in Q} U_{q} (g) [β]$ .

For every unital associative $Q (q)$ -algebra $B$ , any assignment of elements $E_{i}^{'}, F_{i}^{'}, K_{h}^{'} \in B$ satisfying these relations extends uniquely to a $Q (q)$ -algebra homomorphism $U_{q} (g) \to B$ (A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring).

Facts & Assumptions

Given: A symmetrizable Cartan datum and its root lattice $Q$ ; the displayed symbols and relations are formed over $Q (q)$ .

[F1]

The root lattice is freely generated by the simple roots, and $⟨ α_{i}, h ⟩ \in Z$ for $h \in P^{\lor}$ (Symmetrizable Cartan data for quantum groups, Kac Moody root lattice height and positive cone).

[F2]

$q_{i} = q^{d_{i}}$ with $d_{i} > 0$ , so $q_{i} - q_{i}^{- 1} \neq 0$ in $Q (q)$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F3]

The tensor algebra is a direct sum of finite words with concatenation product; assigning group degrees to its generator symbols gives the direct-sum grading by total degree. A linear map from the generating vector space extends uniquely to an algebra homomorphism (Tensor algebra of a vector space, Universal property of the tensor algebra).

[F4]

The two-sided ideal generated by the listed relations is an ideal, its quotient ring is defined by cosets, and maps whose kernel contains that ideal factor uniquely through the quotient (The ideal generated by a subset and principal ideals, The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ , A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring).

Verification

technique · Check the toral inverse, homogeneity of the relations, and the universal factorization

1.1F2F4algebra

The relations $K_{h} K_{- h} - K_{0}$ and $K_{0} - 1$ give $K_{h} K_{- h} = 1 = K_{- h} K_{h}$ in the quotient. In particular $K_{i}^{- 1} = K_{- d_{i} h_{i}}$ is a well-defined two-sided inverse, and the mixed commutator relation has a defined coefficient by [F2].

1.2F1algebra

Give each generator its stated degree in $Q$ . The toral relations have degree $0$ ; the $K_{h} E_{i}$ and $K_{h} F_{i}$ relations have degrees $α_{i}$ and $- α_{i}$ ; the $E_{i} F_{j}$ relation is homogeneous of degree $0$ if $i = j$ and degree $α_{i} - α_{j}$ if $i \neq j$ ; and the two Serre sums have degrees $(1 - a_{i j}) α_{i} + α_{j}$ and its negative. Thus every generator of $I_{D J}$ is homogeneous.

2.1step 1.2F3F4algebra

Every element of the two-sided ideal is a finite sum of products $u r v$ with $r$ a homogeneous defining relation. Decomposing $u$ and $v$ into their finite homogeneous components shows that each homogeneous component of every element of $I_{D J}$ again belongs to $I_{D J}$ . Hence the ideal is homogeneous and the quotient has the direct-sum root-lattice grading stated above.

3.1F3F4algebra∎

Given an assignment into $B$ satisfying the relations, [F3] extends its values to a unique $Q (q)$ -algebra homomorphism $T \to B$ . Every generator of $I_{D J}$ maps to zero, so the ideal lies in the kernel; [F4] then gives the unique ring factor $U_{q} (g) \to B$ . Since the quotient map and the original map preserve scalars, surjectivity of the quotient map makes the factor preserve scalars as well. Conversely, any such factor is determined by the images of the generators because they generate the quotient.

Depends on

Used by

Dependency tree · two levels

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Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)